Non-linear dynamical approaches for characterizing multi-sector climate impacts under irreducible uncertainty

Abstract

Internal climate variability (ICV) remains a major source of uncertainty in climate projections, complicating impact assessments across critical sectors, especially at stakeholder-relevant scales. Given that ICV emerges from the nonlinear interactions of the climate system, we argue that nonlinear dynamical (NLD) approaches can improve its characterization, providing physically interpretable insights that strengthen adaptation strategies and support multisector decision-making. However, despite their suitability for such problems, NLD approaches remain largely underutilized in the analysis of initial condition large ensembles (LEs). We argue that a diverse suite of NLD approaches offers a promising pathway for systematically extracting robust insights from LEs. If effectively applied and systematically integrated, these methods could fully harness the potential of LEs, uncovering underlying patterns and variability across ensemble members to refine fundamental insights from climate projections. This will help bridge the gap between complex climate dynamics and practical resilience strategies, ensuring that decision-makers, resource managers, and infrastructure planners have a more reliable foundation for navigating irreducible uncertainty.

Introduction

A conservation biologist studying ectothermic species faces a persistent challenge: climate model projections yield markedly different extinction risks and timelines. In some projections, population decline occurs within decades, while in others, species persist despite climate change (Fig. 1d, SI 1a, 2). How should conservation strategies account for such variability? Similarly, an infrastructure planner designing flood defenses must reconcile conflicting estimates of once-in-a-century storm events, raising critical questions about risk-informed decision-making (Fig. 1e, SI 1b). Meanwhile, policymakers managing freshwater resources contend with divergent projections ranging from severe water scarcity to widespread flooding (Fig. 1f, SI 1c) (See “Methodology” for more details.) These divergent outcomes arise due to minute differences in initial conditions (ICs) that grow over time, amplified by the climate system’s nonlinear and chaotic dynamics, even when using the same model and emissions scenario. These nonlinear dynamics lead the climate system’s intrinsic, unforced fluctuations to unfold differently across simulations, resulting in multiple plausible trajectories, the range of which is referred to as internal climate variability^1,2,3 (ICV) (see Box 1). ICV exerts its strongest influence on regional scales and over seasonal-to-decadal horizons, where it can delay or intensify emerging climate signals and, in doing so, impact the reliability of climate projections^1,4,5,6. Especially at these scales, characterizing ICV is essential for developing adaptation strategies, designing resilient infrastructure, and managing resources in ways that remain robust across a range of plausible futures.

Fig. 1: Sensitivity of earth system models to initial conditions and implications for multi-sector impact assessment.

The variability in the spatial pattens of average decadal changes in a temperature over Australia, b precipitation in the United States, and c freshwater availability (calculated as the difference between precipitation and evaporation) in South America in the 2040s (2040–2049), relative to the 2020s (2020–2029) from two CMIP6 climate models (CESM 2, MRI-ESM2-0) and 3 initial conditions under SSP585. The role of internal climate variability in impact assessments is shown for d projected population density of Macrolophus pygmaeus (where a decline to zero indicates local extinction), e IDF curves for a 24-h precipitation event at the Earth system model grid point corresponding to Boston, and f decadal changes in mean runoff in the Amazon Basin (analysis for runoff is shown only for CESM2). These results highlight the strong influence of choice of initial conditions on impact assessments and the need to account for internal climate variability in adaptation planning. (See Methodology and Supplementary Information for details).

ICV constitutes one of three primary sources of uncertainty in climate projections, alongside uncertainties from model structure and emissions scenarios. Model uncertainty can be narrowed through advances in physical parameterizations and process representation, while scenario uncertainty may shrink as emissions trajectories and socio-economic pathways become better constrained. In the meantime, these uncertainties can be quantified and managed for more reliable fundamental insights and risk assessments using established methods such as model weighting and ensemble averaging for model uncertainty^7,8,9,10, and frameworks rooted in deep uncertainty for scenario uncertainty¹¹. ICV, however, presents a distinct challenge. As a manifestation of the climate system’s intrinsic dynamics, it is irreducible and fundamentally unpredictable beyond a few years to a decade^12,13. Because it cannot be reduced, ICV must instead be explicitly characterized, yet it is often underrepresented in conventional uncertainty assessment frameworks used in climate research and planning^14,15,16,17. Initial-condition large ensembles^18,19 (LEs) provide a systematic modeling framework for investigating ICV by generating a spread of trajectories through the introduction of perturbations to ICs, while holding models and forcings fixed. Although LEs have been used to examine shifts in mean climate^1,4, extremes²⁰, and sectoral impacts^21,22,23, most analyses focus on descriptive statistics, such as ensemble means or variances, to extract insights and characterize uncertainty. Unlocking the full potential of LEs will require expanding beyond these measures toward tools that can capture dynamical structures, causal relationships, and physically interpretable signals embedded within ensemble trajectories. Utilizing such methods would enable insights that are scientifically robust and actionable for planning and resilience, even in the presence of irreducible uncertainty in climate simulations.

Nonlinear dynamical (NLD) approaches^24,25 offer a pathway to address the limitations of current LE analyses and deepen our understanding of ICV. Statistical methods including extreme value theory (EVT), and process-based models have long been essential for characterizing climate variability and assessing impacts, while machine learning (ML) techniques are increasingly applied to extract complex, often non-intuitive patterns from large climate datasets^26,27, including output from Earth system model (ESM)–based LEs. However, statistical and EVT-based tools are primarily designed for quantifying distributions and extremes rather than probing underlying dynamics, while many ML applications emphasize predictive performance, which can come at the expense of physical interpretability. This leaves an urgent need for complementary approaches that can directly capture the nonlinear, high-dimensional behavior of the climate system and, in doing so, provide deeper insights into ICV. NLD approaches are well-suited to this role because they are designed to describe complex systems governed by nonlinear interactions and can reveal how such systems evolve, behave, and generate variability. While NLD methods have been successfully applied in select areas of climate research²⁸, their potential for systematically studying ICV within ESM-based LEs remains largely untapped, presenting a significant opportunity for advancing both understanding and application. To explore this opportunity, we adopt a slightly broader use of the term NLD than is standard: while NLD typically refers strictly to methods rooted in dynamical systems theory, we include within its scope selected tools capable of capturing the behavior of nonlinear, high-dimensional systems more generally (See Box 1). Together, these methods provide a physics-based framework for examining the evolution, predictability, and structural properties of the climate system, enabling deeper understanding of the mechanisms driving internal variability and ensemble spread in LEs. In addition to this direct capability, NLD can complement statistical and ML approaches, enhancing the extraction of meaningful patterns and improving the interpretability of ICV. Together, LEs and NLD form a powerful intersection, opening new opportunities to investigate the internal dynamics of the climate system with greater process-level understanding. Building on these strengths, this paper discusses a motivational framework for systematically integrating NLD approaches with LEs to advance the study of ICV.

