Route to hyperbolic hyperchaos in a nonautonomous time-delay system

TL;DR

This paper investigates how a nonautonomous time-delay system can transition from non-hyperbolic to hyperbolic hyperchaos through parameter variations, revealing a detailed transition scenario involving intermittency and chaotic dynamics.

Contribution

It introduces a novel mechanism for inducing hyperbolic hyperchaos in a time-delay oscillator by tuning delay and excitation parameters, and analyzes the transition process.

Findings

Identification of a transition scenario from non-hyperbolic to hyperbolic hyperchaos.

Demonstration of hyperbolic hyperchaos emergence via phase doubling and delay tuning.

Analysis of intermittency and chaotic bursts during the transition.

Abstract

We consider a self-oscillator whose excitation parameter is varied. Frequency of the variation is much smaller then the natural frequency of the oscillator so that oscillations in the system are periodically excited and decay. Also a time delay as added such that when the oscillations start to grow at a new excitation stage they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to a nonlinearity the seeding from the past arrives with a doubled phase so that oscillation phase changes from stage to stage according to chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period we affect a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with often visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; (d) transformation of hyperchaos to hyperbolic form.