Introduction

Lowland rivers are dynamic systems whose evolution is largely governed by the gradual growth of meander bends and episodic cutoff events that abruptly reshape channel geometry and reduce sinuosity1,2,3. Cutoffs play a critical role in both local and nonlocal channel dynamics, accelerating bend migration and triggering reorganizations of the planform that extend well beyond the immediate cutoff reach4,5,6,7. When cutoffs cluster or cascade along the channel, they can trigger additional nearby cutoffs, producing avalanching sequences of channel adjustments that resemble self-organized critical behavior3,4,8. In contrast, some rivers exhibit cutoff dynamics that remain spatially localized and do not produce such chain effects9,10.

Nonlinear interactions arising from cutoff events have been proposed as a driver of chaotic dynamics in meandering rivers3,11, although conflicting studies report no definitive signatures of chaos12,13. Related forms of deterministic chaos have been demonstrated in other geomorphic systems. Ecomorphodynamic models exhibit aperiodicity and sensitivity to initial conditions arising from vegetation-sediment-flow feedbacks14, and coupled delta-network models develop strange attractors even though isolated bifurcations remain strictly periodic15.

In chaotic systems, infinitesimal differences in initial conditions grow exponentially, measured by the Lyapunov exponent16,17, leading to diminishing predictability and a finite forecast horizon for river migration. Detecting such sensitivity in natural rivers is challenging because long, high-resolution observational records are scarce11,15. Consequently, simplified deterministic models provide a controlled setting for reproducing core elements of landscape evolution and for systematically examining the mechanisms that govern predictability15,18,19. Previous studies investigating meander dynamics and their predictability have used diverse observables and state-space representations. Analyses of both natural and simulated river planforms have revealed signatures consistent with chaos and self-organization, including nonlinear time-series diagnostics and spatial clustering of cutoffs associated with fractal planform geometries3,11,20. Recent work has further shown that cutoff clustering and floodplain patterns display well-conserved statistical structure, where cutoff dynamics influence long-term river morphology7,21. However, mechanistic simulations have not produced positive Lyapunov exponents or scale-invariant cutoff intervals when analyzed through channel sinuosity12,13. These inconsistencies underscore the need for controlled testing within a minimal deterministic framework to establish whether meandering and cutoff dynamics alone act as a sufficient condition for chaos.

We investigate this question using meanderpy22, a kinematic meander model23. In this framework, lateral migration is governed by curvature. While natural rivers are influenced by sediment transport6,24, bank strength25, vegetation26, and flow variability27, we treat these as higher-order effects to isolate the intrinsic dynamics of the planform. Kinematic models represent the channel as a migrating centerline whose length and node count evolve over time, increasing as bends grow and decreasing as cutoffs remove segments22,23. As a result, the Lagrangian state vector is high-dimensional and its dimension changes in time, complicating direct comparisons between channel configurations.

Cutoffs intermittently reset local channel geometry, raising the questions of whether an evolving planform ever re-enters configurations realized earlier and whether trajectories initiated from infinitesimally different channels diverge over time. A direct Lagrangian test of either property is not possible because node insertions, deletions, and cutoff events continually change the dimensionality of the Lagrangian state vector. In computational fluid dynamics, variable-topology moving interfaces are commonly embedded into fixed Eulerian grids using interface-capturing representations28,29. We adopt the same core idea by mapping the evolving Lagrangian planform onto a fixed Eulerian grid of M cells, assigning each cell a binary label indicating channel (1) or floodplain (0). The resulting binary field g(t) ∈ {0, 1}M preserves planform geometry while providing a fixed-dimensional state suitable for direct comparison.

Within this fixed-dimensional Eulerian representation, we quantify sensitivity to initial conditions by adopting the notion of damage spreading from statistical physics30,31,32. Two replicas of the system, identical except for a localized perturbation, are advanced in parallel, and their separation is measured by the Hamming distance dH(t) = ∥S*(t) − S(t)∥1, which counts the number of cells assigned different channel or floodplain states. In discrete dynamical systems, exponential growth of this distance is a hallmark of deterministic chaos, serving the same diagnostic role as out-of-time-order decorrelators or velocity-dependent Lyapunov exponents in continuous settings31,33,34. Thus, the evolution of the Hamming distance provides a spatiotemporal record of damage propagation from which we infer geometric divergence, diagnose sensitivity to initial conditions, and estimate effective Lyapunov growth rates35.