The integration of NLD approaches with LEs creates opportunities to advance both fundamental and applied understanding of ICV (Fig. 2). Realizing this potential requires progress in several key areas where current approaches fall short. In this perspective, we highlight five priorities for research, spanning theoretical advances and applications such as regional risk assessments and long-term adaptation planning in sectors including water, agriculture, and infrastructure, where NLD methods are particularly well-suited to drive progress:

• 1. Characterizing climate variability, connectivity, and causality, particularly in characterizing ICV.

• 2. Predictability and prediction skill, as the chaotic nature of the climate system places fundamental limits on how far into the future certain phenomena can be anticipated.

• 3. Uncertainty characterization and quantification, where traditional methods are not suited for investigating ICV.

• 4. Change detection and tipping points, since internal variability can mask or amplify emerging climate signal or lead to abrupt transitions.

• 5. Climate decision support and communication, where decision-makers must interpret projections that include irreducible uncertainty, complicating adaptation and mitigation planning.

Fig. 2: Climate insights for flexible adaptation measures and decision-making.

Incorporating multiple initial condition simulations within conventional climate modeling frameworks, combined with nonlinear dynamical approaches, enables robust insights into climate system behavior from Earth system and global climate models. The framework accounts for uncertainty from model spread (reducible), internal variability (irreducible), and emission scenarios (deep uncertainty). Nonlinear dynamical tools complement and, in some cases, enhance predictive insights from conventional statistical and machine learning methodologies. These advancements improve impact assessments and inform flexible adaptation strategies. They also support robust decision-making and policy formulation aimed at enhancing climate resilience under irreducible uncertainty.

This paper presents a motivational framework for exploring how NLD approaches, when paired with LEs, could advance the study of ICV. Rather than proposing a prescriptive methodology, we synthesize established applications of NLD in climate science (Table 1), such as Lyapunov exponents, entropy measures, attractor reconstruction, and bifurcation analysis and draw on the few existing examples of LE-NLD integration to illustrate their promise within the five research priorities highlighted in this perspective article. To ground this framework, we discuss how NLD could generate more actionable insights through proof-of-concept use cases spanning ecological, hydrological, and infrastructure contexts, connecting to the examples introduced at the start of this perspective article.

Table 1 Selected nonlinear system approaches that may be suitable for solving earth system tasks related to predictability, change detection, uncertainty, teleconnections, and prediction and predictive skill

Box 1 Conceptual framing

Internal Climate Variability: In this paper, internal climate variability (ICV) refers to the range of plausible climate outcomes represented across ensemble members of an initial condition large ensemble, arising solely from minute perturbations to ICs in ESM simulations. These divergent trajectories emerge because nonlinear interactions among climate system components make the models highly sensitive to their ICs, causing small perturbations to amplify over time. As a result, internally generated fluctuations, such as those giving rise to modes of variability like the El Niño–Southern Oscillation (ENSO) or the Pacific Decadal Oscillation (PDO), manifest at different times and in different sequences across ensemble members. This leads to a diverging set of climate realizations under identical external forcing, generating the irreducible range of plausible outcomes associated with ICV.

Nonlinear Dynamical Approaches: In this paper, nonlinear dynamical (NLD) approaches refer to a suite of methods rooted in dynamical systems theory that characterize the behavior of high-dimensional, nonlinear, and chaotic systems such as the climate system, as well as selected methods that capture nonlinear, high-dimensional behavior of the Earth system. These approaches analyze the evolution of system trajectories to extract physically interpretable insights into variability, predictability, structural transitions, and uncertainty. Below are some of the NLD approaches that we explore in this perspective which have the potential to extract physically meaningful insights from ESM-based LEs.

NLD Approaches	Selected Applications
Lyapunov exponent	Quantify the sensitivity of the climate system to initial conditions, useful for assessing predictability limits
Attractor reconstruction	Reconstruct the state space of the climate system to study its underlying structure and dynamics
Snapshot attractor	Capture the full spread of LE members at a given time, useful for change detection under non-stationary climate conditions
Snapshot EOF	Identify spatial correlations and dominant climate patterns across ensemble members to identify climate change
Recurrence analysis	Detects repeating patterns in reconstructed state space to identify regime shifts, measure stability, and quantify predictability
Bifurcation analysis	Identify structural transitions in the climate system, including tipping points.
Branched manifold analysis	Classify complex system trajectories into qualitatively distinct climate regimes.
Entropy measures	Quantify the level of unpredictability in model simulations, useful for uncertainty quantification and simulation comparison
Climate networks	Analyze spatial and temporal connectivity patterns in the climate system, including teleconnections.
Convergent cross mapping, PCMCI	Detect and quantify causal relationships between climate variables, useful for understanding the connectivity of the climate system
Kullback-Leibler divergence	Measure the difference between spatial patterns or attractor shapes across climate model simulations

Opportunities for applying NLD approaches to climate projections from large ensembles

By capturing the nonlinear interactions and high-dimensional behavior of the climate system, NLD approaches offer capabilities for analyzing ESM-based LEs that extend beyond what conventional statistical or ML tools can provide. Unlike applying NLD to reanalysis or idealized model output, applying NLD tools to LEs enables systematic investigation of internal variability under consistent external forcing, making it possible to isolate and analyze irreducible uncertainty across plausible climate trajectories.

Insights from weather forecasting, where NLD has become foundational, offer a blueprint for how these approaches could be systematically integrated into LE-based climate analysis. The recognition that weather is inherently chaotic²⁹ has driven key developments in NLD, including chaos theory. In turn, NLD tools have been pivotal in studying the chaotic nature of weather, revealing inherent and practical limits and spatial patterns of predictability^30,31 and exploring how these limits shift in a warmer world³². These insights have informed operational weather forecasting, supporting emergency preparedness and response. A similar, systematic application of NLD methods to LEs can yield comparable benefits by isolating internal variability under consistent external forcings, clarifying irreducible uncertainty across climate trajectories, and enhancing the decision relevance of climate projections for adaptation planning and risk assessment. Establishing clear pipelines from analysis to decision-making would ensure that insights derived from LE-based analyses remain both actionable and representative of the full range of plausible futures.

