The Use of Time Varying Eigenvalues to Investigate Nonlinear System Behaviour

This paper uses nonlinear eigenvalues to examine the behaviour of three well studied nonlinear systems that exhibit chaos. The nonlinear eigenvalue analysis method has been widely used in System Dynamics to study the relative dominance of feedback loops. The method used here is to compute nonlinear eigenvalues using a Taylor expansion about the equilibrium solutions with a Hessian matrix expansion to obtain nonlinear eigenvalues with state variables in the algebraic solution and then substitute state values computed via a Simulink model. Examination of limit cycle and chaos are made for forced 2D systems such as the Duffing and Van der Pol equation and for a 3D system, the Lorenz equations. The eigenvalue variation with time shows a repeating root variation for limit cycles that is intermittently varied for the forced chaos conditions. This behaviour is not the same for the Lorenz system. These results illustrate how the roots of a system can change when chaos is present