On the mathematical representation of nonlinearity

TL;DR

The paper explores a constructive approach to representing nonlinear systems using state equations with variable matrices, supported by theoretical and practical arguments from system theory, electronics, and physics.

Contribution

It proposes a new form for nonlinear state equations, emphasizing a more constructive and axiomatic representation of nonlinear systems.

Findings

Supports the use of matrix functions A(t,x) and B(t,x) in nonlinear state equations

Provides arguments from various fields for this representation approach

Highlights the advantages of this form in modeling complex systems

Abstract

The suggestion of writing, for some problems, nonlinear state equations not as dx/dt = F(x,u,t), but as dx/dt = [A(t,x)]x + [B(t,x)]u(t), which is more "constructive", is considered supported by arguments related to: the axiomatization of system theory, the classification of switched circuits as linear and nonlinear, the use of nonlinear sampling for measuring frequency without clock, the consideration of an ensemble of colliding particles in which the establishing of the chaotic movements is seen as a result of the same kind of nonlinearity as in electronic switching circuits, the modeling of a liquid medium as a "system" whose structure is directly influenced by the "input", and some other problems.