Parameter Switching

The Parameter Switching (PS) algorithm allows to approximate numerical attractors of chaotic dynamical systems

The PS algorithm allows to approximate numerical attractors of chaotic dynamical systems depending on a single real control parameter $p\in R$, such as the Lorenz system, Rossler system, Chen system, Lotka-Volterras ystem, Rabinovich-Fabrikant system, Hindmarsh-Rose system, Lu system, classes of minimal networks and many others, which are modeled by the following Initial Value Problem (IVP):

\begin{equation}

\dot{x(t)}=f(x(t))+pAx(t), x(0)=x_0,

\end{equation}

where $t\in[0,T]$, $T>0$, $x_0\in \mathbb{R}^n$, $A\in \mathbb{R}^{n\times n} is a constant matrix, and $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a continuous nonlinear function.

The code can be made via some convergent explicit fixed step-size $h$ numerical scheme, here the standard RK numerical scheme.

If every $h$ one switches $p$ within a chosen set of values, the obtained "switched" attractor $A^*$ approximates the "averaged" attractor $A^0$ obtained for $p$ replaced with the average value of the switched values.

Details on applications and algorithm convergence can be found on e.g.:

[1] Marius-F. Danca, Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo's paradox, Communications in Nonlinear Science and Numerical Simulation, 18(3), 500–510 (2013).

[2] Marius-F. Danca, Michal Feckan, Nikolay Kuznetsov, Guanrong Chen, Attractor as a convex combination of a set of attractors, Communications in Nonlinear Science and Numerical Simulation, 96, 105721 (2021)

[3] Marius-F. Danca, Random parameter-switching synthesis of a class of hyperbolic attractors, CHAOS, 18, 033111 (2008)