Our results demonstrate that the system partitions into two distinct dynamical regimes depending on the activation of cutoffs. In a controlled switch experiment, the cutoff mechanism is enabled or disabled while all other parameters are held fixed. The finite-time Lyapunov exponent, estimated from the slope of \(\log {d}_{H}(t)\) during its linear growth phase36,37,38, reveals sustained exponential divergence exclusively when cutoffs are active. The inferred divergence rate converges under grid refinement, remains insensitive to perturbation magnitude, and appears consistently across diverse initial planforms. Moreover, we find that the magnitude of the Lyapunov exponent is effectively invariant to the cutoff threshold and instead scales with migration rate, which accelerates bend growth and increases the frequency of cutoffs. Collectively, these findings demonstrate that cutoffs alone induce sensitive dependence on initial conditions, impose a finite predictability horizon for planform in kinematic models.

Results

Kinematic meander migration models represent the channel as a migrating centerline whose nodes evolve according to curvature-driven migration laws22,23. Although the initial configuration can be described analytically, numerical methods are required after the first update due to cutoff-induced discontinuities and geometric distortion. Curvature is then evaluated via finite differencing on a uniform centerline grid. As the channel lengthens or shortens during bend growth and cutoffs, the total node count changes, altering the system’s state dimension and challenging nonlinear-dynamics analyses that assume a fixed phase space37,38. We therefore retain the model’s Lagrangian dynamics but transform its evolution into an Eulerian reference frame. Each grid cell is classified as either channel (1) or floodplain (0), depending on whether the centerline intersects that cell. Figure 1 illustrates this transformation for a 6 km-long meander, where panel (a) shows a Lagrangian ensemble at a given time and panels (b–d) show the same state evaluated on Eulerian grids with 10 m resolution (0–2 km), 50 m resolution (2–4 km), and 100 m resolution (4–6 km).

Fig. 1: Lagrangian and Eulerian representations of the meander model.
Fig. 1: Lagrangian and Eulerian representations of the meander model.The alternative text for this image may have been generated using AI.
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a Lagrangian ensemble of 100 realizations, each initialized from a common planform and perturbed by a single-node transverse displacement. b–d Eulerian representation of the same state, evaluated on fixed grids spanning the full 6 km domain with spatial resolutions of 10, 50, and 100 m, respectively.

Separation metric and growth-rate estimation

We quantify geometric divergence by comparing two simulations that are identical except for a small, localized perturbation applied at t = 0. Both the reference and perturbed realizations are then integrated forward with identical numerical parameters. At each output time, both initial planforms are mapped onto the fixed Eulerian grid, and their separation is defined using the Hamming distance dH(t) = ∥S*(t) − S(t)∥1, where S(t) and S*(t) are the binary channel-occupancy fields of the reference and perturbed trajectories. This distance counts the number of grid cells in which the two configurations differ and therefore captures geometric differences directly30,31,32,36. The finite-time Lyapunov exponent (FTLE) quantifies the rate of separation over a finite interval and is estimated as \({\lambda }_{FT}={({t}_{2}-{t}_{1})}^{-1}{{{\rm{ln}}}}[{d}_{H}({t}_{2})/{d}_{H}({t}_{1})]\), where (t1, t2) spans the period of approximately exponential growth in dH(t). A positive λFT indicates that small perturbations amplify exponentially, demonstrating sensitive dependence on initial conditions within this model.

Cutoffs control divergence at fixed grid resolution

We begin with a controlled on–off experiment in which all parameters are identical except for the cutoff switch. Two trajectories are initialized from the same initial planforms, with one perturbed transversely at a single interior node that is sufficiently small compared to both the nominal node spacing and the Eulerian grid resolution. Figure 2 compares the paired simulations at four times, from t1 to t4. The left column shows the case with cutoffs disabled, and the right column shows the case with cutoffs enabled, where the cutoff distance is set to dc = W, corresponding to a neck cutoff in which two channel banks physically touch. We adopt dc = W here because it represents the canonical geometric condition for neck removal. Choices of the form dc = nW with n > 1 produce the same qualitative behavior and, as shown in later analyses, do not alter the presence of chaos or the magnitude of the Lyapunov exponent in our model. Each panel presents a fixed-grid Eulerian overlay, where white marks cells occupied by both trajectories, red by the reference only, and blue by the perturbed run.

