Finite-Time Synchronization in Complex Dynamical Systems

Finite-time synchronization refers to the coordinated evolution of coupled nonlinear systems such that all states or outputs converge to an identical trajectory within a predetermined or bounded time interval. Unlike asymptotic synchronization, which merely ensures convergence as time approaches infinity, finite-time approaches guarantee that the synchronisation error vanishes in a finite settling time. A related concept, fixed-time synchronization, further removes any dependence of that settling time on initial conditions. The study of these phenomena draws on Lyapunov stability theory, differential inclusion frameworks, Razumikhin and Krasovskii functionals, and sliding-mode or impulsive control strategies. Applications span secure communications, coordinated control of power grids, neuronal network modelling, and multi-agent robotics. Recent advances have extended finite-time criteria to networks with time delays, stochastic perturbations, fractional dynamics, discontinuous activations and memristive connections, thereby broadening both theoretical insight and practical impact across engineering and physical sciences.

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Recent work on cluster and community networks has developed novel state-feedback controllers for semi-Markovian switching fuzzy complex dynamical systems. By combining Takagi-Sugeno fuzzy modelling, stochastic analysis and Lyapunov–Krasovskii functionals, these studies demonstrate global finite-time and fixed-time convergence across discontinuous network nodes under switching protocols.

Investigations into impulsive effects in complex dynamical networks have yielded a new fixed-time stability lemma that ensures settling times independent of initial conditions. Controllers based on 1-norm Lyapunov functions and relaxed Hölder continuity conditions have been shown to synchronise networks with impulsive couplings more accurately and robustly than earlier methods.

In fractional-order and memristor-based fuzzy cellular neural networks, researchers have established fixed-time synchronization criteria under time-varying delays by utilising discontinuous differential inclusion techniques. Constructing appropriate Lyapunov functionals and feedback laws extends synchronisation results to systems with non-Lipschitz nonlinearities and generalised memory effects, thereby enhancing design flexibility and performance guarantees.

Finite-Time Synchronization in Complex Dynamical Systems publication trend

The graph below shows the total number of articles in finite-time synchronization in complex dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Finite-time synchronization: Convergence of coupled systems to a common trajectory within a bounded time that may depend on initial conditions.

Fixed-time synchronization: Synchronization achieved within a predetermined time bound independent of initial states.

Settling time: The maximum time required for the synchronisation error to reach zero under a given control scheme.

Lyapunov function: A scalar function used to assess the stability of a system, whose decay rate ensures convergence.

Differential inclusion: A generalisation of differential equations allowing discontinuous or set-valued right-hand sides.

Impulsive effect: A sudden, discrete change in system state at specified instants, modelled by impulsive differential equations.

Memristor: A nonlinear element whose resistance depends on the history of electrical charge, endowing networks with memory.

Fuzzy neural network: A computational model combining fuzzy logic rules with neural network architectures to handle uncertainty and nonlinearity.

References

1. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete and Continuous Dynamical Systems - B (2017).
2. Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes. AIMS Mathematics (2022).
3. Fixed-Time Synchronization of Complex Dynamical Network With Impulsive Effects. IEEE Access (2020).
4. Fixed-Time Synchronization of Delayed Fractional-Order Memristor-Based Fuzzy Cellular Neural Networks. IEEE Access (2020).