论文标题
耦合Landau-Stuart振荡器中分布延迟的分叉和振幅死亡和混乱的参数强制强制范德尔·波利利系统
Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System
论文作者
论文摘要
著名的耦合Landau-Stuart系统中引入了由“弱通用内核”建模的分布式延迟，以及带有参数强迫的混乱范围Pol-rayleigh系统。系统通过“线性链技巧”接近。研究了启动振荡的HOPF分叉的系统和条件的线性稳定性分析，包括在分叉时得出正常形式，并推论所得的极限周期吸引子的稳定性。 HOPF分叉的延迟参数的值$ a = a_ {hopf} $在所有三个系统中都挑出振幅死亡（AD）的发作，其振荡在较大的值（对应于较弱的延迟）时。在Landau-Stuart系统中，HOPF生成的极限周期$ a> a_ {hopf} $在超出HOPF分叉点以外的所有其他系统参数的情况下非常稳定，并且不会进一步进行对称性，柔性，柔性，柔软，柔性，柔性，转换，临界，批评，neimark或neimark bifurcations bifurcations bifurcercationsercercation。这是可以预期的，因为相应的未延迟系统是其各自参数的宽范围内的强大振荡器。数值模拟揭示了相位空间中极限循环的强烈失真和旋转，因为参数被推到了HOPF后策略中，并揭示了其他特征，例如，随着极限循环吸引子的延迟和其他参数的变化，物理变量的振荡振幅和物理变量的振荡幅度和时间段如何变化。对于混乱的系统，非常强大的延迟仍可能导致振荡的停止和AD的发作（即使是针对相对较大的系统强迫值，这往往会反对这种现象）。另一个重要的系统参数（参数激发）的变化导致了一系列动态行为，分叉从一个制度（或吸引子类型）引入下一个策略（或类型的吸引子类型）。
Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for $a>a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the corresponding undelayed systems are robust oscillators over wide ranges of their respective parameters. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD (even for relatively large values of the system forcing which tends to oppose this phenomenon). Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.
