Switching Dynamical Systems Overview

• Switching Dynamical Systems (SDSs) are mathematical frameworks that integrate discrete regime switching with continuous state evolution to capture complex, time-evolving behaviors.
• They encompass formulations such as Markov-switching models, piecewise-linear ODEs, and hybrid stochastic processes that effectively model regime shifts and multistability.
• Applications span systems biology, neuroscience, robotics, and network control, with ongoing research in scalable inference, optimal control, and causal discovery.

A switching dynamical system (SDS) is a mathematical framework for modeling time-evolving processes where the dynamics can abruptly change among a finite or countable set of regimes, modes, or subsystems according to a rule—deterministic, stochastic, or state-dependent. SDSs fundamentally interleave discrete mode selection with continuous or discrete state evolution, allowing them to capture behaviors such as regime shifts, hybrid computation, synchronization, and emergent multistability across engineering, biological, and networked systems. Their modern formalizations encompass piecewise-smooth and piecewise-linear ODEs, Markov-switching state-space models, stochastic differential equations with regime change, infinite-dimensional switched PDEs, and networked agents with time-varying topology.

1. Mathematical Formulations and Model Classes

At the core of SDS modeling is the interaction of a continuous (vector or function-valued) state process with a discrete-valued regime process. Canonical forms include:

Discrete-time and continuous-time Markov switching state-space models. Here, a latent regime $z_t \in \{1, \ldots, K\}$ follows a Markov chain, modulating dynamics such as $x_{t+1} = A_{z_t} x_t + w_t$ (with Gaussian $w_t$), and observations $y_t$ (which may also be regime-modulated), yielding the Switching Linear Dynamical System (SLDS) (Linderman et al., 2016, Georgatzis et al., 2015). In continuous time, Markov jump processes $z(t)$ drive SDEs in each regime: $dx(t) = f_{z(t)}(x(t), t) dt + G_{z(t)} dW(t)$ (Köhs et al., 2022, Köhs et al., 2021).

Piecewise deterministic and piecewise linear/affine ODEs. Trajectories evolve as $\dot{x}(t) = f_{\sigma(x,t)}(x(t))$, with $\sigma$ a piecewise-constant switching signal (possibly state- or threshold-triggered) (Ontanon-Garcia et al., 2016, Korotkov et al., 2023, Conradi et al., 2010). This subsumes systems generating multiscroll chaos and heteroclinic networks.

Switching diffusion and hybrid stochastic processes. General SDSs allow for interaction between a diffusion process $X(t)$ and a switching process $\alpha(t)$, possibly with delayed or history-dependent rates, on countable or infinite state spaces (Nguyen et al., 2017, Li et al., 2020).

Networked and multi-agent SDSs. In large-scale settings, each agent or subsystem follows local dynamics, with regime changes induced by network structures or interactions that themselves evolve (e.g., via dynamic graphs) (Liu et al., 2023, Mouyebe et al., 1 Apr 2025, Qin et al., 2018).

Infinite-dimensional PDE-SDSs. Here, the regime switching modulates generators of strongly continuous semigroups in Banach or Hilbert space, modeling switching in distributed-parameter systems (Hante, 2018).

2. Inference, Learning, and Identifiability in SDSs

Exact and approximate inference. Classical SLDSs are intractable to marginalize exactly due to the combinatoric growth in possible regime paths and coupling with latent states. Methods range from forward-backward message passing, Kalman and particle-based filtering, and block Gibbs samplers alternating over trajectories and mode sequences (often leveraging conjugate priors and Polya-Gamma augmentation to maintain tractability (Linderman et al., 2016, Fox et al., 2010, Köhs et al., 2022, Nassar et al., 2018)).

Nonparametric regime discovery. To infer the number of dynamical regimes, hierarchical Dirichlet process (HDP) priors on transition matrices (the “sticky HDP-SLDS” framework) jointly learn both the dynamics and the switching process (Fox et al., 2010, Sieb et al., 2018). This enables segmentation and discovery of movement primitives or sub-behaviors with unknown complexity.

Discriminative and hybrid strategies. Discriminative SDSs learn predictive classifiers (e.g., random forests) for switches, conditioning latent inference on feature representations of observed data, with improved detection of transient or artifact regimes (Georgatzis et al., 2015).

Variational and hybrid inference in continuous time. Variational methods with pathwise KL objectives, moment ODE relaxations, and Gaussian process surrogates enable scalable inference in hybrid diffusion–jump models (Köhs et al., 2021).

Identifiability. Under non-degeneracy, unique indexing, and analyticity conditions, non-linear Markov switching models and their nonlinear decoders are provably identifiable up to affine and permutation transformations, as shown for general non-linear SDS latent variable models (Balsells-Rodas et al., 2023). This ensures meaningful recovery of dynamical mechanisms in unsupervised segmentation and causal discovery.

