Hybrid Dynamical Systems Reduction

• Hybrid dynamical systems are models that combine continuous evolution with discrete transitions to capture complex dynamic behaviors.
• They enable rigorous dimensional reduction and robust control design, as illustrated by applications in robotics and cyber-physical systems.
• The reduction process leverages nondegeneracy and smoothing techniques to simplify analysis and support data-driven modeling of rhythmic behaviors.

Hybrid dynamical systems are mathematical models that integrate both continuous-time evolution—governed by differential equations on smooth manifolds—and discrete, event-triggered transitions or state resets. This dual structure is essential for capturing systems that switch between distinct dynamic regimes due to impacts, contact events, mode switches, or discrete control updates. Hybrid systems are foundational in modeling robotics (particularly locomotion and manipulation with impacts), cyber-physical systems, and biological networks exhibiting bursty or phase-dependent changes.

1. Foundational Structure and Reduction Near Periodic Orbits

A hybrid dynamical system is formalized as a tuple $(D, F, G, R)$, where $D = \coprod_{j \in J} D_j$ is the disjoint union of smooth manifolds (the continuous domains), $F$ specifies the smooth vector field(s) on each $D_j$, $G$ is the “guard”—a codimension one subset of the domain boundaries where transitions occur—and $R$ is the reset map applied upon reaching $G$. Near a periodic orbit, the hybrid system’s Poincaré map $P$ encapsulates one full cycle of continuous flow and discrete resets.

A central result is that, given a neighborhood of an exponentially stable periodic orbit, if the iterated Poincaré map $P^m$ has constant rank $r$ (where $m$ is the minimal dimension among domains), then trajectories initiated near the orbit contract (exactly or approximately) to a constant-dimensional hybrid submanifold $M \subset D$ with $\dim(M) = r+1$. This reduction is constructed by applying the Inverse Function Theorem to $P^m$ and propagating the resulting set through one hybrid cycle. Importantly, this mechanism ensures that high-dimensional hybrid systems arising in practice (e.g., locomotion with many degrees of freedom) generically collapse, after a finite number of cycles, onto much lower-dimensional invariant sets.

2. Exponential Stability and Superexponential Contraction

For exponentially stable periodic orbits—characterized by the spectral radius $\rho(D P(\xi)) < 1$ of the linearization—trajectories contract superexponentially to the reduced invariant submanifold. In local coordinates, the Poincaré map splits as $\tilde{P}(z, \zeta) = (Az, S(z,\zeta))$, where $A$ is invertible and $S$’s linearization in nilpotent (inactive) directions captures superexponential decay. More precisely, after $k$ cycles,

$\|P^k(z, \zeta) - (A^k z, 0)\| \leq C^k \epsilon$

for a sufficiently small neighborhood, ensuring extremely rapid convergence to the active (dynamically persistent) subspace. This contraction property underpins the observed robustness of many hybrid behaviors (such as stable walking or hopping gaits) and justifies aggressive dimension reduction for analysis and control design.

3. Nondegeneracy, Rank, and Smoothing Hybrid Transitions

The reduction process is contingent on a nondegeneracy (rank) condition: the iterated map $P^m$ must have rank $r$ at the periodic orbit and in a neighborhood. Typically, due to loss in the reset map, $P$ has at most the minimal domain dimension minus one in rank. The nondegeneracy condition ensures that the “inactive” subspace resulting in this rank loss is dynamically decoupled or collapses rapidly, justifying the elimination of extraneous variables from the effective description.

After reduction onto $M$, discrete hybrid transitions persist. The work shows that when each reduced domain $M_j$ has matching dimension and the reset $R$ acts diffeomorphically on the involved boundaries, a topological quotient can be taken—identifying $x \in G$ with $R(x)$—that yields a smooth manifold $\bar{M}$ supporting continuous-time flows only. This “smoothing” process constructs an equivalent continuous dynamical system whose trajectories correspond bijectively to executions of the original hybrid system.

