---
title: Submersive Resets in Dynamical Systems
url: https://www.emergentmind.com/topics/submersive-resets
type: topic
---

# Submersive Resets in Dynamical Systems

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 [2401.11919, 2401.14476, 2402.14921].

## 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 [2401.11919]. 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 $S$ satisfies $\operatorname{rank} \Delta_* < \operatorname{dim} S$, effecting a submersion onto its image [2401.14476]. 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 $P_{\mathrm{ss}}(x)$ interpolate between the Laplace-like distribution for $a=0$ and a wide Gaussian as $a \to 1$. The explicit steady-state characteristic function is
\[
\widehat{\rho}_{\mathrm{ss}}(k) = \prod_{j=0}^\infty \frac{1}{1 + (D/r)a^{2j}k^2}
\]
with the limiting Gaussian width $\Sigma(a)^2 = \frac{2D/r}{1-a^2}$ as $a \to 1$ [2401.11919]. 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 $\Phi(x)$ and the background potential $V(x)$. For harmonic traps $\Phi(x) = \frac{1}{2}\kappa_R x^2$, the steady-state work rate $W(a)$ is independent of $a$ for all $a<1$. In contrast, for anharmonic traps $\Phi(x) = \frac{\kappa_R}{\zeta}|x|^\zeta$ with $\zeta>2$, $W(a)$ typically increases with $a$, and for $\zeta<2$ it decreases. For all cases, $W(1)=0$, consistent with detailed balance.

## 3. Geometric and Optimal Control Aspects in Hybrid Systems

In the realm of hybrid optimal control, submersive resets engender fundamentally new phenomena. A submersive reset map $\Delta$ causes the push-forward $\Delta_* : T_x S \to T_{\Delta(x)}M$ to be rank-deficient. The consequent adjoint (costate) jump conditions,
\[
p^+ \circ \Delta_* - p^- \in \operatorname{Ann}(T_x S), \quad H^+ - H^- = 0,
\]
may no longer admit unique solutions. Necessary geometric consistency conditions require that the pre-impact costate $p^-$ annihilates the fiber directions of the submersion, i.e., $p^- \in \operatorname{Ann}(\ker \Delta_*)$.

For reset events, the set of candidate post-reset costates $p^+$ forms an affine space of dimension $\operatorname{dim} M - \operatorname{rank}\Delta_* - 1$, 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 [2401.14476].

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 $f$ of oscillators at rate $\lambda$—can nonlocally induce global order [2402.14921].

The Ott-Antonsen reduction for the non-reset subpopulation leads to a mean-field equation with an additive reset term:
\[
\frac{dZ}{dt} = -(\sigma - i\omega_0 + \lambda)Z + \frac{K}{2}(Z - Z^*Z^2) + \lambda f,
\]
where $Z$ is the complex order parameter. For zero mean frequency ($\omega_0 = 0$), even infinitesimal resetting fraction ($f \to 0^+$) ensures a nonzero stationary synchrony ($R_{st}$), 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 $a$ in partial resetting display distinct physical regimes. Strong resetting ($a \to 0$) yields classical resetting behavior with minimal memory of prior states and leads to sharp, typically non-Gaussian NESS. Weak resetting ($a \to 1$) 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 [2401.11919].
- Complex solution structures in hybrid optimal control, requiring boundary-value and root-finding algorithms for admissible trajectory selection [2401.14476].
- Efficient control mechanisms for collective synchronization with minimal intervention in complex networks [2402.14921].

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)| $\dim M - \operatorname{rank}\Delta_* - 1$         | 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.

Source: https://www.emergentmind.com/topics/submersive-resets