Yongli Song

Stability Switches and Hopf Bifurcations in a Pair of Delay-Coupled Oscillators

In this paper, we consider a pair of delay-coupled limit-cycle oscillators. Regarding the arithmetical average of two delays as a parameter, we investigate the effect of time delays on its dynamics. We show that there exist stability switches for time delays under certain conditions, which do not occur for the corresponding coupled system without time delays. A similar result has been reported for the same delay by Ramana Reddy et al. (Physica D, 129 [1999]), but in the present paper we give more detailed and specific conditions determining the amplitude death for different delays. On the other hand, we also investigate Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction. In the region where the stability switches may occur, we not only specifically determine the direction of Hopf bifurcations but also show that the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulation results are also given to support the theoretical predictions.

Spatiotemporal Dynamics of the Diffusive Mussel-Algae Model Near Turing-Hopf Bifurcation

Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reaction-diffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.

Stability and Bifurcation Analysis of Differential Equations and Its Applications

1Department of Mathematics, Tongji University, Shanghai 200092, China 2Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4 3Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China 4College of Sciences, University of Shanghai for Science and Technology, Shanghai 20093, China 5Department of Mathematics, Swinburne University of Technology, Melbourne, VIC 3122, Australia