Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents
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ORIGINAL PAPER
Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents Ivan R. Garashchuk · Alexey O. Kazakov Dmitry I. Sinelshchikov
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Received: 19 March 2020 / Accepted: 29 July 2020 © Springer Nature B.V. 2020
Abstract We study nonlinear dynamics in a system of two coupled oscillators, describing the motion of two interacting microbubble contrast agents. In the case of identical bubbles, the corresponding symmetry of the governing system of equations leads to the possibility of existence of asymptotically stable synchronous oscillations. However, it may be difficult to create absolutely identical bubbles and, moreover, it can be hard to observe in experiments regimes that are unstable with respect to perturbations of equilibrium radii of bubbles. Therefore, we investigate the stability of various synchronous and asynchronous dynamical regimes with respect to the breaking of this symmetry. We show that the main factors determining stability or instability of a synchronous attractor are the presence/absence and the type of an asynchronous attractor coexisting with the synchronous attractor. On the other hand, asynchronous hyperchaotic attractors are stable with respect to the symmetry breaking in all the situations we have studied. Therefore, they are likely to be observed in physically
I. R. Garashchuk · D. I. Sinelshchikov National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected] D. I. Sinelshchikov e-mail: [email protected] A. O. Kazakov (B) National Research University Higher School of Economics, Nizhny Novgorod, Russia e-mail: [email protected]
realistic scenarios and can be beneficial for suitable applications when chaotic behavior is desirable. Keywords Strange attractors · Multistability · Hyperchaos · Nonlinear dynamics · Ultrasound contrast agents
1 Introduction In this work we study a nonlinear dynamical system consisting of two coupled forced nonlinear oscillators. This system describes the dynamics of two interacting microbubble contrast agents under the influence of an external periodic force. Microbubble contrast agents are micrometer size gas bubbles that are encapsulated into a visco-elastic shell. They are currently used for enhancing blood flow visualization, see [1–3], and there are also several possible further biomedical applications like noninvasive therapy and targeted drug delivery, see [4,5]. It is known that nonlinear dynamics of microbubbles can be very complicated and depend on both control parameters and initial conditions (see, e.g., [6–11] and references therein). On the other hand, various types of bubbles dynamics can be both beneficial and undesirable depending on a particular application [3,9]. Thus, it is important to thoroughly study the whole variety of possible stable dynamical regimes of contrast agents, transitions between them, and their dependence on the control parameters and initial conditions.
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