Broader adoption of NLD methodologies in LE analysis presents an opportunity for user-driven innovation. As the demand for tailored climate information grows, user needs could shape development of NLD tools, fostering a reciprocal relationship where scientific progress and practical applications drive methodological refinement, as has occurred with nonlinear approaches in weather forecasting and ecology.

Here, we explore how selected NLD methods, including snapshot attractors and climate networks, can address the five key challenges that were identified earlier in the context of LEs: climate characterization, predictability, change detection, uncertainty quantification, and decision support. While the five challenges we identify are motivated by key gaps in LE-focused climate research, the application of NLD methods to address them in the context of LEs is still emerging. Most of the methods discussed here have seen limited use in the context of ESM-based LEs, though they have demonstrated utility in other areas of climate science. To ground this discussion, we review existing LE applications that align with the five challenges, summarizing selected studies in Table 2, detailing models used, methodologies employed, and key insights gained. Among these, we highlight the few instances where NLD tools have already been paired with LEs. We further illustrate the relevance of NLD methods by linking their applications in other disciplines^33,34,35,36 to potential insights^37,38,39,40 for LEs in Fig. 3. This figure highlights how bifurcation analysis, attractor reconstruction, Complex networks (CNs), and snapshot attractors could be applied to improve the analysis of ICV in climate projections from LEs. Taken together, this section outlines how NLD methods, guided by prior work and plausible extensions, can facilitate the analysis of ICV and irreducible uncertainty, supporting both fundamental understanding and practical applications of LE-based climate research.

Fig. 3: Investigating the climate system with nonlinear dynamical methods.

Nonlinear dynamical (NLD) methods, applied in climate analysis and other deciplines, can be leveraged to enhance climate predictability and climate system understanding when applied to initial condition large ensembles (LEs). The right column presents potential insights (not direct applications other than in the case of the snapshot attractor) from NLD methods applied to LEs^37,38,39,40, the middle column shows examples from other discipline^33,34,35,36, and the left column illustrates the NLD techniques themselves. While some of these methods have already provided insights in climate science, their application to LEs remains limited. a–c Bifurcation (example application: tipping points in species populations³³) can be used to characterize tipping points climate model projections from LEs³⁷. d–f Attractor reconstruction (example application: solar cycle forecasting³⁴) can potentially be used for nonlinear climate forecasting³⁸. g–i complex networks (example application: air transportation networks³⁵) can be used to identify climate teleconnections³⁹. j–l Snapshot attractor³⁶ has been used for climate change detection in the presence of internal climate variability⁴⁰. (Figures used with copyright permissions from the relevant publishers).

Table 2 Selected studies that obtain climate insights related to four of the key challenges identified in the paper by applying nonlinear dynamical approaches to observations, conceptual models and global circulation and Earth system models

Challenge 1: Climate characterization, connectivity, and causality

Characterizing the behavior of the climate system, including its connectivity and causal structure as simulated by ensemble members of LEs, is a critical first step in understanding how ICV manifests within these simulations. The climate system exhibits complex connectivity structures, including proximity-based dependencies (e.g., between neighboring grid points) and teleconnections (long-distance relationships such as those between the tropical Pacific and midlatitudes), with varying levels of detectable correlations (statistical associations) and causal relationships (directional influences that can be inferred through specialized methods). A well-known example is the El Niño Southern Oscillation (ENSO), which alters global climate patterns through ocean–atmosphere interactions. NLD approaches provide a robust framework for analyzing these properties, offering physically interpretable methods to assess the intrinsic characteristics and causal mechanisms embedded in data from LEs.

In LEs, small perturbations to ICs generate a range of plausible climate trajectories, resulting in differences in the magnitude and spatial patterns of climate variables across ensemble members. These variations suggest that underlying connectivity and causal structures may also differ within the ensemble, requiring systematic methods to quantify and compare these differences. Understanding how internal variability influences large-scale climate modes and their associated teleconnections is critical for both process-based studies and climate risk assessments.

One established approach to studying these connectivity structures is climate network (CN) analysis, which emerged from the intersection of climate science and network theory⁴¹. CNs represent pair-wise spatio-temporal relationships between climate variables as networks, where the nodes of the CN correspond to grid points and links capture statistical dependencies, such as correlations between the climate timeseries associated with nodes. This allows for the identification of key climate hubs (regions with many connections), and associated teleconnections, such as ENSO’s influence on rainfall in distant areas. In climate research, CNs have been used to explore the internal structure of the climate system under climate change and have been successfully applied to study teleconnections (Fig. 3g–i^35,39), identify community structures, hubs, and synchronized regions within the climate network (i.e., spatially and dynamically coherent subregions), detect climate change, and establish causal relationships. However, applications of CNs in higher-complexity climate models have largely been limited to multi-model ensembles⁴² and direct observations⁴³. Expanding CN-based approaches to LEs would enable a more systematic investigation of ICV and its influence on climate connectivity across ensemble members⁴⁴, providing a broader and more robust understanding of internal variability-driven changes in network structure and causality.

Complementary to CNs, Convergent cross mapping⁴⁵ (CCM) provides a nonlinear time series–based approach for detecting causal relationships. CCM identifies whether changes in one variable systematically leave a footprint on another, even in complex, nonlinear systems, making it a powerful method for inferring causal influence. It can also detect lagged and asymmetric interactions that arise from system inertia and feedback mechanisms, providing deeper insights into how ICV shapes climate variability.

PCMCI (Peter and Clark Momentary Conditional Independence) offers another causal discovery framework designed for complex, high-dimensional climate data^46,47. PCMCI identifies robust, potentially time-lagged causal links between variables, even in the presence of nonlinear relationships and autocorrelation. This method has been used to map causal relationships and their strengths in both climate observations and model simulations. For example, PCMCI has been used to reconstruct key components of the Walker circulation in the tropical Pacific and to identify its causal connections to remote regions such as the Atlantic^46,47. Within LEs PCMCI can differentiate the causal networks among IC ensemble members⁴⁸, providing a powerful framework for characterizing individual members based on their internal dynamical structure. This opens the possibility of mapping how different modes of variability unfold across a LE, leading to a range of plausible outcomes and potentially enabling novel approaches for uncertainty characterization (see “Challenge 4”).

