Tassilo Küpper

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

AbstractIn this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.

Qualitative bifurcation analysis of a non-smooth friction-oscillator model

Abstract.We numerically study the bifurcation structure (in dependence of the amplitude and frequency of the forcing) of a non-smooth friction-oscillator system with dry friction.

Topological entropy for divergence points

Let X be a compact metric space, f a continuous transformation on X , and Y a vector space with linear compatible metric. Denote by M ( X ) the collection of all the probability measures on X . For a positive integer n , define the n th empirical measure

Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation

L_{n}:X\mapsto M(X)

Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials

as \[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \] where

Existence and Bifurcation Theorems for Nonlinear Elliptic Eigenvalue Problems on Unbounded Domains

\delta_{x}

Non-Smooth Dynamical Systems: An Overview

denotes the Dirac measure at x . Suppose

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

\Xi:M(X)\mapsto Y

Invariant cones for non-smooth dynamical systems

is continuous and affine with respect to the weak topology on M ( X ). We think of the composite \[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \] as a continuous and affine deformation of the empirical measure L n . The set of divergence points of such a deformation is defined as \[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \] In this paper we show that for a continuous transformation satisfying the specification property, if

The lowest point of the continuous spectrum as a bifurcation point

\Xi(M(X))