Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agents

TL;DR

This paper explores complex nonlinear dynamics, including hyperchaotic and strange attractors, in a model of three interacting microbubbles used as ultrasound contrast agents, revealing new bifurcation mechanisms.

Contribution

It introduces the first observation of hyperchaotic attractors with an additional zero Lyapunov exponent in bubble oscillation models and identifies mechanisms for their emergence.

Findings

Discovery of hyperchaotic attractors with three positive Lyapunov exponents.

Identification of bifurcation scenarios leading to complex attractors.

Association of attractor types with specific bifurcation mechanisms.

Abstract

We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyaponov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.