The snapshot attractor framework^40,49,50 offers a particularly powerful approach for analyzing climate variability in non-stationary systems and is one of the few NLD tools that have been directly applied to ESM-based LEs (Fig. 3j–l^36,40) in multiple studies. Unlike traditional attractors, which assume stationarity and rely on long time series (i.e., they assume the climate system has stabilized into a consistent long-term pattern), snapshot attractors are constructed using values from initial condition ensemble members at a single time instant. This unique property allows them to capture transient dynamics (short-term, evolving responses that have not converged to a stable pattern) while representing the full range of plausible climate states at each timestep. This enables the snapshot attractor framework to capture the qualitative behavior of the mean and variability of climate variables at a specified time and separate internal variability from the forced climate change signal⁵¹, making them particularly well-suited for studying the behavior of the climate system within LEs.

A notable methodological innovation within this framework is the development of instantaneous correlation coefficients (ICC) and snapshot empirical orthogonal function (SEOF) analysis⁵², which modifies standard statistical measures and combines them with NLD tools. This represents a concrete example of integrating statistical methodologies with NLD approaches to fit an ensemble framework. Conventional approaches compute correlations and empirical orthogonal functions over time within a single model simulation. However, ICC and SEOF are calculated across ensemble members at each time instant, leveraging the additional information provided by LEs. This ensemble-based snapshot approach enables the tracking of the time evolution of dominant climate modes and reveals potential changes in their relationships with key climate variables. These methodologies have been applied to investigate teleconnections associated with the Arctic Oscillation (AO) and the El Niño–Southern Oscillation (ENSO), providing insights into how ICV influences their spatial and temporal expression under climate change^51,52,53,54. Broader application of these methods to ESM-based LEs could further improve the understanding of how internal variability modulates dominant climate modes, underscoring the potential of snapshot attractors and related innovations to advance insights into ICV.

By applying these NLD methodologies within LEs, it becomes possible to quantify differences in internal variability across ensemble members, improve the interpretability of climate projections, and enhance both process-level understanding and communication of findings. This enables a more diagnostic use of LEs, moving beyond statistical summaries and toward identifying the underlying dynamical pathways that generate ensemble spread.

Challenge 2: Predictability and prediction skill

The extreme sensitivity of climate models to ICs fundamentally limits their predictability and predictive skill, and the predictability limits of the climate system still remain an open question⁵⁵. Estimating these limits and evaluating the predictive skill of climate models are essential for understanding model strengths and weaknesses, ultimately enabling the optimized use of model outputs.

LEs provide a robust framework for systematically studying how small differences in ICs lead to divergent climate trajectories, enabling the evaluation of spatial and temporal patterns in predictability. The Lyapunov exponent, which measures how quickly small perturbations grow in a system over time, provides a standard tool for assessing the total predictability limits of a system⁵⁶. It has been widely applied to assess predictability limits at weather, seasonal, and decadal scales, and its integration with LEs could unlock new insights by leveraging multiple realizations of plausible futures. The nonlinear finite-time Lyapunov exponent⁵⁷ and nonlinear local Lyapunov exponent⁵⁶ (NLLE) both offer complementary tools for analyzing predictability. Together, these Lyapunov-based diagnostics form a foundation for assessing the growth of errors and the spatial and temporal distribution of predictability limits. NLLE captures nonlinear error growth in chaotic systems, providing more realistic temporal–spatial distributions of predictability limits. It accounts for variations across regions, altitudes, and seasons, offering a more detailed depiction of predictability for different climate variables⁵⁶. Applications of NLLE^56,58,59 demonstrate that this approach can reveal predictability limits beyond weather scales, identifying regions and processes that enhance predictability, as well as asymmetries in the predictability limits of climate phenomena such as ENSO during different stages of its lifecycle.

Beyond Lyapunov-based diagnostics, state-dependent indicators, including local dimension and inverse persistence, extend the suite of tools for diagnosing predictability by capturing the instantaneous structure and stability of the system’s attractor^60,61. Unlike average attractor properties, which describe the system under stationary conditions, these instantaneous measures capture transient states, including those that lead to extreme events. Here, the local dimension captures the number of active degrees of freedom in the climate system at a given time and inverse persistence measures the average time a configuration is maintained before the system transitions away from it. These indicators have been shown to correlate with ensemble spread in weather forecasting, acting as proxies for predictability and supporting the evaluation of forecast uncertainty^60,61. Applying these tools to ESM-based LEs can extend their utility to climate-scale variables, helping to assess predictability limits, clarify how variability manifests across ensemble members, and identify regions where improved process representation is needed to enhance the predictive skill of climate models.

Unlike in weather forecasting, where predictability limits are typically measured in days, climate models offer the potential to leverage sources of extended-range predictability. Slowly varying climate components, such as sea surface temperature, and periodic nonlinear oscillations, such as the ENSO, serve as key sources of extended-range predictability on climate timescales³⁸. These sources could be further leveraged within climate modeling frameworks to enhance predictive skill. NLD approaches, including CN methodologies and phase space reconstruction-based approaches⁶² (Fig. 3d–f^34,38), are particularly well-suited for analyzing the behavior of these long-range connections by reconstructing system dynamics from time series data. By capturing nonlinear dependencies and complex interactions, these NLD tools offer significant potential for improvement of model performance by revealing how these predictors influence large-scale climate variability.

Complementing these diagnostics, NLD methods such as attractor reconstruction provide another perspective on predictability, which can be further enhanced by integration with ML. Given that climate models represent high-dimensional dynamical systems, they can be conceptualized as attractors in phase space (i.e., the multi-dimensional space representing the allowable states of the evolving climate system). Attractor reconstruction, based on embedding theorems, allows for the reconstruction of the climate system’s attractor, providing insights into geometry (shape of the attractor) and dynamics relevant for predictability studies^62,63. These properties help define the boundaries of predictable behavior, identifying the allowable states of the system and revealing how the system evolves in response to perturbations. Beyond traditional methods, there is an opportunity to advance these insights by combining NLD with ML. In particular, ML techniques can be employed for feature extraction from reconstructed attractors, uncovering novel patterns and structures that influence predictability.

