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The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered. In this case, the Poincaré map is constructed as a one-dimensional map: a circle map. The parameter boundary between a torus-generating region where the circle map is a diffeomorphism and a chaos-generating region where the circle map has extrema is derived explicitly, without solving the implicit equations, by adopting some novel ideas. On the bifurcation diagram, intermittency and a saddle-node bifurcation from the periodic state to the quasi-periodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified. [ABSTRACT FROM AUTHOR] more...

Abstract: Two coexisting stable canards are discussed in a periodically driven singularly perturbed van der Pol equation, where the amplitude of the driving force is extremely small. Canard breakdown into chaos via period-doubling bifurcations is also observed. [Copyright &y& Elsevier] more...

Abstract: The authors discuss the “embryology” of successive torus doubling via the bifurcation theory, and assert that the coupled map of a logistic map and a circle map has a structure capable of generating infinite number of torus doublings. [Copyright &y& Elsevier] more...

The response of a circuit which generates the duck solution is studied, where an extremely small periodic perturbation is applied. This perturbation breaks down the duck solution eventually, and then chaos occurs afterwards. To analyze this phenomenon in detail, the piecewise linear technique combined with the degenerate technique is employed. Through this technique, a one-dimensional Poincaré map is derived rigorously, hence chaos generation mechanism is well understood. The bifurcation diagrams are obtained for the circuit, and the parameter boundary between torus generating region and chaos generating region is derived exactly. From these bifurcation diagrams, periodic state breaks down when the forcing term is extremely small. [Copyright &y& Elsevier] more...