Center Manifold Reduction of the Hopf-hopf Bifurcation in a Time Delay System

In this work, a differential delay equation (DDE) with a cubic nonlinearity is analyzed as two parameters are varied by means of a center manifold reduction. This reduction is applied directly to the case where the system undergoes a Hopf-Hopf bifurcation. This procedure replaces the original DDE with four first-order ODEs, an approximation valid in the neighborhood of the Hopf-Hopf bifurcation. Analysis of the resulting ODEs shows that two separate periodic motions (limit cycles) and an additional quasiperiodic motion are born out of the Hopf-Hopf bifurcation. The analytical results are shown to agree with numerical results obtained by applying the continuation software package DDE-BIFTOOL to the original DDE. This system has analogues in coupled microbubble oscillators. Introduction Delay in dynamical systems is exhibited whenever the system’s behavior is dependent at least in part on its history. Many technological and biological systems are known to exhibit such behavior ; coupled laser systems, high-speed milling, population dynamics and gene expression are some examples of delayed systems. This work analyzes a simple differential delay equation that is motivated by a system of two microbubbles coupled by acoustic forcing, previously studied by Heckman et al. [1] [2]. The propagation time of sound in the fluid gives rise to a time delay between the two bubbles. The system under study has the same linearization as the equations previously studied, but the sophisticated nonlinear interaction terms in the bubble equations have been replaced by a cubic term in order to provide first insights into the full bubble equations. In particular, the system of coupled microbubbles has been witnessed to exhibit damped oscillation, excited oscillations (i.e. a stable limit cycle created as a result of a supercritical Hopf bifurcation), and quasiperiodic oscillations. These latter dynamics are unexplained by previous work, but it has been previously suggested that these dynamics are the result of a Neimark-Sacker bifurcation. This work explores this possibility by analyzing the dynamics of an analogous system by means of a center manifold reduction. However, in contrast with previous work, this reduction will analyze the Hopf-Hopf bifurcation that results when two parameters (corresponding to the speed of sound in the fluid and the delay propagation time) are varied. 1. Center Manifold Reduction The system under analysis is motivated by the Rayleigh-Plesset Equation with Delay Coupling (RPE), as studied by Heckman et al. [1] [2]. The equation of motion for a spherical bubble contains quadratic nonlinearities 1 Field of Theoretical & Applied Mechanics ; Cornell University ; Ithaca, NY USA 2 Department of Applied Mathematics ; University of Washington ; Seattle, WA USA 3 Department of Mathematics and Department of Mechanical & Aerospace Engineering ; Cornell University ; Ithaca, NY USA c © EDP Sciences, SMAI 2013 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201339008 58 ESAIM: PROCEEDINGS and multiple parameters quantifying the fluids’ mechanical properties ; the equations studied in this work are designed to capture salient dynamical properties while simplifying analysis. The system is : κẍ+ 4ẋ+ 4κx+ 10ẋ(t− T ) = x. (1) Eq. (1) has the same linearization as the RPE, with a cubic nonlinear term added to it. This system has an equilibrium point at x = 0 that will correspond to the local behavior of the RPE’s equilibrium point as a result. In order to put Eq. (1) into a form amenable to treatment by functional analysis, we draw on the method used by Kalmár-Nagy et al. [3] and Rand [4], [5]. The operator differential equation for this system will now be developed. Eq. (1) may be written in the form : ẋ(t) = L(κ)x(t) + R(κ)x(t− τ) + f(x(t),x(t− τ), κ) where x(t) = ( x(t) ẋ(t) ) = ( x1 x2 ) L(κ) = ( 0 1 −4 −4/κ ) , R(κ) = ( 0 0 0 −10/κ )