4. Applications: Mechanisms and Control of Rhythmic Hybrid Behavior

Several concrete systems illustrate theoretical results:

• Vertical Hopper: A 2-mode (aerial and ground) model with plastic impact shows, via Poincaré analysis, contraction to a one-degree-of-freedom subsystem; precisely, if the spectrum of $D P(\xi) \approx 0.57$, the rank hypothesis for exact reduction is satisfied.
• Multi-Leg Polyped: For an underactuated $3+2n$ DOF model, an explicit feedback law decouples the closed-loop Poincaré map into a non-nilpotent “template” part and a nilpotent remainder. High-dimensional hybrid behavior thus reduces to a three DOF subsystem, irrespective of leg count.
• Deadbeat Control: By enforcing a full-rank condition on the Poincaré map’s derivative with respect to discrete event-driven input $\theta$, the system can be controlled to hit the fixed point in one (or a few) cycles. If $D_\theta P(\xi,\theta)$ is invertible, a feedback law $\psi(x)$ exists to make all trajectories reach the periodic orbit in finite time; deadbeat and approximate deadbeat designs are structurally robust.
• Hybrid Floquet Normal Forms: After reduction and smoothing, a generalized Floquet normal form is obtained:

$$\dot{z} = A(\theta) z, \quad \dot{\theta} = 2\pi/\tau,$$

providing rigorous coordinates for phase and amplitude in hybrid oscillators and a framework for data-driven low-dimensional modeling.

5. Feedback and Event-Driven Control Structures

The hybrid system framework accommodates controls that are updated only at discrete events (e.g., at guard crossings). Two principal cases arise:

• Exact Deadbeat Law: When $$\operatorname{rank} D_\theta P(\xi, \theta) = \dim \Sigma$$, one may synthesize a smooth feedback $\psi(x)$ such that $P(x, \psi(x)) = \xi$ near the fixed point, yielding finite-time convergence.
• Multi-Cycle Reduction/Approximate Deadbeat: If a single cycle does not yield full rank, multi-step iterates can be considered; if $D_{(\theta_1, ..., \theta_k)} P_k$ achieves full rank at the fixed point, a deadbeat law can be synthesized for convergence in $k$ cycles. When neither is possible, approximate deadbeat control ensures superexponential attraction to the reduced manifold, preserving system robustness under smooth perturbations.

This framework supports synthesis and structural stability analysis for feedback control laws governing rhythmic hybrid systems such as walking, running, or manipulation cycles.

6. Implications for Dimensionality Reduction and Hybrid Normal Forms

The reduction and smoothing framework provides mathematical justification for the observation that complex, high-dimensional hybrid systems often behave as if governed by low-dimensional templates—core to the “templates and anchors” hypothesis in biological locomotion. By proving that, under generic conditions, hybrid systems generically collapse (by exact contraction or superexponential decay) to a subsystem whose dimension is dictated by the rank of the Poincaré iterates, the analysis gives both a theoretical and practical path from intricate models to tractable, template-driven behavior.

Moreover, the construction of normal forms on the quotient manifold $\bar{M}$ enables the deployment of classical dynamical systems theory for stability, bifurcation analysis, and even data-driven identification in hybrid settings.

7. Significance in Synthesis, Analysis, and Future Directions

These results unify hybrid dynamical systems reduction, smoothing, and control under a rigorous geometric and algebraic framework. They explain not just the apparent low effective dimensionality of dynamic locomotion and manipulation, but also provide design principles for synthesizing robust, rhythmic controllers. The findings also indicate a broad class of hybrid systems whose long-term behavior is equivalent—modulo a topological quotient—to smooth flows, facilitating direct application of smooth methods for further analysis.

Open directions include generalization to broader hybrid model classes, extension to non-rhythmic behaviors, and interrogation of how symmetry and input structure affect reduction and control strategies. These insights lay a foundation for continued exploration of low-dimensional structure, robustness, and feedback design in engineered and natural hybrid dynamical systems.