scholarly journals Non-smooth saddle-node bifurcations I: existence of an SNA

2014 ◽  
Vol 36 (4) ◽  
pp. 1130-1155 ◽  
Author(s):  
GABRIEL FUHRMANN
Keyword(s):  
Lyapunov Exponent ◽  
General Result ◽  
Sufficient Conditions ◽  
Positive Lyapunov Exponent ◽  
Hyperbolic Attractor ◽  
Interval Maps ◽  
Skew Products ◽  
Saddle Node ◽  
Invariant Graph

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are${\mathcal{C}}^{2}$-open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

scholarly journals Small C1-smooth perturbations of skew products and the partial integrability property

2020 ◽  
Vol 5 (2) ◽  
pp. 317-328
Author(s):  
L.S. Efremova
Keyword(s):  
Sufficient Conditions ◽  
Skew Product ◽  
Interval Maps ◽  
Skew Products ◽  
Partial Integrability

AbstractIn this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

On Weak Lyapunov Exponent and Sensitive Dependence of Interval Maps

2014 ◽  
Vol 24 (10) ◽  
pp. 1450120
Author(s):  
Xinxing Wu ◽  
Guanrong Chen ◽  
Peiyong Zhu
Keyword(s):  
Lyapunov Exponent ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Interval Maps ◽  
System A ◽  
Sensitive Dependence

By using weak Lyapunov exponent, some sufficient conditions ensuring a system ([a, b], f) to be sensitively dependent on the initial value x0 ∈ [a, b] are obtained.

More General Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
General Result ◽  
Convex Analysis ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
General Function ◽  
Existence Of A Solution ◽  
Generalize Convex ◽  
Monge Problem ◽  
Natural Way

This chapter considers a case with a more general surplus function. It shows that when the scalar-product surplus is replaced by a more general function, much of the machinery put in place in Chapter 6 goes through. In particular, it is possible to generalize convex analysis in a natural way, and to obtain generalized notions of convex conjugates, of convexity, and of a subdifferential that are perfectly suited to the problem. A general result on the existence of dual minimizers is given, as well as sufficient conditions for the existence of a solution to the Monge problem.

On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

2019 ◽  
Vol 7 (4) ◽  
pp. 1157-1172 ◽  
Author(s):  
Alejandro Silva-Juárez ◽  
Carlos Javier Morales-Pérez ◽  
Luis Gerardo de la Fraga ◽  
Esteban Tlelo-Cuautle ◽  
José de Jesús Rangel-Magdaleno
Keyword(s):  
Lyapunov Exponent ◽  
Positive Lyapunov Exponent ◽  
Chaotic Oscillators

scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

scholarly journals Return- and hitting-time distributions of small sets in infinite measure preserving systems

2019 ◽  
Vol 40 (8) ◽  
pp. 2239-2273
Author(s):  
SIMON RECHBERGER ◽  
ROLAND ZWEIMÜLLER
Keyword(s):  
Sufficient Conditions ◽  
Finite Measure ◽  
Hitting Time ◽  
Independent Random Variables ◽  
Limit Laws ◽  
Interval Maps ◽  
Infinite Measure ◽  
Sufficient Conditions For Convergence ◽  
Set Up ◽  
The One

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

scholarly journals RETURN TIME STATISTICS OF INVARIANT MEASURES FOR INTERVAL MAPS WITH POSITIVE LYAPUNOV EXPONENT

2009 ◽  
Vol 09 (01) ◽  
pp. 81-100 ◽  
Author(s):  
HENK BRUIN ◽  
MIKE TODD
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measures ◽  
Return Time ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Normal Fluctuations ◽  
Return Time Statistics ◽  
Absolutely Continuous Invariant

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a "polynomial Gibbs property" for these systems, and that the convergence to the entropy in the Ornstein–Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.

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