Bilinear systems and chaos. (English) Zbl 0823.93026

Motivated by the chaotic behavior of Lorenz equations, the authors presents a conjecture on chaos in a bilinear system in \(\mathbb{R}^ 3\) of the form \(\dot x= Ax+ Bxu\), \(x\in \mathbb{R}^ 3\), \(u\in \mathbb{R}\). The main results of this article are two theorems on the existence of a pair of symmetric homoclinic orbits and on the chaotic behavior of a generalized Lorenz equation (GLE). The computer simulation shows that the GLE has a chaotic behavior similar to the Lorenz equation although there is some difference between the theoretical analysis and the numerical simulation.


93C15 Control/observation systems governed by ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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