Fig. 2: Cutoffs control divergence in a kinematic meander model.
Fig. 2: Cutoffs control divergence in a kinematic meander model.The alternative text for this image may have been generated using AI.
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Side-by-side Eulerian maps compare paired simulations with cutoffs disabled (left; panels a, c, e, g) and enabled (right; panels b, d, f, h). Each panel shows channel occupancy on a fixed grid: white marks cells shared by both runs, red by the reference only, and blue by the perturbed run (black = floodplain). Panels a,b show identical initial conditions. Panels c, d show meander growth prior to any geometric separation. In panel e, the no-cutoff channel self-intersects, whereas in panel (f), the cutoff-enabled channel has already diverged. Panels g, h show continued evolution at the final stage. Without cutoffs, the two trajectories remain coincident throughout the run (dH(t) = 0; see Supplementary Fig. S1).

In the no-cutoff case, the two trajectories remain coincident on the Eulerian grid for the entire run, so the Hamming distance dH(t) stays identically zero. At t1, t2, t3, and t4 (Fig. 2a, c, e, g), the fields are uniformly white, reflecting identical channel occupancies at all times. By t3-t4 (Fig. 2e, g), the centerline starts to self-intersect, an unphysical configuration for a physical river. This zero-separation behavior is invariant under grid refinement, as the occupancies coincide across all tested spatial resolutions (Supplementary Fig. S1), and the corresponding differing-cell counts remain identically zero (Supplementary Fig. S2).

When cutoffs are enabled, a neck is severed once the minimum distance between nonadjacent segments falls below dc. The intervening loop is excised, the ends are reconnected, and the channel shortens, resetting local geometry. By t3-t4 (Fig. 2f, h), geometric differences appear as expanding red-blue regions in the right column, indicating divergence on the fixed grid. Because only cutoff-enabled runs produce measurable and sustained separation (Supplementary Fig. S2), Lyapunov exponents are reported exclusively for those cases. A time-lapse rendering of the paired trajectories at multiple grid resolutions is provided in Supplementary Video 1.

Dependence on Eulerian grid resolution

To quantify how Eulerian discretization affects Lyapunov-rate estimates, we simulate a channel with initial length 10 km and width W = 100 m migrating at 1 m yr−1 for 10,000 yr. We impose neck cutoffs with cutoff distance dc = W. We represent planform geometry on an Eulerian grid defined in physical units, with square cells of side length Δ, and we vary Δ over 1, 5, 10, 50, 100, 500, and 1000 m (spanning resolutions both finer than and coarser than W).

Across all Δ, ln dH(t) shows an initial transient, an approximately linear growth interval, and a saturation regime (Fig. 3a). Because dH is evaluated on an Eulerian grid, coarse discretizations merge nearby channel segments into the same cells, delaying the first nonzero separation and biasing λFT low. Consistent with this, the estimated finite-time Lyapunov exponents for Δ = {1, 5, 10, 50, 100, 500, 1000} m are {2.26, 2.07, 1.86, 1.57, 1.56, 0.472, 0.496} × 10−3 yr−1, respectively. The strongest reduction occurs once the grid becomes coarser than the channel width (Δ > W = 100 m): at Δ = 500 − 1000 m, λFT ≈ (0.47 − 0.50) × 10−3 yr−1, roughly a factor of  ~ 3 smaller than the Δ = 50 − 100 m estimates, indicating that resolving the channel at approximately its width is sufficient to avoid substantial underestimation of divergence.

Fig. 3: Resolution dependence of Lyapunov-rate estimates.
Fig. 3: Resolution dependence of Lyapunov-rate estimates.The alternative text for this image may have been generated using AI.
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a Temporal evolution of ln dH(t) for Eulerian grids with cell sizes Δ = 1, 5, 10, 50, 100, 500, and 1000 m. Coarser grids merge nearby channel segments and delay the first detectable separation, reducing the apparent slope of ln dH(t), whereas finer grids resolve divergence earlier. b Finite-time Lyapunov exponents λFT estimated from the linear growth window of ln dH(t) for each Δ. Growth-rate estimates decrease sharply once the grid becomes coarser than the channel width (Δ > W = 100 m).

