Stable invariant manifolds with impulses and growth rates

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

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Stable invariant manifolds of random diffeomorphisms

Smooth Ergodic Theory of Random Dynamical Systems - Lecture Notes in Mathematics ◽

10.1007/bfb0094312 ◽

1995 ◽

pp. 55-90

Author(s):

Pei-Dong Liu ◽

Min Qian

Keyword(s):

Invariant Manifolds ◽

Stable Invariant Manifolds ◽

Stable Invariant

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ANALYTICITY OF STABLE INVARIANT MANIFOLDS FOR GINZBURG-LANDAU EQUATION

Applied Analysis and Differential Equations ◽

10.1142/9789812708229_0009 ◽

2007 ◽

Author(s):

A.V. FURSIKOV

Keyword(s):

Invariant Manifolds ◽

Landau Equation ◽

Ginzburg Landau ◽

Ginzburg Landau Equation ◽

Stable Invariant Manifolds ◽

Stable Invariant

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Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under(μ,ν)-Dichotomies

Journal of Mathematics ◽

10.1155/2014/989526 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Lijun Pan

Keyword(s):

Differential Equations ◽

Nonlinear Equation ◽

Delay Differential Equations ◽

Invariant Manifolds ◽

Parameter Dependence ◽

Nonlinear Perturbations ◽

Linear Delay ◽

Stable Invariant Manifolds ◽

Delay Differential ◽

Stable Invariant

We obtain the existence of stable invariant manifolds for the nonlinear equationx′=L(t)xt+f(t,xt,λ)provided that the linear delay equationx′=L(t)xtadmits a nonuniform(μ,ν)-dichotomy andfis a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameterλ. Namely, the stable invariant manifolds are Lipschitz in the parameterλ. In addition, we also show that nonuniform(μ,ν)-contraction persists under sufficiently small nonlinear perturbations.

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Stable manifolds for perturbations of exponential dichotomies in mean

Stochastics and Dynamics ◽

10.1142/s0219493718500223 ◽

2018 ◽

Vol 18 (03) ◽

pp. 1850022 ◽

Cited By ~ 1

Author(s):

Luis Barreira ◽

Claudia Valls

Keyword(s):

Nonlinear Dynamics ◽

Banach Space ◽

Exponential Dichotomy ◽

Invariant Manifolds ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Exponential Behavior ◽

Exponential Dichotomies ◽

Stable Invariant Manifolds ◽

Stable Invariant

We establish the existence of stable invariant manifolds for any sufficiently small perturbation of a cocycle with an exponential dichotomy in mean. The latter notion corresponds to replace the exponential behavior in the classical notion of an exponential dichotomy by an exponential behavior in average with respect to an invariant measure. We consider both perturbations of a cocycle over a map and over a flow that can be defined on an arbitrary Banach space. Moreover, we obtain an upper bound for the speed of the nonlinear dynamics along the stable manifold as well as a lower bound when the exponential dichotomy in mean is strong (this means that we have lower and upper bounds along the stable and unstable directions of the dichotomy).

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