Simple driven chaotic oscillators with complex variables
Abstract
Despite a search, no chaotic driven complex-variable oscillators of the form z+f(z)=e(iOmegat) or z+f(z)=e(iOmegat) are found, where f is a polynomial with real coefficients. It is shown that, for analytic functions f(z), driven complex-variable oscillators of the form z+f(z)=e(iOmegat) cannot have chaotic solutions. Seven simple driven chaotic oscillators of the form z+f(z,z)=e(iOmegat) with polynomial f(z,z) are given. Their chaotic attractors are displayed, and Lyapunov spectra are calculated. Attractors for two of the cases have symmetry across the x=-y line. The systems' behavior with Omega as a control parameter in the range of Omega=0.1-2.0 is examined, revealing cases of period doubling, intermittency, chaotic transients, and period adding as routes to chaos. Numerous cases of coexisting attractors are also observed.
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