DERIVATIVES FOR THE STABLE AND UNSTABLE MANIFOLDS OF A Cr Diffeomorphism ofR2

1993 ◽  
Vol 03 (06) ◽  
pp. 1601-1605 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Unstable Manifold ◽  
Taylor Polynomial ◽  
Stable And Unstable Manifolds ◽  
Taylor Coefficients ◽  
Unstable Manifolds

We illustrate how to form the Taylor polynomial for a stable or unstable manifold when considered as a curve in the plane. Our method does not use power series or assume that the manifold is analytic as is done in the method of Poincaré. Instead we use simple calculus to obtain closed form expressions for the Taylor coefficients.

scholarly journals Singularities of the susceptibility of a Sinai–Ruelle–Bowen measure in the presence of stable–unstable tangencies

2011 ◽  
Vol 369 (1935) ◽  
pp. 482-493 ◽  
Author(s):  
David Ruelle
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Discrete Time ◽  
Smooth Function ◽  
Rigorous Analysis ◽  
Physical Measure ◽  
Stable And Unstable Manifolds ◽  
Expectation Value ◽  
Stable Direction ◽  
Unstable Manifolds

Let ρ be a Sinai–Ruelle–Bowen (SRB or ‘physical’) measure for the discrete time evolution given by a map f , and let ρ ( A ) denote the expectation value of a smooth function A . If f depends on a parameter, the derivative δρ ( A ) of ρ ( A ) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ ( z ) at z =1. When f is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ ( z ) has a radius of convergence r ( Ψ )>1, and that δρ ( A )= Ψ (1), but it is known that r ( Ψ )<1 in some other cases. One reason why f may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for ( f , ρ ). The present paper gives a crude, non-rigorous, analysis of this situation in terms of the Hausdorff dimension d of ρ in the stable direction. We find that the tangencies produce singularities of Ψ ( z ) for | z |<1 if d <1/2, but only for | z |>1 if d >1/2. In particular, if d >1/2, we may hope that Ψ (1) makes sense, and the derivative δρ ( A )= Ψ (1) thus has a chance to be defined.

scholarly journals A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

2021 ◽  
Author(s):  
Constantinos Siettos ◽  
Lucia Russo
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Stochastic Systems ◽  
Black Box ◽  
Coarse Grained ◽  
Stationary States ◽  
Macroscopic Description ◽  
Stable And Unstable Manifolds ◽  
Agent Based ◽  
Unstable Manifolds

AbstractWe address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures: the constant unstable dimension case

2015 ◽  
Vol 36 (5) ◽  
pp. 1494-1515 ◽  
Author(s):  
MICHIHIRO HIRAYAMA ◽  
NAOYA SUMI
Keyword(s):  
Unstable Manifold ◽  
Probability Measures ◽  
Stable And Unstable Manifolds ◽  
Srb Measure ◽  
Surface Diffeomorphisms ◽  
Srb Measures ◽  
Almost Everywhere ◽  
Unstable Manifolds ◽  
Math Phys

In this paper we consider diffeomorphisms preserving hyperbolic Sinaĭ–Ruelle–Bowen (SRB) probability measures${\it\mu}$having intersections for almost every pair of the stable and unstable manifolds. In this context, when the dimension of the unstable manifold is constant almost everywhere, we show the ergodicity of${\it\mu}$. As an application we obtain another proof of the ergodicity of a hyperbolic SRB measure for transitive surface diffeomorphisms, which is shown by Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces.Comm. Math. Phys.306(1) (2011), 35–49].

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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