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E. V. Vasil’eva - One of the best experts on this subject based on the ideXlab platform.
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Multidimensional Diffeomorphisms with Stable Periodic Points
Vestnik St. Petersburg University: Mathematics, 2019Co-Authors: E. V. Vasil’evaAbstract:Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed Point are discussed in this paper. It is assumed that at the intersection of stable and unstable manifolds, there are Points that are different from the hyperbolic Point. Such Points are called Homoclinic and are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. It follows from articles by S. Newhouse, L. P. Shil’nikov, B. F. Ivanov, and others, that with a certain method of tangency of a stable manifold with an unstable one, the neighborhood of a non-transversal Homoclinic Point contains an infinite number of stable periodic Points, but at least one of the characteristic exponents at these Points tends to zero with increasing period. The present study is a continuation of previous studies by the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic Point. More precisely, it was assumed that either all eigenvalues are real and the Jacobi matrix is diagonal, or the matrix has only one real eigenvalue less than one in modulus, while all other eigenvalues are various complex integers greater than one in modulus. Within this framework, conditions are obtained for the presence of an infinite set of stable periodic Points with characteristic exponents separated from zero in an arbitrary neighborhood of a non-transversal Homoclinic Point. It is assumed in this paper that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic Point. In this case, the conditions are obtained for the existence of an infinite set of stable periodic Points whose characteristic exponents are separated from zero in the neighborhood of the non-transversal Homoclinic Point. The conditions are imposed, first of all, on the method of tangency of a stable manifold with an unstable one; however, the proof of the theorem essentially uses the properties of the eigenvalues of the Jacobi matrix at a hyperbolic Point.
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Stability of Periodic Points of Diffeomorphisms of Multidimensional Space
Vestnik St. Petersburg University Mathematics, 2018Co-Authors: E. V. Vasil’evaAbstract:We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed Point at the origin and a nontransversal Homoclinic Point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal Homoclinic Point can contain an infinite set of stable periodic Points, but at least one of the characteristic exponents of those Points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal Homoclinic Point of the studied diffeomorphism contains an infinite set of stable periodic Points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal Homoclinic Point contains an infinite set of stable periodic Points with characteristic exponents separated from zero.
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Smooth diffeomorphisms of the plane with stable periodic Points in a neighborhood of a Homoclinic Point
Differential Equations, 2012Co-Authors: E. V. Vasil’evaAbstract:We consider self-diffeomorphisms of the plane of the class C ^ r (1 ≤ r < ∞) with a fixed hyperbolic Point and a nontransversal Point Homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a Homoclinic Point contains countably many stable periodic Points with characteristic exponents bounded away from zero.
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Diffeomorphisms of multidimensional space with infinite set of stable periodic Points
Vestnik St. Petersburg University: Mathematics, 2012Co-Authors: E. V. Vasil’evaAbstract:Multidimensional diffeomorphisms with a hyperbolic fixed Point and its Homoclinic Point are considered. It is shown that the neighborhood of the Homoclinic Point can contain an infinite set of stable periodic Points whose characteristic exponents are bounded away from zero.
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Diffeomorphisms of the plane with stable periodic Points
Differential Equations, 2012Co-Authors: E. V. Vasil’evaAbstract:We consider self-diffeomorphisms of the plane with a hyperbolic fixed Point and a nontransversal Homoclinic Point. We show that a neighborhood of the Homoclinic Point may contain countably many stable periodic sets whose characteristic exponents are bounded away from zero.
Klaus Schmidt - One of the best experts on this subject based on the ideXlab platform.
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Algebraic actions of the discrete Heisenberg group: expansiveness and Homoclinic Points
Indagationes Mathematicae, 2014Co-Authors: Martin Göll, Klaus Schmidt, Evgeny VerbitskiyAbstract:Abstract We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan’s local principle. Furthermore, we present a first example of an absolutely summable Homoclinic Point for a nonexpansive action of the discrete Heisenberg group and use it to construct an equal-entropy symbolic cover of the system.
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Expansive algebraic actions of discrete residually finite amenable groups and their entropy
arXiv: Dynamical Systems, 2006Co-Authors: Christopher Deninger, Klaus SchmidtAbstract:We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma $ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma $-action by means of a `fundamental Homoclinic Point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $\Gamma_n$-fixed Points, where $(\Gamma_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma $ with trivial intersection.
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Common Homoclinic Points of commuting toral automorphisms
Israel Journal of Mathematics, 1999Co-Authors: Anthony Manning, Klaus SchmidtAbstract:The Points Homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a ℤ2-action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same Homoclinic Points or have no Homoclinic Point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs.
