Bifurcations ``cycle-chaos-hyperchaos'' in some nonideal electroelastic systems (Q2094218)

This paper describes how deterministic chaos can evolve in a dynamical system consisting of a piezoceramic transducer and an LC generator. This represents an oscillatory system with a source of excitation and an oscillatory load. Such systems are called nonideal when the power of the oscillation source is of the same order as the power consumed by the oscillatory load. Unexpected steady-state behavior can arise in systems like this because of the interaction between the source of oscillation excitation and the oscillatory subsystem. This can lead to the emergence of deterministic chaos. The authors' goal here is to study bifurcations of the transition to chaos in lower power systems of the sort they describe. A mathematical model of the transduce-generator system was derived in [\textit{T. S. Krasnopolskaya} and \textit{A. Yu. Shvets}, ``Chaos in vibrating systems with a limited power-supply'', Chaos 3, No. 3, 387--395 (1993; \url{doi:10.1063/1.165946}); Nonlinear Dyn. Syst. Theory 6, No. 4, 367--387 (2006; Zbl 1123.37308)]. Their model consists of the system: \begin{align*} \ddot \varphi + \omega_0^2 \dot\varphi &= a_1 \dot\varphi + a_2\dot\varphi^2- a_3 \dot\varphi ^3 - a_4 V,\\ \ddot V + \omega_1^2 V &= a_5 \varphi + a_6\dot \varphi - a_7 \dot V, \end{align*} where \(V(t)\) is the voltage in the electrodes of the transducer as a function of time, \(\varphi(t) = \int _{0}^{t} (e_g - E_g) dx\), \(e_g\) is the tube grid voltage, and \(E_g\) is the constant component of voltage \(e_g\). Having reduced these model equations to normal form with dimensionless variables, the authors proceed to find numerical solutions using various selections of system parameters. The authors discovered several interesting behaviors, including hyperchaos (chaos with at least two positive Lyapunov exponents). They claim to have been the first to discover generalized intermittency in a transducer-generator system. ``Generalized intermittency'' refers to transitions between different regimes of chaotic behavior, something that goes beyond ordinary intermittency that refers to a transition between regular and chaotic behavior. For the entire collection see [Zbl 1485.37001].