Sensitivity to initial perturbation magnitude

Using the same physical simulation setup as in the grid-resolution analysis, we next evaluate whether the Lyapunov growth rate depends on the magnitude of the initial perturbation. The perturbation is applied as a transverse displacement to the middle node, with δ0 ranging from 100 to 10−10 m, spanning ten orders of magnitude in physical length. Using a converged Eulerian grid with Δ = 10 m, we compute the temporal evolution of ln dH(t) over the 10,000 yr simulation (Fig. 4a), and estimate λFT from the linear growth interval in physical time.

Fig. 4: Dependence on initial perturbation magnitude.
Fig. 4: Dependence on initial perturbation magnitude.The alternative text for this image may have been generated using AI.
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a Temporal evolution of ln dH(t) for paired runs with transverse perturbations δ0 ranging from 100 to 10−10 m. b Estimated finite-time Lyapunov exponents λFT as a function of δ0 (reported in units of 10−3 yr−1). For δ0≤10−1 m, λFT remains \({{{\mathcal{O}}}}(1{0}^{-3})\,{yr}^{-1}\), whereas the largest perturbation (δ0 = 100 m) yields a substantially higher growth rate.

Decreasing δ0 delays the onset of detectable separation but does not produce a systematic change in the inferred growth rate once exponential divergence is resolved. For δ0≤10−1 m, the estimated exponents remain \({{{\mathcal{O}}}}(1{0}^{-3})\,{yr}^{-1}\), spanning (2.0 − 4.3) × 10−3 yr−1 (Fig. 4b). In contrast, the largest perturbation tested, δ0 = 100 m, yields a substantially higher value (λFT = 1.08 × 10−2 yr−1), indicating that meter-scale displacements exceed the small-perturbation regime and instead probe finite-amplitude divergence. In all cases, ln dH(t) saturates once the finite Eulerian grid is fully filled, and fits are restricted to the pre-saturation interval.

Sensitive dependence recurs across initial planforms

We next examine whether sensitive dependence persists across distinct channel geometries and whether the Lyapunov exponent remains stable when sampled repeatedly along a long trajectory. Using the same physical setup as before (H = 10 km, W = 100 m, 1 m yr−1 migration rate, dc = W, and a 10 m Eulerian grid), we initialize simulations with four Kinoshita curves with amplitudes θ0 = 0.5,  1.0,  1.5, and 2.0 (Eq. (6)), spanning a representative range of meander shapes (Fig. 5a). For each initial planform we apply a transverse perturbation of magnitude δ0 = 10−5 m and integrate the paired trajectories for 10,000 yr, from which a finite-time Lyapunov exponent is estimated from the linear growth interval of ln dH(t).

Fig. 5: Robustness across initial planforms.
Fig. 5: Robustness across initial planforms.The alternative text for this image may have been generated using AI.
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a Four initial geometries drawn from a Kinoshita family with amplitudes θ0 = 0.5,  1.0,  1.5, and 2.0. b Distributions of finite-time Lyapunov exponents obtained from ten perturb-reset cycles for each initial planform (each cycle spanning 10,000 yr), using a 10 m Eulerian grid and a perturbation magnitude of δ0 = 10−5 m.

At the end of each 10,000 yr segment, the perturbed trajectory is reset to coincide exactly with the reference channel, while the reference trajectory continues uninterrupted and retains its evolving geometry. An identical perturbation is then applied, and both trajectories are advanced for another 10,000 yr. This perturb–reset procedure follows the renormalized Lyapunov-exponent approach (Benettin algorithm), in which repeated perturbations along a single evolving reference trajectory provide repeated samples of the local divergence rate39. Performing ten cycles yields ten Lyapunov-exponent estimates for each initial geometry and a total integration time of 100,000 yr per planform (Supplementary Fig. S3).

Across all four Kinoshita geometries, the resulting distributions cluster at \({{{\mathcal{O}}}}(1{0}^{-3})\,{yr}^{-1}\) (Fig. 5b). Median values are 3.58, 2.45, 2.62, and 3.39 × 10−3 yr−1 for θ0 = 0.5,  1.0,  1.5, and 2.0, respectively, with sample variances of 5.93, 0.959, 9.93, and 0.688 × 10−6 yr−2. Although these medians differ numerically, Lyapunov exponents operate on an exponential scale, so the observed spread corresponds to modest shifts in predictability horizon rather than a change in dynamical regime. This consistency indicates that the strength of sensitive dependence is not tied to any particular initial planform, but is an intrinsic feature of the cutoff-driven dynamics.