Quantifying predictability limits and their variability across ensemble members is essential not only for model evaluation but also for communicating projection reliability to stakeholders. Applying these NLD methodologies to ESM-based LEs would enable the quantification of confidence intervals for predictability limits under climate change and yield metrics of variation in prediction skill across ensemble members. Such insights would not only improve understanding of the sources and limits of predictability but also enhance the communication of model reliability to stakeholders. By providing clearer indications of where and when climate projections are most reliable, these methodologies could support better-informed decisions in sectors dependent on climate information, such as disaster preparedness, infrastructure planning, and resource management.

Challenge 3: Change detection and tipping points

Anticipating when and where the climate system may cross critical thresholds is central to building resilience. Detecting both gradual shifts and abrupt, potentially irreversible transitions is vital for informing adaptation and mitigation strategies. NLD tools offer a pathway for detecting and quantifying both gradual and abrupt climate changes, enabling the identification of precursors that serve as early warnings for tipping points.

Many conventional change-detection methods assume a stationary climate, averaging over time or across ensemble members, which can mask critical transitions and smooth signals from ICV. Snapshot attractors overcome these limitations by characterizing the instantaneous state of the system across LE members, allowing robust detection of emerging changes in non-stationary climates without the biases introduced by temporal averaging⁶⁴ (Fig. 3j–l^36,40). This capability has already been demonstrated in simplified models, where tracking the shape and size of the snapshot attractor provides a qualitative measure of climate change over time. Measures such as the Wasserstein distance, which determines the distance between a reference attractor and subsequent attractors, have been employed to quantify these changes⁶⁵. Extending these approaches to ESM-based LEs will enable robust detection and quantification of the magnitude and timing of climate change, offering more reliable estimates of when and where climate impacts are likely to emerge, insights critical for engineering applications and policy decisions.

Beyond gradual change detection, NLD tools can also identify early-warning signals and critical thresholds linked to abrupt transitions which are critical for timely intervention and response (Fig. 3a–c^33,37). Tipping points occur when gradual, small, or rapid changes push the system into a new, often irreversible state, resulting in abrupt shifts in the climate system that can drive serious, global-scale impacts. One well-known example is the Atlantic Meridional Overturning Circulation (AMOC), a bistable system that has experienced abrupt transitions in the past and which could, under continued warming, reach another tipping point in the future leading to its collapse⁶⁶. Such abrupt shifts can arise through multiple mechanisms⁶⁷. In some cases, gradual changes bring the climate system close to a critical threshold, after which it abruptly transitions to a new state (bifurcation-induced tipping). In other cases, random short-term fluctuations can trigger a shift even if the system isn’t near a threshold (noise-induced tipping). A third possibility is when changes in the climate system occur too quickly for the system to adapt (rate-dependent tipping). In reality, these mechanisms often interact, making it more challenging to anticipate when a tipping point might be approaching or how the system might respond⁶⁷. This complexity underscores the need for NLD approaches, which can disentangle forced signals from ICV, diagnose early-warning indicators, and track how thresholds emerge across different ensemble trajectories.

Experiments with simplified models illustrate how small perturbations to ICs can yield drastically different outcomes, with the climate system transitioning between a snowball Earth and a warm climate⁶⁸ or triggering abrupt changes in the AMOC⁶⁹. Applying similar NLD analyses to ESM-based LEs would enable a more comprehensive understanding of tipping thresholds, the distinct mechanisms that lead to tipping, and the ways in which these mechanisms can interact, all in the presence of ICV. Such understanding can also improve the ability to detect early warning signals ensuring that emergent risks are identified in time to inform adaptive actions.

Beyond snapshot attractors, other NLD-based methodologies have also been used for tipping point detection in observations and simplified models. For instance, changes in attractor properties⁶⁶, branched manifold analysis through homologies^70,71, and snapshot attractor–tipping probability assessments using bifurcation analysis⁷² have been applied to detect tipping points in observations and simplified models. Network-based indicators such as normalized degree, average path length, and betweenness centrality have also been used to detect tipping points at global scales, revealing early warning signals of potential system shifts⁷³. Applying these methodologies to ESM-based LEs could yield deeper insights into tipping points in the presence of ICV, revealing how internal variability influences the timing and likelihood of critical transitions. Integrating these approaches with analyses of the underlying dynamics of the climate system (Challenge 1) would enhance our understanding of tipping point dynamics and support the early detection of warning signals. Additionally, there is a significant opportunity to combine NLD with ML for a more robust analysis of tipping points. While NLD tools can identify early warning signals based on attractor properties, ML techniques can facilitate efficient detection and classification of these changes in large datasets. Such integration ensures a comprehensive exploration of plausible futures, including critical worst-case scenarios that might otherwise be overlooked.

Given the potentially catastrophic consequences of crossing tipping points and the possibility that gradual climate change may surpass the adaptive capacity of natural and manmade systems and critical services, these advancements could play a pivotal role in informing climate policy and guiding effective mitigation and adaptation strategies.

Challenge 4: Uncertainty characterization and quantification

Identifying and delineating sources of uncertainty, as well as defining the range of their magnitudes in climate projections, is critical for obtaining robust scientific insights from LEs in the presence of irreducible uncertainty, particularly when internal variability can obscure or amplify the forced climate signal. NLD methods offer novel ways to achieve this by capturing both statistical differences and dynamical complexities across ensemble members. Accurate uncertainty quantification not only increases the interpretation of insights from climate models (including those from Challenges 1–3) but also plays a pivotal role in informing climate risk assessments, guiding infrastructure design, and supporting policy decisions (Challenge 5). This is particularly important when the magnitude of internal variability is comparable to that of the forced signal due to increased greenhouse gas emissions. NLD methods provide promising opportunities for characterizing and quantifying uncertainty in climate projections, offering a pathway toward more actionable and reliable climate information.

In climate research, entropy-based diagnostics quantify the complexity and uncertainty of the climate system, using measures rooted in information theory (e.g., Shannon entropy) and dynamical systems approaches (e.g., Kolmogorov–Sinai entropy). These metrics capture aspects of system dynamics that traditional statistical measures, such as variance or autocorrelation, are not designed to represent. Applied to LEs, these metrics can capture the full range and structure of ICV across ensemble members, avoiding the pitfalls of analyzing single realizations and ensuring that natural fluctuations are not conflated with model error.

