一类耦合非线性振子同步混沌Hopf分岔及其电路仿真

Hopf bifurcation from synchronous chaos and its circuit simulation in a coupled nonlinear oscillator system

对于一类同时存在扩散耦合和梯度耦合的非线性振子系统 ,通过空间傅里叶变换 ,得到具有不同波矢的各运动模式的相互独立的运动方程 .计算各横截模的Lyapunov指数 ,可在耦合参数平面上确定同步混沌的稳定区域 .在稳定区域边界 ,一对共轭横截模式失稳 ,导致同步混沌的Hopf分岔 .对耦合Lorenz振子系统进行了数值模拟 ,并设计了耦合Lorenz振子系统的电路 ,进行耦合振子系统同步混沌Hopf分岔的电路仿真实验 .计算和仿真的结果表明 ,Hopf分岔的特征频率等于失稳横截模式的振荡频率 .

For a coupled nonlinear oscillator system with diffusion and gradient couplings, spatial Fourier transformation is performed and the dynamic equations of various space modes are derived. By calculating the Lyapunov exponents of the transverse modes, one can determine the stable region of the synchronous chaos on the plane of coupling parameters. On the boundary of the stable region, a couple of conjugate transverse modes destabilize, and a Hopf bifurcation takes place. Numerical simulations are carried out for the coupled Lorenz oscillator system. An electronic circuit is designed for simulating the bifurcation in the system. Results from the simulations show that the frequency created by the Hopf bifurcation is equal to the oscillation frequency of the destabilized transverse modes.