General Solution of the Rayleigh Equation for the Description of Bubble Oscillations Near a Wall

Abstract

Abstract. We consider a generalization of the Rayleigh equation for the description of the dynamics of a spherical gas bubble oscillating near an elastic or rigid wall. We show that in the non-dissipative case, i.e. neglecting the liquid viscosity and compressibility, it is possible to construct the general analytical solution of this equation. The corresponding general solution is expressed via the Weierstrass elliptic function. We analyze the dependence of this solution properties on the physical parameters.

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

“…After some simplifications, the dynamics of two coupled bubbles is described by a system of five ordinary differential equations. In this work we show that, in addition to regular and chaotic regimes which are typical for models of one bubble [6,[14][15][16][17][18], this system exhibits quasiperiodic and, what is more interesting, hyperchaotic types of motion. While there are many known examples of multi-dimensional nonlinear systems demonstrating quasiperiodic and chaotic dynamics with two or more positive Lyapunov exponents, to the best of our knowledge, neither quasiperiodic nor hyperchaotic oscillations of two coupled bubbles have been studied previously.…”

“…Despite the great interest, there are only few works devoted to studying of nonlinear dynamics of gas bubbles. For example, nonlinear dynamics of a single bubble described by one of the Rayleigh-Plesset-like models was studied in [6,[14][15][16][17][18], where it was shown that oscillations of a single bubble can be either regular or chaotic and routes to the corresponding attractors were studied. Some bifurcations of two and N coupled bubbles were studied in [9,13,16].…”

“…Another interesting property of the considered system is its multistability. It has recently been shown [17,18] that the dynamics of a single encapsulated bubble can be multistable, i.e. several attractors can co-exists at the same values of parameters.…”