Entropy has numerous applications in evaluating historical simulations of climate models. For example, variants of entropy (relative entropy, approximate entropy, and sample entropy, etc.) have been used to assess uncertainty in multi-model ensemble simulations (including those with multiple ICs) by comparing them against observations^74,75,76. CNs and causal discovery tools complement entropy measures by capturing dynamical structure and process-level interactions in climate models. Community structures derived from CNs provide a dynamics-based approach for model comparison and have been applied to compare general dynamics within models, enabling evaluations that reflect underlying system behavior rather than relying solely on bulk statistical properties⁷⁷. Causal discovery algorithms such as PCMCI, enable process-oriented model evaluation by capturing the direction and time lags of key interactions, for example the Walker circulation, directly from observational data and benchmarking them against climate model output⁴⁸. Models whose causal fingerprints, i.e., network patterns summarizing the direction, strength, and timing of key variable interactions, more closely resemble those derived from observations tend to exhibit smaller climatological biases. These fingerprints are typically more consistent across IC ensemble members of a given model than across different climate models especially when there is less overlap in the model development pathways. This consistency highlights how causal networks can distinguish variability driven by internal dynamics from structural model differences. Both these approaches can be adapted for climate simulations from LEs, with entropy measures capturing statistical differences and CNs capturing dynamical variations across ensemble members. Utilizing these tools within LEs would enable a comprehensive quantification of ICV, providing uncertainty bounds for climate projections. Such insights are crucial for robust decision-making, helping to avoid maladaptation by ensuring that climate risks are assessed within the full range of plausible futures.

Beyond quantifying uncertainty, entropy-based measures, CN-derived community structures and causal networks can also improve model evaluation under irreducible uncertainty. Conventional climate model evaluation methods typically compare a single model realization against observations. However, the chaotic nature of the climate system means that unforced fluctuations may evolve differently in the real world compared to any single model run. This discrepancy can lead to an over- or underestimation of model accuracy, resulting in false confidence in insights that may not fully capture the range of plausible outcomes¹⁵. To address these limitations, the concept of observational LEs has been introduced, where ensembles are constructed from observational or reanalysis data to mimic the variability captured in ESM-based LEs⁷⁸, providing a framework to better account for internal variability when evaluating climate models. Additionally, artificial neural networks have been applied to LE evaluation⁷⁹, demonstrating potential but with limited broader adoption so far. Building on these advances, we suggest that entropy-based measures and community structures derived from CNs could offer a streamlined and interpretable approach for model evaluation using ESM-based LEs. By capturing both statistical properties and dynamical behavior using NLD methods, this framework would enable a more robust assessment of model performance. Such an approach would provide clearer guidance on how to appropriately interpret model output, ensuring that users can better understand the strengths and limitations of climate projections. Ultimately, these insights could support more reliable climate risk assessments, informed infrastructure planning, and evidence-based policy decisions, enhancing the practical utility of LE-based climate information.

NLD approaches can also identify when forced climate signal becomes distinguishable from internal variability (i.e., time of emergence), a critical factor for climate impact and risk assessments. Snapshot attractor approaches⁵¹ and Shannon entropy in conjunction with mutual information⁸⁰ have already provided a pathway to delineate the forced signal from ICV in a LEs. Kullback-Leibler divergence criteria, which provides the distance between probability distributions, have already been used for model comparisons⁷⁴. This metric can also be adapted to assess when forced signals emerge from ICV by comparing individual ensemble members (representing the superposition of internal variability and the forced signal) with that of the multi-model ensemble mean (representing the forced signal when the ensemble is sufficiently large⁸¹). This approach allows for estimating the ratio between internal variability (noise) and the forced signal, helping to determine when the forced climate signal will emerge from ICV. These insights improve preparedness for future “climate surprises,” where ICV may amplify the forced signal, and increase the accuracy of detecting mitigation benefits.

Taken together, these approaches enhance the ability to evaluate the performance of LEs and quantify uncertainty in ways that are more consistent with the nonlinear and ensemble-based nature of climate model output. By capturing how ICV and the resulting irreducible uncertainty manifest across multi-model, multi-IC ensembles, NLD-informed metrics can support more robust evaluations and improve confidence in both fundamental understanding and risk-relevant climate insights obtained from ESM-based LEs.

Challenge 5: Climate decision support and communication

Despite the availability of LEs and widespread recognition of the importance of ICV in climate projections, research and decision-making communities continue to rely predominantly on multi-model ensembles and emission scenarios, often overlooking ICV. This oversight is exacerbated by the lack of established frameworks for seamlessly integrating ICV-related insights into policy, engineering design, and climate risk assessment tools. As a result, the full range of plausible futures is frequently excluded from adaptation planning, leading to misallocated resources, misplaced efforts, and a false sense of preparedness for future conditions. To address this gap, insights derived from NLD analyses in Challenges 1–4 such as predictability limits, early-warning indicators for tipping points, and refined uncertainty quantification must be systematically integrated into decision frameworks to enhance flexibility and resilience.

Flexible adaptation measures centered on concepts of deep uncertainty¹¹ (i.e., where the most likely future scenarios or outcomes cannot be agreed upon) are essential for addressing the inherent uncertainties in climate projections. NLD-enabled outputs from earlier challenges can identify where uncertainty is low or reducible and where it is irreducible, helping decision-makers deploy flexibility where it is most needed while applying targeted solutions where uncertainty is better understood. For example, better process understanding (Challenge 1) can lead to model developments that reduce structural uncertainty in climate simulations. Predictability limits (Challenge 2) can inform planning horizons and reliability windows while early-warning indicators for tipping points (Challenge 3) can guide contingency planning, mitigation strategies and the implementation of monitoring systems. Meanwhile, refined uncertainty quantification (Challenge 4) can support resource allocation and prioritization. Embedding these diagnostics into risk assessment tools ensures that adaptation strategies remain evidence-based and robust across a full spectrum of plausible outcomes.