What controls the Lyapunov exponent?

A fundamental question concerns the physical processes that govern the strength of sensitive dependence in cutoff-driven meander dynamics. In our kinematic model, cutoffs act as the source of non-invertibility, providing the topological folding necessary to bound the system, while curvature-driven migration drives the stretching of trajectories. From first principles, the timing of these reset events depends on both the lateral migration rate (how fast bends evolve) and the cutoff distance dc (the geometric threshold for loop removal). To disentangle these factors, we performed a suite of simulations varying migration rates in {1, 5, 10, 15, 20} m yr−1 and cutoff thresholds dc ∈ {W, 5W, 10W, 15W, 20W}, where dc = W corresponds to neck cutoffs and dc = nW (n > 1) corresponds to chute cutoffs. Simulation durations were scaled inversely with migration rate to ensure comparable statistical sampling of cutoff events across regimes.

Figure 6a shows that the cutoff rate rc decreases systematically as dc increases. This trend is initially counterintuitive from the perspective of an isolated bend, where a larger proximity threshold would ostensibly shorten the time to cutoff. In a coupled planform, however, dc regulates the geometric complexity that can accumulate before a reset. A small threshold (dc = W) allows bends to mature into high-sinuosity loops, leading to higher migration rates, with frequent non-adjacent interactions, whereas a large threshold (dc = 20W) prunes bends early, maintaining a lower-curvature planform that migrates more slowly. Consequently, the median cutoff rate drops by an order of magnitude from rc ≈ 0.31 yr−1 at dc = W to rc ≈ 0.045 yr−1 at dc = 20W.

Fig. 6: Migration rate controls stretching, while cutoff threshold controls reset frequency.
Fig. 6: Migration rate controls stretching, while cutoff threshold controls reset frequency.The alternative text for this image may have been generated using AI.
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Heatmaps showing dependence on migration rate ({1, 5, 10, 15, 20} m  yr−1) and cutoff threshold n (dc = nW). a Cutoff rate rc (yr−1) decreases with n due to suppression of sinuosity. b Finite-time Lyapunov exponent λFT (yr−1) scales with migration rate but is insensitive to n. c The topological predictability horizon, defined as the number of cutoffs per Lyapunov time (Nc = rc/λFT). Nc is invariant with respect to migration rate but decreases for chute cutoffs, identifying a maximum horizon of  ~ 10 events in the neck-cutoff regime. Color scales in (a, b) are logarithmic.

Figure 6b reveals that the finite-time Lyapunov exponent λFT is controlled almost exclusively by the migration rate, exhibiting no consistent dependence on the cutoff threshold. Median λFT values increase from  ≈ 2.7 × 10−3 yr−1 at 1 m  yr−1 to  ≈ 8 × 10−2 yr−1 at 20 m  yr−1, while showing negligible sensitivity to dc for a fixed migration rate. This indicates that the migration rate sets the dominant timescale for trajectory stretching, whereas dc acts only as a discrete trigger for topological resetting without altering the intrinsic rate of geometric divergence.

Figure 6c presents the ratio Nc = rc/λFT, which represents the expected number of cutoff events within one Lyapunov time. This dimensionless quantity defines a topological predictability horizon. We find that Nc is invariant with respect to migration rate, as increasing the migration speed accelerates both cutoff frequency and Lyapunov exponent proportionally. Instead, Nc depends strongly on the cutoff threshold. For neck cutoffs (dc = W), the horizon is maximized, with a median of Nc ≈ 7.4 events. For chute cutoffs (dc = 20W), this horizon contracts to Nc ≈ 1.9. This reduction occurs because the suppression of bend growth at high dc slows the event frequency rc significantly more than it slows the chaotic stretching rate λFT. Thus, while the migration rate scales the speed of evolution, the cutoff mode determines the topological limit of predictability.

