Algorithm for localizing Chua attractors based on the harmonic linearization method
Consider systems of the form \[ {dx\over dt}= Px+ \psi(x),\tag{1} \] where \(P\) is a constant \(n\times n\)-matrix \(\psi\), is a continuous vector function with \(\psi(0)= 0\). In order to find a periodic solution close to a harmonic oscillation and to obtain the attractor of system (1), the authors introduce a matrix \(K\) and a finite sequence of continuous functions \(\varphi^j\) \((j= 0,1,\dots, m)\) such that the matrix \(P_0= P+ K\) has a pair of purely imaginary eigenvalues \(\pm\omega_0\) \((\omega_0> 0)\), where the other eigenvalues of \(P_0\) have negative real parts, and the graphs of two neighboring functions of \(\{\varphi^j(x)\}\) differ little in a certain sense, \(\varphi^m(x):= \psi(x)- Kx= \varphi(x)\), and \(|\varphi^0(x)|\) is small. Then system (1) can be rewritten as \[ {dx\over dt}= P_0x+ \varphi(x).\tag{2} \] The authors introduce the systems \[ {dx\over dt}= P_0 x+\varphi^j(x),\quad j= 0,1,2,\dots,m,\tag{3} \] and use these systems to search for an attractor of system (2). The corresponding procedure is applied to localize an attractor of the Chua system with a scalar nonlinearity.
scientific article; zbMATH DE number 5809148