New approach to non-linear instability and chaos based on generalized tellegen's principle

This paper deals with dissipativity, internal stability, instability, chaotic behavior and related structural properties of a relatively broad class of finite dimensional strictly causal systems. The class of nonlinear systems under consideration is described in the state-space representation form. System properties are investigated by a new approach based on a geometric interpretation of a new abstract state energy concept, and on a proper generalization of the well known Tellegen's theorem as a form of the energy conservation principle. The resulting energy metric function is induced by the output signal power and determines both, the structure of a proper system representation as well as the corresponding system state space topology. The state minimality, as well as parameter minimality requirements play a crucial role in the proposed method. Several typical problems are solved in detail, and results of simulation examples are shown for illustration of fundamental ideas and basic attributes of the proposed method.