Convergence of discretized attractors for parabolic equations on the line.
Decay of geometry for Fibonacci critical covering maps of the circle
Dependence of hidden attractors on non-linearity and Hamilton energy in a class of chaotic system
Non-linearity is essential for occurrence of chaos in dynamical system. The size of phase space and formation of attractors are much dependent on the setting of nonlinear function and parameters. In this paper, a three-variable dynamical system is controlled by different nonlinear function thus a class of chaotic system is presented, the Hamilton function is calculated to find the statistical dynamical property of the improved dynamical systems composed of hidden attractors. The standard dynamical...
Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation
We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup does exist for any bounded ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.
Differentiable semiflows for differential equations with state-dependent delays.
Dissipative dynamics of reaction diffusion equations in
Dynamical properties for a relaxation scheme applied to a weakly damped non local nonlinear Schrödinger equation.
Dynamics of a Lotka-Volterra map
Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior...
Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions
We prove the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time, more rapidly for larger acoustic speeds and smaller...
Dynamics of stochastic approximation algorithms
Entropic approximation in kinetic theory
Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys. 83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global...
Entropic approximation in kinetic theory
Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular...
Entropy of scalar reaction-diffusion equations
We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms....
Ergodicity for the stochastic complex Ginzburg–Landau equations
Existence and approximation results for gradient flows
This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space
Existence of doubly-weighted pseudo almost periodic solutions to non-autonomous differential equations.
Existence of weak solutions to doubly degenerate diffusion equations
We prove existence of weak solutions to doubly degenerate diffusion equations by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains with Dirichlet or Neumann boundary conditions. The function can be an inhomogeneity or a nonlinearity involving terms of the form or . In the appendix, an introduction to weak differentiability...
Exotic galilean symmetry and non-commutative mechanics.
Exponential attractor for a first-order dissipative lattice dynamical system.
Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls.