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Chaotic period-doubling and OGY control for the forced Duffing equation. (English) Zbl 1253.34048

The authors study the Duffing equation modified by an additional piecewise constant term \(\nu(t,t_0,\mu)\), where \(t_0\) is the initial time and \(\mu\) is a real parameter. The term \(\nu(t,t_0,\mu)\) takes two values, and the sequence of switching times is determined as a solution of the logistic equation with coefficient \(\mu\). Note that the dynamics of the composite system is of hybrid type. The authors consider also the \(n\)-dimensional analogue of such a system.
They prove that these systems exhibit a chaotic behavior through a period-doubling cascade. They also use the so-called OGY method to stabilize a periodic orbit of the system, by constructing a sequence of perturbed values of the parameter around the critical value.
Numerical examples and simulations are given.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34H10 Chaos control for problems involving ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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