Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

Abstract

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

Keywords

• Chaotic and nonchaotic dynamics
• Lorenz-like systems
• Darboux theory of integrability
• Invariant algebraic surface
• Darboux invariant
• Stable and unstable manifolds

Supporting Institution

State of São Paulo Foundation (FAPESP)

Project Number

Grant Number 2019/10269-3

Thanks

The first author was supported by State of São Paulo Foundation (FAPESP) grant number 2019/10269-3 and by CNPq/Brazil grant number 311355/2018-8. The second author was supported by CAPES/Brazil through a PhD fellowship.

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