The perturbation of attractors of skew-product flows with a shadowing driving system
P.E. Kloeden Victor S. Kozyakin
Discrete & Continuous Dynamical Systems - A 2001, 7(4): 883-893 doi: 10.3934/dcds.2001.7.883
The influence of the driving system on a skew-product flow generated by a triangular system of differential equations can be perturbed in two ways, directly by perturbing the vector field of the driving system component itself or indirectly by perturbing its input variable in the vector field of the coupled component. The effect of such perturbations on a nonautonomous attractor of the driven component is investigated here. In particular, it is shown that a perturbed nonautonomous attractor with nearby components exists in the indirect case if the driven system has an inflated nonautonomous attractor and that the direct case can be reduced to this case if the driving system is shadowing.
keywords: perturbations attractors Skew product flow shadowing.
Uniform nonautonomous attractors under discretization
P.E. Kloeden Victor S. Kozyakin
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 423-433 doi: 10.3934/dcds.2004.10.423
A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets of which converge upper semi continuously to their continuous time counterparts as the maximum time step decreases to zero. The proof involves a Lyapunov function characterizing the uniform pullback attractor of the original system.
keywords: perturbations discretization. Cocycle dynamical systems attractors
Arnold tongues for bifurcation from infinity
Victor S. Kozyakin Alexander M. Krasnosel’skii Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 107-116 doi: 10.3934/dcdss.2008.1.107
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$. The map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude $n$-periodic trajectories exist for a fixed $n$. In the related problems on small periodic orbits near zero, similarly defined parameter sets, known as Arnold tongues, are more narrow.
keywords: Arnold tongue positively homogeneous nonlinearity discrete time system bifurcation at infinity Poincare map. Periodic trajectory saturation

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