On the theory of one-sided models in spaces with arbitrary cones

The paper presents a way of constructing quasimonotone nonautonomous systems ensuring x-stability of the nonautonomous system. There are described extensions quasimonotone with respect to an arbitrary cone, Perron condition and invariant surface stability under perturbations U-stability on the set of non wandering points is proved to imply u-stability of quasimonotone nonlinear system and exponential u-stability on minimal attraction center provides u-stability of the total systems Examples are available.

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Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

International Journal of Differential Equations ◽

10.1155/2013/740980 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Yūki Naito ◽

Mervan Pašić

Keyword(s):

Differential Equations ◽

Asymptotic Behaviour ◽

Zero Solution ◽

Nonautonomous System ◽

Linear Differential Equations ◽

Solution Curve ◽

Nonautonomous Systems ◽

Polynomial Coefficients ◽

The Matrix ◽

Linear Differential

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at .

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Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

Abstract and Applied Analysis ◽

10.1155/2008/756934 ◽

2008 ◽

Vol 2008 ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Francesca Faraci ◽

Antonio Iannizzotto

Keyword(s):

Nonlinear System ◽

Linear System ◽

Boundary Conditions ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

Accumulation Point ◽

Second Order ◽

Nontrivial Solutions ◽

Nonautonomous Systems ◽

Second Order Hamiltonian Systems

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a functionu, and prove that the set of bifurcation points for the solutions of the system is notσ-compact. Then, we deal with a linear system depending on a real parameterλ>0and on a functionu, and prove that there existsλ∗such that the set of the functionsu, such that the system admits nontrivial solutions, contains an accumulation point.

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Dynamics of Weakly Mixing Nonautonomous Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501232 ◽

2019 ◽

Vol 29 (09) ◽

pp. 1950123 ◽

Cited By ~ 3

Author(s):

Mohammad Salman ◽

Ruchi Das

Keyword(s):

Dynamical System ◽

Sufficient Conditions ◽

Unit Interval ◽

Nonautonomous System ◽

Probability Measures ◽

Topological Transitivity ◽

Weak Mixing ◽

Nonautonomous Systems ◽

Nonautonomous Dynamical System ◽

Weakly Mixing

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

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Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy

Abstract and Applied Analysis ◽

10.1155/2014/725098 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Yong-Hui Xia ◽

Xiaoqing Yuan ◽

Kit Ian Kou ◽

Patricia J. Y. Wong

Keyword(s):

Existence And Uniqueness ◽

Exponential Dichotomy ◽

Nonlinear Term ◽

Nonautonomous System ◽

Uniqueness Of Solution ◽

Suitable Assumption ◽

Essential Condition ◽

Prove Theorem ◽

Nonautonomous Systems ◽

Nonuniform Exponential Dichotomy

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies|f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying|f(t,x)|≤μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be|f(t,x)|≤μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption|f(t,x)|≤μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.

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