scholarly journals Cherry flow: physical measures and perturbation theory

2016 ◽  
Vol 37 (8) ◽  
pp. 2671-2688 ◽  
Author(s):  
JIAGANG YANG
Keyword(s):  
Perturbation Theory ◽  
Periodic Orbits ◽  
Invariant Measures ◽  
Saddle Connection ◽  
Physical Measure ◽  
Hyperbolic Flow ◽  
Physical Measures ◽  
Full Volume ◽  
Math Phys

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows.Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

On the convergence to equilibrium states for certain non-hyperbolic systems

1997 ◽  
Vol 17 (4) ◽  
pp. 977-1000 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Periodic Orbits ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Markov Property ◽  
Standard Technique ◽  
Equilibrium States ◽  
Convergence To Equilibrium ◽  
Invariant Density ◽  
Frobenius Operator ◽  
Uniformly Bounded

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

scholarly journals Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Awadhesh Kumar Poddar ◽  
Divyanshi Sharma
Keyword(s):  
Perturbation Theory ◽  
Rigid Body ◽  
Coordinate System ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
General Perturbation ◽  
Triaxial Rigid Body ◽  
Three Dimensional Coordinate

AbstractIn this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.

scholarly journals GEOMETRICAL AND MEASURE-THEORETIC STRUCTURES OF MAPS WITH A MOSTLY EXPANDING CENTER

2021 ◽  
pp. 1-41
Author(s):  
Jiagang Yang
Keyword(s):  
Exponential Decay ◽  
Combinatorial Structure ◽  
Residual Subset ◽  
Hölder Functions ◽  
Hyperbolic Diffeomorphisms ◽  
Exponential Decay Of Correlations ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Full Volume

Abstract In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume. We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

2015 ◽  
Vol 36 (6) ◽  
pp. 1865-1891 ◽  
Author(s):  
STEFANO GALATOLO ◽  
ISAIA NISOLI
Keyword(s):  
Invariant Measures ◽  
Wasserstein Distance ◽  
Continuous Invariant Measure ◽  
Local Dimension ◽  
Physical Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Rigorous Computation ◽  
Absolutely Continuous Invariant

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

scholarly journals Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems

2021 ◽  
Author(s):  
Peter Ashwin ◽  
Julian Newman
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Autonomous Systems ◽  
Chaotic Attractors ◽  
Physical Measure ◽  
Multiple Attractors ◽  
The Past ◽  
Rate Dependent ◽  
Typical Behaviour ◽  
Autonomous Dynamical Systems

AbstractPhysical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems. In this paper, we make some steps towards extending this notion to more general nonautonomous (time-dependent) dynamical systems. There are barriers to doing this in general in a physically meaningful way, but for systems that have autonomous limits, one can define a physical measure in relation to the physical measure in the past limit. We use this to understand cases where rate-dependent tipping between chaotic attractors can be quantified in terms of “tipping probabilities”. We demonstrate this for two examples of perturbed systems with multiple attractors undergoing a parameter shift. The first is a double-scroll system of Chua et al., and the second is a Stommel model forced by Lorenz chaos.

scholarly journals Physical measures of discretizations of generic diffeomorphisms

2016 ◽  
Vol 38 (4) ◽  
pp. 1422-1458 ◽  
Author(s):  
PIERRE-ANTOINE GUIHÉNEUF
Keyword(s):  
Numerical Experiments ◽  
Invariant Measures ◽  
The Other ◽  
Spatial Discretization ◽  
Generic Point ◽  
Physical Measures ◽  
Generic Points ◽  
Ergodic Behaviour

What is the ergodic behaviour of numerically computed segments of orbits of a diffeomorphism? In this paper, we try to answer this question for a generic conservative $C^{1}$-diffeomorphism and segments of orbits of Baire-generic points. The numerical truncation is modelled by a spatial discretization. Our main result states that the uniform measures on the computed segments of orbits, starting from a generic point, accumulate on the whole set of measures that are invariant under the diffeomorphism. In particular, unlike what could be expected naively, such numerical experiments do not see the physical measures (or, more precisely, cannot distinguish physical measures from the other invariant measures).

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