Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems

Fusco, Giorgio; Oliva, Waldyr M.

doi:10.1007/bf01047768

Cited by 21 publications

(26 citation statements)

References 11 publications

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“…For the Neumann case ðNÞ; this property has been discovered independently by Henry [Hen85] and Angenent [Ang86]. For the S 1 -equivariant periodic case ðPÞ; the Morse-Smale property is established in Section 3 following Fusco and Oliva [FuOl90]. Loosely speaking it states that hyperbolicity of E,F,R alone is sufficient to imply transverse intersections of stable and unstable manifolds.…”

Section: Introductionmentioning

confidence: 90%

“…Claim (b), (3.5) involves a decomposition of the Floquet eigenspaces, according to zero number z ¼ 2k; very similar to-but less explicit than-(3.11) above. See [AF88] and also [FuOl90] for details. The lap number cðvÞ ¼ zðv x Þ enters diacritically, because v t ¼ cv x is the Floquet eigenfunction of the trivial Floquet multiplier 1, and thus separates the strongly stable from the strongly unstable Floquet eigenspace.…”

Section: Article In Pressmentioning

confidence: 97%

“…Loosely speaking it states that hyperbolicity of E,F,R alone is sufficient to imply transverse intersections of stable and unstable manifolds. This automatic transversality between stable and unstable manifolds of hyperbolic periodic orbits has been shown to hold in some special classes of finite-dimensional systems, [Pal78,FuOl90]. The infinite-dimensional case has recently been considered in [Oli02], where a class of retarded functional differential equations is studied.…”

Section: Introductionmentioning

confidence: 98%

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Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Fiedler¹,

Rocha²,

Wolfrum³

2004

Journal of Differential Equations

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We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves. r

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Section: Introductionmentioning

confidence: 90%

Section: Article In Pressmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

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Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Fiedler¹,

Rocha²,

Wolfrum³

2004

Journal of Differential Equations

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“…In the case when the linear system (1.3) is time-periodic in t, Smith [34] studied the Floquet theory by using an integer-valued Lyapunov function σ, first defined by Smillie [33] (see also similar forms by Mallet-Paret and Smith [17], Fusco and Oliva [8,9], and Mallet-Paret and Sell [16]), and related the values of σ to the Floquet multipliers of the linear periodic system. This function σ is not defined everywhere but only on an open and dense subset Λ of R n on which it is also continuous (see section 2).…”

Section: Introductionmentioning

confidence: 99%

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

Fang¹,

Gyllenberg²,

Wang³

2013

SIAM J. Math. Anal.

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Abstract. We consider a general time-dependent linear competitive-cooperative tridiagonal system of differential equations in the framework of skew-product flows and obtain canonical Floquet invariant bundles which are exponentially separated. Such Floquet bundles naturally reduce to the standard Floquet space when the system is assumed to be time-periodic. We apply the Floquet theory so obtained to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitivecooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

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“…For a generic perturbation R the associated parabolic flow Ψ t is a Morse-Smale flow [21]. The (finite) collection of rest points {u a } and periodic orbits {γ b } of Ψ t then yields a Morse decomposition of Inv(N ), and the Morse inequalities are…”

Section: Lemma 36 An Arbitrary Parabolic Flow On a Bounded Relative mentioning

confidence: 99%

Morse theory on spaces of braids and Lagrangian dynamics

Ghrist

Berg

Vandervorst

2003

Inventiones Mathematicae

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ABSTRACT. In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a MorseConley homotopy index.In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits. PRELUDEIt is well-known that under the evolution of any scalar uniformly parabolic equation of the form (1) u t = f (x, u, u x , u xx ) ; ∂ uxx f ≥ δ > 0, the graphs of two solutions u 1 (x, t) and u 2 (x, t) evolve in such a way that the number of intersections of the graphs does not increase in time. This principle, known in various circles as "comparison principle" or "lap number" techniques, entwines the geometry of the graphs (u xx is a curvature term), the topology of the solutions (the intersection number is a local linking number), and the local dynamics of the PDE. This is a valuable approach for understanding local dynamics for a wide variety of flows exhibiting parabolic behavior with both classical [54] and contemporary [41,5,10,19] implications. This paper is an extension of this local technique to a global technique. One such well-established globalization appears in the work of Angenent on curve-shortening [4]: evolving closed curves on a surface by curve shortening isolates the classes of curves dynamically and implies a monotonicity with respect to number of self-intersections.In contrast, one could consider the following topological globalization. Superimposing the graphs of a collection of functions u α (x) gives something which resembles the projection of a topological braid onto the plane. Assume that the "height" of the strands above the page is given by the slope u α x (x), or, equivalently, that all of the crossings in the projection are of the same sign (bottom-over-top): see Fig. 1[left]. Evolving these functions under a parabolic equation (with, say, boundary endpoints fixed) yields a flow on a certain space of braid diagrams which has a topological monotonicity: linking can be destroyed but not created. This establishes a partial ordering on the semigroup of positive braids which is respected by parabolic dynamics. The idea of topological braid classes with this partial ordering is a globalization of the lap number (which, in braid-theoretic terms becomes the length of the braid in the braid group under standard generators).1.1. Parabolic flows on spaces of braid diagrams. In this paper, we initiate the study of parabolic flows on spaces of braid diagrams. The particular braids in question wi...

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Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems

Cited by 21 publications

References 11 publications

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

Morse theory on spaces of braids and Lagrangian dynamics

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