Study of the passage from a dissipative to a conservative state in two-dimensional nonlinear systems of ordinary differential equations

Burov, D. A.; Golitsyn, D. L.; Ryabkov, O. I.

doi:10.1134/s0012266112030159

Cited by 3 publications

(4 citation statements)

References 2 publications

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“…In Refs. [12,13] the given approach has been applied and strictly proved by continuation along parameter of solutions from dissipative into conservative areas by means of the Magnitskii method of stabilization of unstable periodic orbits [1] at research bifurcations and chaos in the Duffing-Holmes equation…”

Section: Dynamical Chaos In Hamiltonian and Conservative Systemsmentioning

confidence: 99%

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Bifurcation Theory of Dynamical Chaos

Magnitskii¹

2018

Chaos Theory

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The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.

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Section: Dynamical Chaos In Hamiltonian and Conservative Systemsmentioning

confidence: 99%

“…Corresponding bifurcation diagrams in a plane (ε, μ) of existence of cycles of various periods down to a conservative case at μ = 0 are shown in [12][13][14]. Application of Magnitskii approach has revealed the essence of dynamical chaos in Hamiltonian and simply conservative systems.…”

Section: Dynamical Chaos In Hamiltonian and Conservative Systemsmentioning

confidence: 99%

Bifurcation Theory of Dynamical Chaos

Magnitskii¹

2018

Chaos Theory

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“…The developed methods have been tested on some model systems and in particular used to prove the existence of periodic trajectories in a system that describes the model of a space pendulum and was studied in detail in [5]. Thus, the suggested algorithm does not require extra computational and time expenditures and can be implemented on standard computers.…”

Section: Practical Use Of the Algorithmmentioning

confidence: 99%

Algorithm for the numerical proof of the existence of periodic trajectories in two-dimensional nonautonomous systems of ordinary differential equations

2013

Self Cite

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We suggest an algorithm that permits one to prove the existence of limit periodic trajectories (cycles) in two-dimensional nonautonomous dissipative systems with periodic coefficients with the use of computational methods alone. We prove a fixed point theorem for two-dimensional mappings and describe methods of its application to two-dimensional nonautonomous systems with the use of the Poincaré mapping and interval arithmetics.

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“…This scenario of transition to chaos takes place also in many partial differential equations and systems, such as Brusselator and Kuramoto-Tsuzuki (time dependent Ginzburg-Landau) equations, reaction-diffusion and FitzHugh-Nagumo type systems of equations, nonlinear Schrodinger and Kuramoto-Sivashinskii equations [1][2][3][12][13][14], and others. Moreover, this scenario also describes the laminar-turbulent transitions in any tasks for Navier-Stokes equations and transition to chaos in Hamiltonian and conservative systems, such as conservative Croquette and Duffing-Holmes equations, Mathieu-Magnitskii and Yang-Mills-Higgs Hamiltonian systems [2,3,[15][16][17][18][19][20][21][22], and others. The listed systems of equations describe a variety of complex natural, social, scientific and technical processes and phenomena in physics, chemistry, biology, economics, medicine and sociology, which emphasizes the universal applicability of the considered bifurcation approach.…”

Section: Introductionmentioning

confidence: 99%

Universal Bifurcation Chaos Theory and Its New Applications

Magnitskii

2023

Mathematics

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In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory. The obtained results once again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar–turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations and that chaos in the system can be confirmed only by detection of some main cycles or tori in accordance with the universal bifurcation diagram presented in the article.

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Study of the passage from a dissipative to a conservative state in two-dimensional nonlinear systems of ordinary differential equations

Cited by 3 publications

References 2 publications

Bifurcation Theory of Dynamical Chaos

Bifurcation Theory of Dynamical Chaos

Algorithm for the numerical proof of the existence of periodic trajectories in two-dimensional nonautonomous systems of ordinary differential equations

Universal Bifurcation Chaos Theory and Its New Applications

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