Completely integrable dynamical systems of Hopf–Langford type

•Nonlinear dynamical systems.•Generalized Hopf–Langford system.•Nonlinear Duffing equation.•Complete integrability.•Exact analytical solutions. In this work we consider a three-dimensional autonomous system of nonlinear ordinary differential equations, which may be thought of as a generalization of...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2021-01, Vol.92, p.105464, Article 105464
Main Authors: Nikolov, Svetoslav G., Vassilev, Vassil M.
Format: Article
Language:English
Subjects:
Complete integrability
Differential equations
Duffing oscillators
Dynamical systems
Elliptic functions
Exact analytical solutions
Generalized Hopf–Langford system
Mathematical analysis
Nonlinear differential equations
Nonlinear Duffing equation
Nonlinear dynamical systems
Nonlinear equations
Oscillators
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Summary:•Nonlinear dynamical systems.•Generalized Hopf–Langford system.•Nonlinear Duffing equation.•Complete integrability.•Exact analytical solutions. In this work we consider a three-dimensional autonomous system of nonlinear ordinary differential equations, which may be thought of as a generalization of the well-known Hopf–Langford system introduced about forty years ago. This dynamical system turned out to be equivalent to the nonlinear force-free Duffing oscillator. In three special cases, it is found to be completely integrable. To the best of our knowledge, these facts have not been noticed so far in the rich literature on the subject. In the aforementioned three special cases, the general solutions of the respective systems are expressed in explicit analytical form by means of elementary and Jacobi elliptic functions depending on the values of the system parameters. This allowed us to characterize in details the dynamics of the regarded systems.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2020.105464