3. Regime Switching Mechanisms and Bifurcation Theory

SDSs admit various regime switching mechanisms:

• Markovian switching: Dynamics transition according to a Markov chain or jump process, with possibly countable or infinite states (Nguyen et al., 2017, Köhs et al., 2022).
• State- or observation-dependent switching: Transition probabilities or gating functions depend on the continuous latent state, as in recurrent SLDS (rSLDS) (Linderman et al., 2016) or tree-structured recurrence (TrSLDS) (Nassar et al., 2018).
• Threshold, piecewise, or delay switching: Switching is triggered by thresholds or boundaries in the state-space or depends on a segment of prior state (history) (Ontanon-Garcia et al., 2016, Korotkov et al., 2023, Li et al., 2020).

Bifurcation theory provides rigorous analytic criteria for switching behavior, especially in systems biology and network dynamics. Saddle-node bifurcations correspond to “switchpoints” where a defective zero eigenvalue in the Jacobian signifies a regime-change, and these can be detected by solving explicit systems of linear inequalities even in high-dimensional mass-action ODE models (Conradi et al., 2010).

4. Stability, Control, and Optimization in Switching Systems

Stability under switching. Uniform stability—exponential, asymptotic, or in moments—requires Lyapunov functions common across all regimes or classes of signals. Classical results establish that the existence of a common Lyapunov function is necessary and sufficient for global uniform exponential stability (GUES), both in finite and infinite dimensions (Hante, 2018, Mouyebe et al., 1 Apr 2025).

Optimal control. Mixed-integer optimal control for SDSs (discrete switches and continuous controls) is computationally hard; relaxations via outer convexification (“sum-up-rounding”), adjoint-based gradient methods (switching-time and mode-insertion gradients), and second-order SQP schemes enable tractable, near-optimal scheduling of switches (Stellato et al., 2016, Hante, 2018).

Feedback and sampled-data control. In regime-switching stochastic diffusion models, stabilization criteria for delay- and sample-based feedback controls can be derived, with explicit bounds on the sampling interval and controller delay required for exponential or moment stability, closely tied to Lyapunov exponents and the stationary distribution of the switching process (Li et al., 2020).

Coupled, networked, and hybrid systems. Coupling-induced stabilization depends conjointly on agent dynamics, the spectrum of switching graph Laplacians, and the switching signal. Sufficient conditions for stability are given in terms of degree-based inequalities, master-stability functions, and graph-theoretic properties (e.g., connectivity, bipartiteness) (Mouyebe et al., 1 Apr 2025, Qin et al., 2018).

5. Applications and Empirical Studies

SDSs are extensively used in:

Systems biology and chemical networks. Detection of multistability and switching via analytic conditions in mass-action networks (e.g., cell cycle regulation, biochemical switches) (Conradi et al., 2010).

Neuroscience and excitable systems. Modeling of switching activity and winnerless competition in neural populations via heteroclinic cycle dynamics, yielding robust sequential activation interpreted as transient neural coding (Korotkov et al., 2023).

Robotics and trajectory segmentation. Nonparametric SLDSs enable unsupervised discovery of movement primitives from demonstrations, forming the foundation for movement libraries and modular control (Sieb et al., 2018).

Time series segmentation and health monitoring. Discriminative and hybrid SLDSs are used for ICU monitoring (artifact detection, physiological state estimation), achieving high area under the ROC curve for multiple events (Georgatzis et al., 2015).

Networked and multi-agent systems. Graph-based switching models (GRASS) handle interacting objects with latent, dynamic interactions, outperforming independent-object approaches in learning from high-dimensional streaming data and real-world tasks (colliding particles, human motion) (Liu et al., 2023).

Infinite-dimensional and PDE applications. Switched PDE control arises in optimal operation of distributed processes with regime changes, including hyperbolic systems with intermittent damping and mixed-integer switching constraints (Hante, 2018).

6. Extensions, Challenges, and Open Directions

Active research in SDSs addresses:

• Nonparametric and hierarchical regime models: Automatic selection of mode cardinality and hierarchical structure with Bayesian nonparametrics and deep generative mechanisms (Fox et al., 2010, Nassar et al., 2018).
• Scalable inference techniques: Hybrid amortized–exact methods, Polya-Gamma augmentation, variational EM for continuous-time models, and blockwise Gibbs schemes for efficient posterior simulation (Linderman et al., 2016, Köhs et al., 2021, Köhs et al., 2022).
• Interpretability and hierarchical structure: Tree-structured and multi-scale models decompose complex nonlinear flows into hierarchies of locally linear regimes, balancing prediction and interpretability (Nassar et al., 2018).
• Graphical and interacting-object systems: Extension of SDSs to learning time-varying, sparse interaction structures and capturing inter-object regime dependencies (Liu et al., 2023).
• Identifiability, causal inference, and segmental state-space recovery: Formal guarantees for unique recovery of nonlinear regime-structured dynamical models, with implications for causal discovery in complex systems (Balsells-Rodas et al., 2023).
• Synchronization and coordination: Algebraic-geometric analyses of global synchronization under directed switching topologies, with joint connectivity and spectral gap conditions for consensus under switching communication networks (Qin et al., 2018).