Hopf 分岔点的优化
Optimization of Hopf bifurcation points
AI总结：
本文提出一种基于扩展系统约束优化的数值方法，用于控制动力系统中 Hopf 分岔的位置、稳定性及振荡频率，并在生物与流体动力学模型中验证其有效性。
摘要：
我们引入一种数值技术，用于控制动力系统中 Hopf 分岔的位置和稳定性性质。该算法包括求解一个由刻画 Hopf 分岔点的非线性偏微分方程扩展系统所约束的优化问题。该方法的灵活性和鲁棒性使我们能够将 Hopf 分岔提前或推迟到分岔参数的目标值，并且能够相对于系统参数或解所定义区域的形状来控制振荡频率。数值应用在来自生物学和流体动力学的系统中给出，例如 FitzHugh--Nagumo 模型、Ginzburg--Landau 方程、Rayleigh--Bénard 对流问题和 Navier--Stokes 方程，在这些系统中周期解的位置和振荡频率的控制具有很高的研究价值。
英文摘要：
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--Bénard convection problem, and Navier--Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high interest.
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