A Review on Some Bifurcations in the Lorenz System

Algaba, Antonio; Domínguez-Moreno, M. Cinta; Merino, Manuel; Rodríguez-Luis, Alejandro J.

doi:10.1007/978-3-319-66766-9_1

Cited by 4 publications

(2 citation statements)

References 91 publications

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“…which has degree two (so it is quadratic), presents chaotic behavior if s = 10, b = 8 3 and r = 28 (Lorenz 1963). Later on, it was shown by several authors that Lorenz system (2) has chaotic behavior for many other combinations of parameter values (see for instance the nice book (Sparrow 1982) or the more recent review (Algaba et al 2018) and references cited in them).…”

Section: Chaosmentioning

confidence: 99%

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

Messias

SİLVA²

2022

Chaos Theory and Applications

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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.

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Section: Chaosmentioning

confidence: 99%

“…The Lorenz systems (2) is maybe the most famous system which is known to be chaotic for certain parameter values (Algaba et al 2018;Lorenz 1963;Sparrow 1982). Using Theorem 2 we can obtain some subclasses of Lorenz system which are not chaotic.…”

Section: Lorenz Systemmentioning

confidence: 99%

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

Messias

SİLVA²

2022

Chaos Theory and Applications

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A Degenerate Takens–Bogdanov Bifurcation in a Normal form of Lorenz’s Equations

Algaba

Domínguez-Moreno

Merino

et al. 2021

NODYCON Conference Proceedings Series

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A double-zero bifurcation in a Lorenz-like system

Algaba,

Domínguez-Moreno,

Merino

et al. 2023

Nonlinear Dyn

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The Lorenz system presents a double-zero bifurcation (a double-zero eigenvalue with geometric multiplicity two). However, its study by means of standard techniques is not possible because it occurs for a non-isolated equilibrium. To circumvent this difficulty, we add in the third equation a new term, $$Dz^2$$ D z 2 . In this Lorenz-like system, the analysis of the double-zero bifurcation of the equilibrium at the origin guarantees, for certain values of the parameters, the existence of a heteroclinic cycle between the two equilibria located on the z-axis. The numerical continuation in parameter space of the locus of heteroclinic connections allows to detect various degeneracies of codimension two and three, some of which have not been previously studied in the literature. These bifurcations are organizing centers of the complicated dynamics exhibited by this system. Furthermore, studying how the bifurcation sets evolve when D tends to zero, we are able to explain, in the Lorenz system, the origin of several global connections which are related to T-point heteroclinic loops.

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A Review on Some Bifurcations in the Lorenz System

Cited by 4 publications

References 91 publications

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

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

A Degenerate Takens–Bogdanov Bifurcation in a Normal form of Lorenz’s Equations

A double-zero bifurcation in a Lorenz-like system

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