Discussion

Within the kinematic model class studied here, enabling cutoffs yields positive finite-time Lyapunov exponents that are stable with respect to numerical choices and initial conditions. When cutoffs are disabled, the Hamming separation dH(t) = ∣S*(t) − S(t)∣1 remains identically zero for paired trajectories, whereas enabling cutoffs produces macroscopic divergence. Figure 3 shows that, provided the Eulerian grid resolves the channel (Δ ≤ W), Lyapunov-rate estimates avoid the low bias seen on coarser grids and depend only weakly on further refinement, indicating that the measured growth reflects model dynamics rather than discretization artifacts. Figure 4 confirms that, once resolution is fixed, the inferred growth rate becomes independent of the initial perturbation magnitude for δ0 ≲ 10−1 m, consistent with sensitive dependence on initial conditions. Furthermore, Fig. 5 demonstrates that the same exponential divergence recurs across distinct initial planforms, indicating that the Lyapunov exponent reflects an intrinsic property of the dynamics rather than sensitivity to a particular starting geometry.

A key finding of this study is the decoupling of the chaotic divergence rate from the frequency of topological resetting. Figure 6 shows that the Lyapunov exponent scales systematically with migration rate but remains effectively invariant to the cutoff distance dc. In contrast, the cutoff rate decreases strongly as the cutoff distance increases, because larger thresholds (dc ≫ W) suppress the formation of high-sinuosity loops. This disparity creates a topological predictability horizon, defined as the expected number of cutoffs within one Lyapunov time, Nc = rc/λFT. We find that Nc decreases monotonically with increasing cutoff threshold. In the neck-cutoff regime (dc = W), the predictability horizon is maximized, whereas in chute-dominated regimes (dc = 20W) the horizon contracts substantially. This contraction occurs because increasing the cutoff threshold suppresses the event frequency rc far more strongly than it affects the chaotic stretching rate λFT. Thus, while curvature-driven migration sets the absolute magnitude of chaotic growth, the mode of cutoff determines the event-based horizon, limiting long-term forecasts to fewer than two topological updates in straight, chute-dominated channels.

Our findings also connect cutoff-driven meander evolution to event-driven chaos and self-organization in fluvial systems14,15. Previous studies have interpreted cutoff clustering as a self-organized critical mechanism that shifts the system between ordered and irregular planform states3,8,11. The cutoff-on/off experiment introduced here provides an Eulerian formulation of this idea, demonstrating that cutoffs act as topological resets that generate spatial recurrence and maintain sensitive dependence on initial conditions. From a dynamical-systems perspective, meander evolution with cutoffs is a hybrid process in which continuous planform migration is punctuated by state-dependent events that alter topology. Analogous reset-driven chaos has been analyzed in pulse-coupled (spiking) neural network models40,41, and in impact oscillators such as the bouncing ball, where intermittent collisions generate irregular, chaotic trajectories42.

The counterfactual experiment deserves emphasis because it cannot be realized in nature. A real river cannot evolve with cutoffs suppressed, yet a simulator can impose this condition and thereby isolate its influence while holding all other processes fixed. Under these circumstances, the shift from negligible growth to sustained exponential separation provides direct evidence that cutoffs supply the mechanism responsible for practical unpredictability in this model. Although the finite-time Lyapunov exponent carries physical units and yields a nominal predictability horizon of order 1/λFT, this value should not be taken as a forecast limit for natural rivers. Meanderpy omits fluid dynamics, sediment transport, bank-strength contrasts, hydroclimatic variability, and other stochastic or threshold processes that would shorten predictability, often well before a cutoff occurs.

Several caveats are important. The kinematic model abstracts many physical processes, including three-dimensional flow structure43,44, sediment transport6,7, bank-material heterogeneity26, and flood variability27,45,46. These factors strongly influence neck- and chute-cutoff formation in natural rivers, where overbank hydrodynamics, floodplain connectivity, vegetation, and sediment supply play key roles27,45,46. The kinematic formulation employed here reproduces only upstream-skewed, downstream-migrating bends, corresponding to the subresonant regime, which is widely regarded as the most common morphodynamic regime for large, sand-bedded meandering rivers in nature44,47. Superresonant bends, which are downstream-skewed and migrate upstream, are not captured by this class of models and may influence cutoff timing in ways outside the scope of the present analysis. The curvature–migration law used here also excludes nonlinear behavior observed in natural rivers, as migration rates do not always increase monotonically with curvature and may saturate or vary widely at a given curvature due to vegetation, bank strength, and flow–sediment feedbacks22,26,48. This abstraction isolates the mechanism, but it limits the scope of our conclusions to this kinematic model class and parameter ranges; additional nonlinear or time-variable migration processes may introduce further pathways to chaos.