These diagnostics only add value if translated into usable forms for diverse end users. Aligning scientific approaches with practical needs, through active stakeholder engagement, ensures that climate research remains relevant, credible, and tailored to end-user requirements. For instance, policymakers often operate under significant uncertainty⁸², while engineers designing stormwater systems require precise information and uncertainty quantification. Recognizing these differing needs enables scientists to produce actionable information, increasing the likelihood of its use in robust decision-making. This requires open communication and the co-production of knowledge through the collaborative and iterative definition of needs, tools, and use contexts with stakeholders⁸³. In turn, this knowledge exchange can drive innovation within the field of NLD, helping scientists identify new tools and methods most needed to explore the Earth system and inform robust climate decisions. However, scaling these insights into operational decision-support systems will also require addressing key computational barriers, including the demands of applying NLD techniques to high-dimensional climate data and the need to expand the number and diversity of LEs available for analysis. Tackling these challenges is necessary to ensure that NLD-enabled insights can be implemented broadly and efficiently.

A roadmap for climate resilience can be built on this foundation. Embedding insights from NLD-enabled LE analyses into risk assessment and planning frameworks can deliver decision-support systems that help allocate resources efficiently and design flexible strategies that withstand a wide range of futures. With the computational challenges outlined in the next section addressed, these systems can transition from research to practice, ensuring adaptation and mitigation efforts remain both evidence-based and robust.

Data and computational considerations for LE–NLD integration

Data availability continues to limit the broader integration of NLD approaches with ESM-based LEs. While recent efforts such as CMIP6 have an increased number of IC ensemble members, many models either provide only a small number of IC runs or exclude them entirely⁸⁴. No single widely available LE combines multiple models, scenarios, and ICs in a single framework, further restricting the scope of LE-NLD analyses and hindering efforts to disentangle the relative contributions of model uncertainty, scenario uncertainty, and internal variability^5,19. However, recent calls for the expanded integration of IC simulations into CMIP7 offer an opportunity to address these structural gaps¹⁴.

Computational resources also pose a significant barrier. High-resolution LE datasets are large and require substantial storage and preprocessing. NLD methods such as complex network metrics and entropy-related approaches are more computationally demanding than standard climate diagnostics especially when applied across high resolution (currently ~100 km resolution) LEs. Unlike descriptive statistics, which are computed grid-wise, many NLD metrics involve pairwise calculations across spatial nodes, leading to a significant increase in time requirements and computational load. This burden becomes even greater when repeated across multiple ICs, models, and emission scenarios, suggesting a need for the careful design of LE–NLD analyses based on research priorities. For example, the appropriate approach may differ depending on whether the goal is broad system understanding or precise information for decision-making.

Despite these constraints, the application of NLD methods to LEs offers distinct advantages that justify their integration into climate analysis workflows. These approaches reveal dynamics and variability patterns that standard diagnostics often overlook, particularly those arising from internal variability⁴⁴. In doing so, LE–NLD analyses enable novel scientific insights, support better-informed adaptation strategies and reduce the risk of maladaptation by capturing plausible but underrepresented outcomes¹⁶. The added computational burden can therefore often be a worthwhile trade-off, especially when the analysis targets spatial and temporal scales at which ICV plays a dominant role, whether the goal is to advance fundamental understanding or to inform robust decision-making.

NLD methods offer potential solutions to challenges in ecological conservation, infrastructure design and resource management

To illustrate the practical relevance of these methods, we examine three use cases where ICV poses significant challenges to decision-making and where NLD approaches can offer targeted solutions. The presence of irreducible uncertainty makes the traditional reliance on a single “best guess” future unsuitable, as this may lead to adaptation and mitigation strategies that are either insufficient or unnecessary¹⁶. In this section, based on three use cases (Fig. 1, SI 1–5, See “Methodology” for more details), we discuss how NLD tools, when combined with other methodologies, could potentially provide actionable insights and robust solutions for three key sectors: ecological conservation, infrastructure design, and freshwater resource management. Here we discuss the challenges posed by ICV in each use case, then propose how NLD methodologies can address these challenges. These examples are intended to illustrate the prospective value of NLD approaches in LE-based analysis, rather than demonstrate fully developed applications.

Ecological conservation: identifying tipping processes and early warning signals

Ectothermic species experience increased extinction risk as a consequence of long-term climate change, compounded by short-term temperature fluctuations that amplify stress on these populations^85,86. In our first case study (Fig. 1d, SI 1a and 2), this risk is shown to emerge from the combined effects of anthropogenic forcing and ICV, where specific ICs trigger tipping processes that lead to extinction. Similar risks may also affect endothermic species that are highly sensitive to climatic conditions, highlighting that this threat can extend across a broad range of species. Since the extinction of a species is an irreversible loss, there is minimal room for error in the insights guiding species conservation decisions. In such cases, prioritizing worst-case trajectories over a broader envelope of possibilities may be necessary to safeguard ecological stability.

While conservation strategies may need to plan for worst-case scenarios to safeguard ecological stability, improving our understanding of how these tipping processes unfold can help refine and target those interventions. NLD tools applied to LEs provide a means to diagnose the mechanisms leading to extinction, offering early warning indicators of tipping points⁸⁷, such as slowing recovery rates or shifts in system stability. These insights can identify critical thresholds and windows for action, enabling conservation efforts to focus on the species and ecosystems most at risk and to act before irreversible losses occur. Early-warning indicators can also prompt intensified monitoring and the deployment of adaptive, flexible management measures, ensuring conservation responses can scale or shift as risks escalate. By combining worst-case planning with deeper diagnostic understanding, these approaches help conservation strategies remain both precautionary and precise.

Infrastructure design: constraining uncertainty in extreme rainfall projections

The intensity of a 100-year rainfall event, a key threshold for risk assessment⁸⁸, varies significantly among IC ensemble members, especially under higher emission scenarios (Fig. 1e and SI 1b), as shown in our second use case. This irreducible spread complicates the selection of threshold-based design and operational parameters for infrastructure. Resource constraints and concerns about maladaptation⁸⁹ further amplify this challenge.

To capture the full spectrum of possibilities in climate extremes—from the most optimistic to the most severe—ensembles incorporating multiple ICs, models, and emission scenarios are essential. For deeper insights, approaches from NLD that explore the mechanics of extreme events^90,91 and obtain predictive insights in dynamical systems⁹² can be adapted for studying climate extremes. Snapshot attractors⁴⁹, attractor reconstruction, recurrence analysis⁹³, and topological data analysis⁹⁴ provide a robust framework for estimating the magnitude of extremes based on attractor properties such as geometry, density, and trajectory in phase space. Embedding ML approaches within this framework enhances analysis by tracking attractor properties.

