On the response of autonomous sweeping processes to periodic perturbations

If $x_0$ is an equilibrium of an autonomous differential equation $\dot x=f(x)$ and $\det \|f'(x_0)\|\not=0$, then $x_0$ persists under autonomous perturbations and $x_0$ transforms into a $T$-periodic solution under non-autonomous $T$-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form $-\dot u\in N_B(u)+f_0(u),$ $u\in\mathbb{R}^2,$ i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set $B$ of the reduced system is a non-moving unit ball of $\mathbb{R}^2.$ We show that an equilibrium $\|u_0\|=1$ persists under periodic perturbations, if the projection $\overline{f}:\partial B\to\mathbb{R}^2$ of $f_0$ on the tangent to the boundary $\partial B$ is nonsingular at $u_0$

Second-order differential equations with random perturbations and small parameters

International audienceWe consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero

Continuous time crystal in an electron-nuclear spin system: stability and melting of periodic auto-oscillations

Crystals spontaneously break the continuous translation symmetry in space, despite the invariance of the underlying energy function. This has triggered suggestions of time crystals analogously lifting translational invariance in time. Originally suggested for closed thermodynamic systems in equilibrium, no-go theorems prevent the existence of time crystals. Proposals for open systems out of equilibrium led to the observation of discrete time crystals subject to external periodic driving to which they respond with a sub-harmonic response. A continuous time crystal is an autonomous system that develops periodic auto-oscillations when exposed to a continuous, time-independent driving, as recently demonstrated for the density in an atomic Bose-Einstein condensate with a crystal lifetime of a few ms. Here we demonstrate an ultra-robust continuous time crystal in the nonlinear electron-nuclear spin system of a tailored semiconductor with a coherence time exceeding hours. Varying the experimental parameters reveals huge stability ranges of this time crystal, but allows one also to enter chaotic regimes, where aperiodic behavior appears corresponding to melting of the crystal. This novel phase of matter opens the possibility to study systems with nonlinear interactions in an unprecedented way.Comment: 12 figures, 17 page