论文标题
HOPF分叉点的优化
Optimization of Hopf bifurcation points
论文作者
论文摘要
我们引入了一种数值技术，用于控制动态系统中HOPF分叉的位置和稳定性。该算法包括解决一个由非线性部分微分方程的扩展系统限制的优化问题，该系统表征了HOPF分叉点。该方法的灵活性和鲁棒性使我们能够将HOPF分叉推向分叉参数的目标值，并相对于系统的参数或定义解决方案的域的形状来控制振荡频率。数值应用是在生物学和流体动力学引起的系统中介绍的，例如Fitzhugh-Nagumo模型，Ginzburg-Landau方程，Rayleigh-Bénard对流问题以及Navier-stokes等方程，在该方程式中，定期解决方案的位置控制和振荡频率很高。
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.