These limitations point to straightforward next steps. The same Eulerian framework and Hamming separation metric can be applied to morphodynamic models that incorporate additional physics to test whether positive finite-time Lyapunov exponents persist or change in magnitude. Long image sequences from remote sensing could, in principle, be mapped into the same fixed-dimensional state, although the limited 50-year satellite record and slow migration rates of most rivers make it difficult to observe the long-term cutoff dynamics relevant here. For management and hazard assessment, explicitly incorporating a predictability horizon into probabilistic frameworks may be more informative than seeking ever longer deterministic forecasts49, especially in settings where cutoff frequency is high.

Conclusion

We used a fixed-dimensional Eulerian state and the Hamming distance to quantify sensitive dependence in a kinematic meander model. Three key results stand out. First, within this model class and parameter ranges, enabling cutoffs yields positive finite-time Lyapunov exponents, while disabling cutoffs removes measurable growth. The counterfactual experiment that suppresses cutoffs isolates their role and cannot be performed in nature, which underscores the value of the modeling approach. Second, the inferred Lyapunov growth rate is robust. It converges under grid refinement, is independent of the initial perturbation magnitude once the perturbation is sufficiently small, and appears consistently across diverse initial planforms. Third, we disentangled the controls on chaotic growth and topological resetting. The magnitude of the Lyapunov exponent is effectively invariant to the cutoff distance and instead scales with migration rate, demonstrating that curvature-driven migration governs the strength of sensitive dependence. However, the cutoff distance remains consequential because it sets the frequency of reset events, thereby controlling the event-based predictability horizon: the number of cutoffs expected within one Lyapunov time decreases monotonically as the cutoff distance increases, imposing a stricter topological limit on predictability in chute-dominated regimes compared to neck-cutoff regimes.

Together, these findings imply a practical predictability limit for planform evolution, where the horizon scales as 1/λFT. Beyond this horizon, deterministic forecasts of channel position degrade even when initial states differ only infinitesimally. This predictability horizon is conditional on both the simplified dynamics of the kinematic model and the Eulerian observation scale Δ; additional stochastic forcing, parameter variability, and unresolved processes in natural rivers should, in principle, shorten it. The Eulerian framework is broadly applicable: it provides a resolution-controlled, geometry-preserving state in which separation is well defined and amenable to convergence testing. As such, it offers a general route for detecting sensitive dependence in Earth-surface systems that undergo topological change and helps connect dynamical-systems concepts with geomorphic modeling in an interpretable, operational way.

Methods

Kinematic meander model

We simulate channel evolution using meanderpy22, an open-source deterministic model that represents the river as a planar centerline x(s, t) = [x(s, t), y(s, t)] parameterized by arc length s and time t. The update rules contain no stochastic forcing; for a prescribed initial centerline and fixed parameters, the evolution is fully reproducible. The unit tangent and normal vectors are \(\widehat{{{{\bf{t}}}}}=\partial {{{\bf{x}}}}/\partial s\) and \(\widehat{{{{\bf{n}}}}}={\widehat{{{{\bf{t}}}}}}^{\perp }\), and curvature is κ(s, t) = ∂θ/∂s, where θ(s, t) is the local flow direction and R = ∣κ∣−1 is the radius of curvature.

Nodes migrate in the local normal direction at a speed proportional to a nominal curvature-derived migration rate and an adjusted migration rate that incorporates upstream influence22,23. The nominal rate is

$${R}_{0}(s,t)={k}_{\ell }\,W\,\kappa (s,t),$$
(1)

where W is channel width and kℓ is a lateral-migration coefficient. The adjusted migration rate is

$${R}_{1}(s,t)=\Omega \,{R}_{0}(s,t)+\Gamma \left[{\int }_{0}^{\infty }{R}_{0}(s-\xi ,t)\,G(\xi )\,d\xi \right]{\left[{\int }_{0}^{\infty }G(\xi )d\xi \right]}^{-1},$$
(2)

where Ω and Γ are weighting parameters. In the implementation used here, the adjusted rate is further scaled by sinuosity as \({R}_{1}\leftarrow {{{{\mathcal{S}}}}}^{-2/3}{R}_{1}\), where \({{{\mathcal{S}}}}\) is the ratio of centerline length to end-to-end distance. The kernel controlling upstream influence is

$$G(\xi )={e}^{-\alpha \xi },\,\alpha =\frac{2k\,{C}_{f}}{D},\,k=1,$$
(3)

and depends on a friction coefficient Cf and flow depth D.