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Homoclinic Points of algebraic ℤ^{}-actions
Journal of the American Mathematical Society, 1999Co-Authors: Douglas Lind, Klaus SchmidtAbstract:An algebraic zd-action is an action of Zd by (continuous) automorphisms of a compact abelian group. The dynamics of a single group automorphism have been investigated in great detail over the past several decades. More recently, the study of algebraic Zd-actions for d > 2 has revealed a striking interplay between these actions and commutative algebra. In ?2 we summarize those parts of this interaction needed here. The purpose of this paper is to study the Homoclinic Points of algebraic Zd_ actions. Let al be an algebraic Zd-action on the compact abelian group X, and let Ox denote the additive identity of X. A Point x E X is Homoclinic for ce if c nX -? Ox as llnll -oo. The set Ai(X) of all Homoclinic Points for Oc is clearly a subgroup of X which we call the Homoclinic group of oz. In ?3 we discuss some elementary properties of the Homoclinic group, including counltability of ZA (X) whenever oz is expansive. Our two main results are contained in ?4. These are that if oz is an expansive algebraic Ed-action, then (1) A,(X) is nontrivial if and only if oz has (strictly) positive entropy, and (2) A, (X) is nontrivial and dense in X if and only if oz has completely positive entropy. The second result is proved by first establishing in Lemma 4.5 the density of A, (X) for certain "principal" expansive actions by use of Fourier analysis. For these actions A, (X) is generated by a single fundamental Homoclinic Point which can be computed explicitly. This lemma is then combined with some commutative algebra to prove (2). Recent work of Kaminker and Putnam [3], [11] has suggested a general duality in the K-theory of C*-algebras. For a principal expansive action al on X we show that A, (X) is isomorphic to the dual group of X, providing a class of examples to which their duality theory applies. Ruelle has investigated expansive topological Zd-actions which satisfy an orbit tracing property called specification (see [12] and [13]), showing that there is a thermodynamic formalism for such actions. In ?5 we show that expansive algebraic Zd-actions with completely positive entropy always satisfy very strong specification properties, thereby providing an extensive class of examples to which the thermodynamic formalism applies.
Ray Brown - One of the best experts on this subject based on the ideXlab platform.
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Horseshoes in the measure-preserving Hénon map
Ergodic Theory and Dynamical Systems, 1995Co-Authors: Ray BrownAbstract:We show, using elementary methods, that for 0 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric Homoclinic Point is transversal. To obtain our results, we show that for a branch, C", of the unstable manifold of a hyperbolic fixed Point of H, C crosses the line y = — x and that this crossing is a Homoclinic Point, Xc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of W(p ) and W s (p) at Xc is degenerate then the slope of C at this crossing is one. Following this we show that if Xc is a degenerate Homoclinic its JC-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of C at H" 1 ^) must be both positive and negative, thus we conclude that Xc is transversal. Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as 'the curvature lemma', is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C 2 .
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Horseshoes in the measure-preserving Hénon map
Ergodic Theory and Dynamical Systems, 1995Co-Authors: Ray BrownAbstract:AbstractWe show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric Homoclinic Point is transversal.To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed Point of H, Cu crosses the line y = − x and that this crossing is a Homoclinic Point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate Homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.
Daniel Stoffer - One of the best experts on this subject based on the ideXlab platform.
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Transversal Homoclinic Points of the H
2020Co-Authors: Urs Kirchgraber, Daniel StofferAbstract:Using shadowing techniques we prove that the H´ enon map Ha,b(x, y) = (a − x 2 + by, x) admits a transversal Homoclinic Point for a set of parameters which is not small. For the area and orientation preserving H´ enon map (corresponding to b =− 1) we prove that a transversal Homoclinic Point exists for a ≥ 0.265625. Applying a computer-assisted version of our scheme we show that the result holds even for a ≥− 0.866. This supports an old conjecture due to Devaney and Nitecki dating back to 1979, see (4), claiming that the Hmap in the case b =− 1 admits a transversal Homoclinic Point for a > −1.
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Transversal Homoclinic Points of the Hénon map
Annali di Matematica Pura ed Applicata, 2020Co-Authors: Urs Kirchgraber, Daniel StofferAbstract:Using shadowing techniques we prove that the Henon map \(H_{a,b}(x,y)=(a-x^{2}+by,x)\) admits a transversal Homoclinic Point for a set of parameters which is not small. For the area and orientation preserving Henon map (corresponding to b=-1) we prove that a transversal Homoclinic Point exists for a≥0.265625. Applying a computer-assisted version of our scheme we show that the result holds even for a≥-0.866. This supports an old conjecture due to Devaney and Nitecki dating back to 1979, see [4], claiming that the Henon map in the case b=-1 admits a transversal Homoclinic Point for a>-1.
Eihab M. Abdel-rahman - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear dynamics of a resonant gas sensor
Nonlinear Dynamics, 2010Co-Authors: Ali H. Nayfeh, Slim Choura, Hassen M. Ouakad, Fehmi Najar, Eihab M. Abdel-rahmanAbstract:We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which Homoclinic Points appear, we conduct a Melnikov analysis. We show the presence of a Homoclinic Point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.