DCDS
Extended gradient systems: Dimension one
Siniša Slijepčević
Discrete & Continuous Dynamical Systems - A 2000, 6(3): 503-518 doi: 10.3934/dcds.2000.6.503
We propose a general theorey of formally gradient differential equations on unbounded one-dimensional domains, based on an energy-flow inequality, and on the study of the induced semiflow on the space of probability measures on the phase space. We prove that the $\omega$-limit set of each point contains an equilibrium, and that the $\omega$-limit set of $\mu$-almost every point in the phase space consists of equilibria, where $\mu$ is any Borel probability measure invariant for spatial translation.
keywords: return times Gradient semiflows recurrent orbits Lyapunov function periodic orbits invariant measures.
DCDS
The Aubry-Mather theorem for driven generalized elastic chains
Siniša Slijepčević
Discrete & Continuous Dynamical Systems - A 2014, 34(7): 2983-3011 doi: 10.3934/dcds.2014.34.2983
We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We first show that the union of supports of all space-time invariant measures, denoted by $\mathcal{A}$, projects injectively to a dynamical system on a 2-dimensional cylinder. We also prove existence of space-time ergodic measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set $\mathcal{A}$ attracts almost surely (in probability) configurations with bounded spacing. In the DC case, $\mathcal{A}$ consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally invariant probability measures on the state space, which counts intersections.
keywords: twist maps synchronization reaction-diffusion equation uniformly sliding states Poincaré-Bendixson theorem Aubry-Mather theory Frenkel-Kontorova model space-time invariant measure attractors minimizing measures space-time entropy.
DCDS
Stability of invariant measures
Siniša Slijepčević
Discrete & Continuous Dynamical Systems - A 2009, 24(4): 1345-1363 doi: 10.3934/dcds.2009.24.1345
We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.
keywords: Invariant measure exponential stability optimal transport problem Liapunov stability attractors. weak* topology

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