The centerline motion is purely normal:

$$\frac{\partial {{{\bf{x}}}}}{\partial t}(s,t)={R}_{1}(s,t)\,\widehat{{{{\bf{n}}}}}(s,t).$$
(4)

A cutoff is triggered when the minimum distance between nonlocal pairs of centerline nodes (after blanking a diagonal band of nearby along-stream neighbors) first falls below a threshold dc:

$${\min }_{| i-j| > \Delta {i}_{\min }}\parallel {{{{\bf{x}}}}}_{i}(t)-{{{{\bf{x}}}}}_{j}(t)\parallel < {d}_{c},$$
(5)

with \(\Delta {i}_{\min }=\lfloor ({d}_{c}+20\Delta s)/\Delta s\rfloor\) in the code.

When a cutoff occurs, the enclosed loop is excised, the open ends are reconnected along the shortest path, and the centerline is reparameterized by arc length. Nodes are redistributed to restore near-uniform spacing Δs, ensuring consistent curvature estimation. In the implementation shown, reparameterization is performed by spline-based resampling with zero smoothing (s=0); the only explicit smoothing step in the main loop is a Savitzky–Golay filter applied to the elevation coordinate z (used for the optional autoaggradation calculation), not to the planform coordinates x, y.

Our experiments initialize the channel using a Kinoshita curve. The orientation angle varies sinusoidally with downstream coordinate s:

$$\theta (s)={\theta }_{0}\,\sin \left(\frac{2\pi s}{\lambda }\right)+{\theta }_{base},$$
(6)

where θ0 is initial amplitude, λ is meander wavelength, and θbase is a uniform offset. The planform coordinates follow by integrating the unit tangent:

$$x(s)={\int }_{0}^{s}\cos \theta (\xi )\,d\xi ,$$
(7)
$$y(s)={\int }_{0}^{s}\sin \theta (\xi )\,d\xi .$$
(8)

Eulerian state representation and separation metric

To enable separation diagnostics and Lyapunov-exponent estimation, we observe the Lagrangian model in a fixed-dimensional Eulerian frame. The planar domain \(\Omega \subset {{\mathbb{R}}}^{2}\) is discretized into an Nx × Ny grid with square cells of side length Δ. At each saved time, the evolving centerline Γ(t) is mapped to a binary occupancy field:

$${S}_{k\ell }(t)=\left\{\begin{array}{ll}1, & \Gamma (t)\cap {C}_{k\ell }\ne {{\emptyset }},\\ 0, & otherwise,\end{array}\right.$$
(9)

where Ckℓ denotes cell (k, ℓ).

For two realizations differing only by a small initial perturbation, their instantaneous separation is defined using the Hamming distance:

$${d}_{H}(t)=\parallel {{{{\bf{S}}}}}^{* }(t)-{{{\bf{S}}}}(t){\parallel }_{1},$$
(10)

where entries of ΔS(t) = S*(t) − S(t) lie in { − 1, 0, 1}, so dH(t) counts the number of cells in which the two realizations differ.

Lyapunov-exponent estimation and experiment design

Two trajectories with Eulerian fields S(t) and S*(t) are initialized from identical planforms except for a small transverse perturbation of magnitude δ0 ≪ ε, where ε is the nominal node spacing. Let \({t}_{\det }\) denote the earliest time at which dH(t) > 0. The finite-time Lyapunov exponent (FTLE) over an interval of exponential growth is

$${\lambda }_{FT}({t}_{1},{t}_{2})=\frac{1}{{t}_{2}-{t}_{1}}{{{\rm{ln}}}}\left(\frac{{d}_{H}({t}_{2})}{{d}_{H}({t}_{1})}\right),\,{t}_{\det } < {t}_{1} < {t}_{2}.$$
(11)

Robustness is assessed by decreasing the perturbation magnitude (with δ0 ≪ ε), refining the Eulerian grid until convergence, and repeating the experiment for multiple initial planforms.