Attractors from one dimensional Lorenz-like maps

Choi, Youngna

doi:10.3934/dcds.2004.11.715

Cited by 7 publications

(5 citation statements)

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“…Suppose E is a proper completely invariant closed set of expanding Lorenz map f . The renormalization R E f defined by (6) in Theorem A is called the reormalization associated with E. And E is called the repelling set associated to the renormalization R E . The interval (e − , e + ), with endpoints e + and e − defined in (4), is called the critical interval of E and R E .…”

Section: Resultsmentioning

confidence: 99%

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Renormalization and $\alpha$-limit set for expanding Lorenz maps

Ding

2011

Discrete &Amp; Continuous Dynamical Systems - A

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We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of periodic orbit of minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and α-limit sets are obtained via consecutive renormalizations. Some properties of periodic renormalizations are collected in Appendix.Date: November 1, 2018. Mathematical classification (2000):37E05, 37F25, 54H20.

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Section: Resultsmentioning

confidence: 99%

“…Lorenz map plays an important role in the study of the global dynamics of families of vector fields near homoclinic bifurcations, see [21,22,26,30,31] and references therein. The expanding condition follows from [13,15,19], which is weaker than many other conditions used in [6,14,26] etc.…”

Section: Introductionmentioning

confidence: 99%

Renormalization and $\alpha$-limit set for expanding Lorenz maps

Ding

2011

Discrete &Amp; Continuous Dynamical Systems - A

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“…Si 𝑓 𝐴 no es transitivo no significa que no existe atractor topológico para 𝑓. De hecho, si 𝑓 satisface la condición 1 y 2, 𝛼 > 1 y 𝛽 > 1; se tiene que 𝑓 𝐴 satisface las condiciones dadas para la existencia de atractores topológicos probado en (Morales & Pujals, 1997) y ver (Choi, 2004) para la estructura del conjunto de puntos periódicos del atractor topológico. De estos dos trabajos aplicados a 𝑓 𝐴 concluimos que: Teorema 5.1.…”

Section: Conclusionesunclassified

Atractores en funciones lineales crecientes por parte en la recta real.

Iñiguez

Leal

2021

NOVASINERGIA

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Las funciones lineales por parte aparecen como modelos matemáticos para describir sistemas provenientes de la ingeniería eléctrica, ciencias físicas y economía, recientemente también aparece en modelos de la actividad neuronal. Se considera una familia a 4 parámetros de funciones lineales crecientes por parte sobre la recta real usando la teoría de funciones continua crecientes por parte sobre intervalos compactos para estudiar la existencia de conjuntos atractores y describir la dinámica del atractor, verificando la existencia de órbitas periódicas, transitividad y la existencia de medidas ergódicas invariantes. Se demuestra específicamente los diferentes valores del parámetro donde la familia exhibe un intervalo atractor. Se prueba las condiciones necesarias y suficientes para que el conjunto atractor sea globalmente atractor, de hecho, en este caso se prueba que la dinámica de dicho atractor se comporta como la dinámica de la rotación de Poincaré del circulo unitario. También, se describe bajo qué condiciones en los parámetros la familia exhibe un atractor topológico. Finalmente se prueba la existencia de medidas invariantes ergódicas absolutamente continua a la medida de Lebesgue asociado a la familia restricta al atractor, inclusive se prueba el caso en que la medida es equivalente a la medida de Lebesgue.

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“…[30]). The existence of an invariant interval I on which j 0 j 4 1 implies the existence of a Lorenz-like attractor [31][32][33][34]. It suffices to consider f since one can take 1 as small as required.…”

Section: Lorenz-like Attractorsmentioning

confidence: 99%

Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Golmakani

Homburg

2010

Dynamical Systems

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Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in R 3 that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.

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Attractors from one dimensional Lorenz-like maps

Cited by 7 publications

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Renormalization and $\alpha$-limit set for expanding Lorenz maps

Renormalization and $\alpha$-limit set for expanding Lorenz maps

Atractores en funciones lineales crecientes por parte en la recta real.

Lorenz attractors in unfoldings of homoclinic-flip bifurcations

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