Optimization of Hopf Bifurcation Points
Optimization of Hopf Bifurcation Points
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DOI：
10.1137/22m1474448
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
N. Boull'e;P. Farrell;M. Rognes
中科院分区：
文献类型：
--
作者：
N. Boull'e;P. Farrell;M. Rognes
介绍了一种控制动力系统Hopf分岔位置和稳定性的数值方法。该算法包括求解一个由具有Hopf分岔点的扩展非线性偏微分方程组约束的优化问题。该方法的灵活性和鲁棒性使我们能够将Hopf分岔提前或延迟到分岔参数的目标值，以及控制相对于系统参数的振荡频率或定义解的域的形状。数值应用在生物和流体动力学系统中，如FitzHugh- Nagumo模型，Ginzburg- Landau方程，Rayleigh- B 'enard对流问题和Navier- Stokes方程，其中周期解的位置和振荡频率的控制是非常有趣的。
We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear partial differential equations that characterizes Hopf bifurcation points. The flexibility and robustness of the method allows us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency with respect to a parameter of the system or the shape of the domain on which solutions are defined. Numerical applications are presented in systems arising from biology and fluid dynamics, such as the FitzHugh--Nagumo model, Ginzburg--Landau equation, Rayleigh--B\'enard convection problem, and Navier--Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high interest.