Novel bifurcation structure generated in piecewise-linear three-LCLC resonant circuit and its Lyapunov analysis

We analyse a piecewise-linear oscillator that consists of a three-LCLC resonant circuit with a hysteresis element. Three sets of two-dimensional linear equations, including a hysteresis function, represent the governing equations of the circuit, and all the Lyapunov exponents are calculated in a remarkably simple manner based on derived explicit solutions. Various dynamical phenomena such as two-torus, three-torus, and hyperchaos with four positive Lyapunov exponents are observed by Lyapunov analysis. We obtained a detailed bifurcation diagram in which novel bifurcation structure which we call a “two-torus Arnold tongue” is observed where two-torus generating regions exist in a three-torus generating region as if periodic states exist in a two-torus generating region.

► We investigate a three-LC resonant circuit with a hysteresis element.
► We calculate all Lyapunov exponents, and obtain a detailed bifurcation diagram.
► We discover novel bifurcation structure we call a “Two-torus Arnold tongue”—where two-torus generating regions exist in a three-torus generating region.
► Hyperchaotic attractor with four positive Lyapunov exponents are observed.