Hyperchaos and Multistability in Nonlinear Dynamics of Two Interacting Microbubble Contrast Agents

Abstract

We study nonlinear dynamics of two coupled contrast agents that are micro-meter size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system various types of complex dynamics can occur, namely, we observe periodic, quasi-periodic, chaotic and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either Feigenbaum's cascade of period doubling bifurcations or Afraimovich--Shilnikov scenario of torus destruction. For the onset of hyperchaotic attractor we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the bubbles' dynamics can be multistable, i.e. various combinations of co-existence of the above mentioned attractors are possible. These cases include co-existence of hyperchaotic regime with any of the other remaining types of dynamics for different parameter values. Thus, the model of two coupled gas bubbles provide a new examples of physically relevant system with multistable hyperchaos.