Submersive Resets in Dynamical Systems

• Submersive resets are partial reset mechanisms in stochastic and hybrid systems, characterized by a submersion parameter that scales the system state.
• They enable tunable non-equilibrium steady states with controlling effects on energy dissipation, diffusion behavior, and phase synchronization in complex systems.
• They also introduce unique challenges in hybrid optimal control, where rank-deficient reset maps require careful geometric consistency and costate selection.

Submersive resets denote a broad class of reset mechanisms in stochastic and hybrid dynamical systems where the reset operation induces a continuous reduction or transformation—often parameterized by a "strength"—rather than a full return to a designated state. The concept arises naturally in the study of stochastic resetting with partial resets and in hybrid systems where the reset map is a submersion: a smooth map whose differential may drop rank, causing dimensional collapse in the post-reset state space. These mechanisms generalize classical resetting protocols and introduce rich non-equilibrium and geometric phenomena, relevant to non-equilibrium thermodynamics, optimal control, and synchronization of complex systems (Olsen et al., 2024, Clark et al., 2024, Majumder et al., 2024).

1. Mathematical Formalism of Submersive Resets

Submersive resets in stochastic processes are typified by protocols in which a system variable $x(t)$ is intermittently reset at random times to a transformed value $a x(t)$, where $a \in [0,1]$ is the submersion parameter controlling reset strength (Olsen et al., 2024). The general stochastic dynamics are described by

$$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$

with $\lambda(t) \in \{0,1\}$ indicating exploration and resetting phases. The prototypical Fokker-Planck equation governing such partial reset dynamics is

$$\frac{\partial P}{\partial t} = D\,\frac{\partial^2 P}{\partial x^2} - r P(x, t) + r a^{-1} P\left(\frac{x}{a}, t\right),$$

where resets occur at Poissonian rate $r$ and each event maps $x \to a x$.

In hybrid dynamical systems, submersive resets are formalized as events where the reset map $\Delta: S \to M$ has non-maximal rank; that is, the differential $\Delta_*$ at the guard $a x(t)$0 satisfies $a x(t)$1, effecting a submersion onto its image (Clark et al., 2024). The result is a drop in effective system dimension after the reset.

2. Thermodynamic and Stationary Properties

For overdamped Brownian motion subject to submersive resets, the steady-state distributions $a x(t)$2 interpolate between the Laplace-like distribution for $a x(t)$3 and a wide Gaussian as $a x(t)$4. The explicit steady-state characteristic function is

$a x(t)$5

with the limiting Gaussian width $a x(t)$6 as $a x(t)$7 (Olsen et al., 2024). These distributions characterize new classes of non-equilibrium steady states (NESS) parametrized by the reset strength.

The thermodynamic work required to maintain these NESS depends nontrivially on both the resetting trap potential $a x(t)$8 and the background potential $a x(t)$9. For harmonic traps $a \in [0,1]$0, the steady-state work rate $a \in [0,1]$1 is independent of $a \in [0,1]$2 for all $a \in [0,1]$3. In contrast, for anharmonic traps $a \in [0,1]$4 with $a \in [0,1]$5, $a \in [0,1]$6 typically increases with $a \in [0,1]$7, and for $a \in [0,1]$8 it decreases. For all cases, $a \in [0,1]$9, consistent with detailed balance.

3. Geometric and Optimal Control Aspects in Hybrid Systems

In the field of hybrid optimal control, submersive resets engender fundamentally new phenomena. A submersive reset map $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$0 causes the push-forward $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$1 to be rank-deficient. The consequent adjoint (costate) jump conditions,

$$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$2

may no longer admit unique solutions. Necessary geometric consistency conditions require that the pre-impact costate $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$3 annihilates the fiber directions of the submersion, i.e., $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$4.

For reset events, the set of candidate post-reset costates $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$5 forms an affine space of dimension $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$6, subject to a further scalar Hamiltonian continuity constraint. Forward propagation under Hamiltonian flow, coupled with consistency at the next reset (a “return map” argument), selects the physically admissible solution trajectory (Clark et al., 2024).

A paradigmatic example is a point-mass with internal variables and impacts resetting some of its coordinates, leading to a family of admissible costates after each impact; unique optimality is recovered by ensuring next-impact consistency.

4. Subsystem and Partial Resetting in Many-Body Dynamics

Subsystem or partial resetting represents a many-body extension where only a subset of system constituents undergo resets while the rest evolve under nominal dynamics. In the Kuramoto model of phase oscillators, subsystem resetting—periodically synchronizing a fraction $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$7 of oscillators at rate $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$8—can nonlocally induce global order (Majumder et al., 2024).

The Ott-Antonsen reduction for the non-reset subpopulation leads to a mean-field equation with an additive reset term: $$\dot x(t) = \sqrt{2D}\,\xi(t) - [1-\lambda(t)]\,\mu\,V'(x) - \lambda(t)\,\mu\,\Phi'(x),$$9 where $\lambda(t) \in \{0,1\}$0 is the complex order parameter. For zero mean frequency ($\lambda(t) \in \{0,1\}$1), even infinitesimal resetting fraction ($\lambda(t) \in \{0,1\}$2) ensures a nonzero stationary synchrony ($\lambda(t) \in \{0,1\}$3), overriding the critical coupling of the bare model. For nonzero mean frequency, the phase diagram exhibits regions with stationary or oscillatory global synchronization.

5. Asymptotic and Limiting Behaviors

The limiting cases of the submersion parameter $\lambda(t) \in \{0,1\}$4 in partial resetting display distinct physical regimes. Strong resetting ($\lambda(t) \in \{0,1\}$5) yields classical resetting behavior with minimal memory of prior states and leads to sharp, typically non-Gaussian NESS. Weak resetting ($\lambda(t) \in \{0,1\}$6) recovers standard diffusive equilibrium, with the work required to enforce resetting vanishing in all cases.

For control-theoretic submersion, the dimension of the admissible costate space after a reset increases as the reset becomes more degenerate (lower rank), requiring additional post-impact constraints to restore determinacy.

6. Physical, Algorithmic, and Operational Implications

Submersive resets generalize classical resetting and resettable hybrid systems by providing tunable interpolation between fully deterministic resets and continuous flow. Their introduction leads to:

• Families of NESS with systematically controllable properties (e.g., width, modality) via the submersion parameter.
• New scaling laws for energy dissipation in NESS maintenance, with possible non-monotonic dependence on reset strength, especially in anharmonic systems (Olsen et al., 2024).
• Complex solution structures in hybrid optimal control, requiring boundary-value and root-finding algorithms for admissible trajectory selection (Clark et al., 2024).
• Efficient control mechanisms for collective synchronization with minimal intervention in complex networks (Majumder et al., 2024).

Operational distinctions versus diffeomorphic resets are summarized in the following table:

Reset Type	Costate Solution Structure	Admissible-Set Dim (per reset)	Algorithmic Selection
Diffeomorphic	Unique	0	Classical jump, no consistency checks needed
Submersive	Affine family (often infinite)	$\lambda(t) \in \{0,1\}$7	Forward-propagate, next-impact “return map” consistency

The rich structure of submersive resets informed by recent research establishes them as a crucial organizing concept at the intersection of non-equilibrium thermodynamics, geometric control, and statistical physics.