Concurrently, integrating EVT into this NLD framework refines the uncertainty bounds derived from LEs^95,96. As IC ensemble members share the same model physics, they can be combined as data from a single distribution, increasing the sample size for extreme value analysis. This integration not only constrains uncertainty estimates but also supports adaptive infrastructure systems resilient to future climate extremes. Moreover, these approaches can be extended to other climate extremes, including heatwaves, cold snaps, and severe wind events.

Resource management: unraveling spatial variability and intersectoral interdependencies

The variability in spatial patterns of freshwater availability, as illustrated in our third use case focused on the Amazon Basin (Fig. 1f and SI 1c), presents a central challenge for water resource management. ICV drives substantial fluctuations in water supply, complicating reservoir operations that must simultaneously deliver reliable water for daily use and manage flood risks during periods of excess. Leveraging LEs combined with multiple models and scenarios, allows this full range of futures to be represented, improving estimates of how reservoirs transition between storage states⁹⁷. These insights support operation policies that remain robust across a range of climate conditions, enabling water systems to consistently deliver supply, generate hydropower, and mitigate flood hazards even in the presence of irreducible uncertainty.

This same framework addresses the linked challenge of maintaining hydropower generation when water scarcity or warming threatens plant operation. By sampling a wide envelope of future freshwater conditions, LEs identify regions at risk of crossing critical thresholds for power generation⁹⁸. Domain expertise, through expert-defined operational limits, then determines where plant capacity may be reduced, temporarily suspended, or decommissioned. Incorporating NLD measures such as the correlation integral and spatio-temporal entropy can further refine these insights by quantifying the spatial coherence and variability of freshwater availability across the ensemble. This combination of LEs projections, expert thresholds, and NLD diagnostics enables operators to target vulnerable regions and design strategies that preserve both water and energy security.

Water resources, ecosystems, and infrastructure such as dams and reservoirs are deeply interconnected, with disruptions in one sector often cascading into others. Complex network approaches can be applied within this LE–NLD framework to map these interconnections, and explore cascading events and impacts across these sectors, providing a pathway to incorporate interdependencies into resilience planning. This integrated approach allows for the assessment of compound risks and the development of robust adaptation strategies, aligning with the growing need for holistic climate resilience planning in interconnected systems.

Conclusions

NLD tools hold significant untapped potential for addressing challenges in understanding and quantifying ICV. This perspective presents a motivational framework for integrating NLD approaches with ESM-based LEs, positioning ICV not as an explosion of uncertainty but as an opportunity to explore the full range of plausible futures⁹⁸ and, in some cases, better constrain uncertainties^95,96. By utilizing NLD approaches in isolation or combining with complementary methodologies such as statistical techniques, ML, and domain knowledge, this framework addresses core needs in climate science. These include improving model evaluation^15,99, isolating forced climate signals^{100,101,102,103}, detecting climate mitigation benefits^17,104, and expanding the tools available for characterizing connectivity, predictability, uncertainty, tipping behavior, and decision support, all of which are central to advancing fundamental understanding and making climate information actionable across the five priorities identified here. Addressing these priorities highlights the need for universal dynamical metrics for ICV, enabling standardized assessments across models and scenarios. Developing such metrics, alongside coordinated investment in computing and data infrastructure, will be essential for applying NLD methods to high-resolution, multi-member ensembles, whose computational demands exceed those of current conventional climate analytics.

These advances could fundamentally reshape how future climate uncertainties are conceptualized and operationalized. By embedding NLD-enabled insights into existing analytical and decision-support frameworks, we can strengthen dynamic climate planning, optimize resource management, minimize maladaptation risks, and build resilience across critical sectors. To meet the escalating demands of a rapidly changing climate, integrating NLD approaches with LEs must now move from promise to practice, becoming a standard part of how climate simulations are analyzed and applied.

Methodology

To illustrate the impact of ICV on climate variables, we conducted a set of simplified proof-of-concept analyses using data from the CMIP6 archive. We used two Earth system models (CESM2 and MRI-ESM2-0) under two emissions scenarios (SSP585 and SSP245). These were selected based on the availability of daily and monthly data at high spatial resolution (~100 km) for at least three initial condition ensemble members for each variable of interest (temperature, precipitation, evaporation, runoff). We focused on the near-term period of 2020–2049, which is of relevance for stakeholders and policymakers. This setup yielded a 12-member “super ensemble” (2 models × 2 scenarios × 3 ICs) incorporating variations across models, ICs, and emissions scenarios. For each ensemble member, we computed decadal changes between 2040–2049 and 2020–2029 in temperature over Australia (Fig. 1a, SI Fig. 3), precipitation over the United States (Fig. 1b, SI Fig. 4), and freshwater availability (defined as precipitation minus evaporation) over South America (Fig. 1c, SI Fig. 5) to illustrate region-specific expressions of ICV. Multi-model and multi-initial condition ensemble means were also calculated for each case so that the behavior of individual ensemble members could be compared to the ensemble-average response. We then explored three illustrative use cases: (1) extinction risk for the ectotherm Macrolophus pygmaeus in Europe, where daily population density was calculated using published thermal tolerance data and methodology^85,105, with local extinction defined as density reaching zero under the assumption of no migration (Fig. 1d, SI 1a), (2) infrastructure stress via intensity-duration-frequency (IDF) curves for extreme precipitation (for a fixed duration of 24 h) in Boston using Generalized Extreme Value Theory (Fig. 1e, SI 1b), and (3) water resource planning through the analysis of decadal changes between 2040–2049 and 2020–2029 for Amazon basin runoff (Fig. 1f, SI 1c). To identify regions with the greatest spread among initial-condition members, we calculated the maximum difference in decadal runoff change as:

$$\mathrm{Maximum}\,\mathrm{difference}\,\mathrm{among}\,\mathrm{initial}\,\mathrm{conditions}=\mathrm{Max}\,\left\{{{\rm{R}}}_{1},{{\rm{R}}}_{2},{{\rm{R}}}_{3}\right\}\,-\,\mathrm{Min}\,\{{{\rm{R}}}_{1},{{\rm{R}}}_{2},{{\rm{R}}}_{3}\}$$

(1)

Where R_i denotes the decadal change in runoff for the i^th ensemble member. For completeness, Supplementary Note 2 describes how the remaining supplementary figures were generated and discusses the results.