Related papers: The Gentlest Ascent Dynamics
The characterization of the "most connected" nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social, biological, and technological networked systems, including their robustness…
The escaping set of an entire function consists of the points in the complex plane that tend to infinity under iteration. This set plays a central role in the dynamics of transcendental entire functions. The goal of this survey is to…
This paper considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem and makes the feasible set forward invariant and asymptotically stable. The invariance of the feasible…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…
Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope --- an influential notion of steepest descent in a nonsmooth setting. We moreover show that for…
Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which \textit{unstable…
Incremental stability of dynamical systems ensures the convergence of trajectories from different initial conditions towards each other rather than a fixed trajectory or equilibrium point. Here, we introduce and characterize a novel class…
A game theory inspired methodology is proposed for finding a function's saddle points. While explicit descent methods are known to have severe convergence issues, implicit methods are natural in an adversarial setting, as they take the…
In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins,…
A general system of particles (of one or several species) on a one dimensional lattice with boundaries is considered. Two general behaviors of such systems are investigated. The stationary behavior of the system, and the dominant way of the…
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are…
In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f (x) = (a x + b)/(c x + d), where a, b, c and d are rational numbers. In particular, we investigate behavior of…
In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction. Isostables are boundaries of specific forward-invariant sets defined by the so-called Koopman operator, which…
Some dynamical properties of non interacting particles in a bouncer model are described. They move under gravity experiencing collisions with a moving platform. The evolution to steady state is described in two cases for dissipative…
We study deterministic escape dynamics in the framework of the discrete Klein-Gordon modelwith a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
We study a pursuit-evasion problem which can be viewed as an extension of the keep-away game. In the game, pursuer(s) will attempt to intersect or catch the evader, while the evader can visit a fixed set of locations, which we denote as the…
These lecture notes can be read in two ways. The first two Sections contain a review of the phenomenology of several physical systems with slow nonequilibrium dynamics. In the Conclusions we summarize the scenario derived from the solution…
We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the…