scholarly journals Lorenz attractors through Šil'nikov-type bifurcation. Part I

1990 ◽  
Vol 10 (4) ◽  
pp. 793-821 ◽  
Author(s):  
Marek Ryszard Rychlik
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Parameter Family ◽  
Double Loop ◽  
Hyperbolic Fixed Point ◽  
Lorenz Attractors ◽  
Two Parameter

AbstractThe main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on ℝ3is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1> 0, λ2> 0, λ3> 0 and |λ2|>|λ1|>|λ3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors.A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.

Horseshoes in the measure-preserving Hénon map

1995 ◽  
Vol 15 (6) ◽  
pp. 1045-1059 ◽  
Author(s):  
Ray Brown
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Recursive Formula ◽  
Homoclinic Point ◽  
Hénon Map ◽  
Henon Map ◽  
Broad Array ◽  
Hyperbolic Fixed Point ◽  
Elementary Methods

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.

Dynamics and bifurcations of a host–parasite model

2017 ◽  
Vol 10 (06) ◽  
pp. 1750089 ◽  
Author(s):  
Ali Atabaigi ◽  
Mohammad Hossein Akrami
Keyword(s):  
Fixed Point ◽  
Difference Equations ◽  
Parameter Family ◽  
Invariant Curve ◽  
Discrete Models ◽  
Nonlinear Difference Equations ◽  
Host Parasite ◽  
Parameter Values ◽  
Two Parameter

A two-parameter family of discrete models, consisting of two coupled nonlinear difference equations, describing a host–parasite interaction is considered. In particular, we prove that the model has at most one nontrivial interior fixed point which is stable for a certain range of parameter values and also undergoes a Neimark–Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter.

From homoclinics to quasi-periodic solutions for ordinary differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1091-1114
Author(s):  
Changrong Zhu
Keyword(s):  
Functional Analysis ◽  
Periodic Solution ◽  
Fixed Point ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Homoclinic Solution ◽  
Unperturbed System ◽  
Perturbed System ◽  
Hyperbolic Fixed Point ◽  
Parameter Values

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

Subcontinua of the closure of the unstable manifold at a homoclinic tangency

1999 ◽  
Vol 19 (2) ◽  
pp. 289-307 ◽  
Author(s):  
MARCY BARGE ◽  
BEVERLY DIAMOND
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Inverse Limit ◽  
Resonance Condition ◽  
Homoclinic Tangency ◽  
Limit Space ◽  
Open Set ◽  
Hyperbolic Fixed Point ◽  
Inverse Limit Space

Suppose that $F$ is a $C^\infty$ diffeomorphism of the plane with hyperbolic fixed point $p$ for which a branch of the unstable manifold, $W^u_+(p)$, has a same-sided quadratic tangency with the stable manifold, $W^s(p)$. If the eigenvalues of $DF$ at $p$ satisfy a non-resonance condition, each nonempty open set of $ \cl( W^u_+(p))$ contains a copy of any continuum that can be written as the inverse limit space of a sequence of unimodal bonding maps.

scholarly journals On the accumulation of separatrices by invariant circles

2021 ◽  
pp. 1-41
Author(s):  
A. KATOK ◽  
R. KRIKORIAN
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Positive Measure ◽  
Symplectic Diffeomorphisms ◽  
Invariant Circles ◽  
Hyperbolic Fixed Point ◽  
Symplectic Diffeomorphism

Abstract Let f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

Embedding diffeomorphisms in flows in Banach spaces

2009 ◽  
Vol 29 (4) ◽  
pp. 1349-1367 ◽  
Author(s):  
XIANG ZHANG
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Vector Field ◽  
Banach Spaces ◽  
Floquet Theory ◽  
Classical Theorem ◽  
Local Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Hyperbolic Fixed Point

AbstractThis paper concerns the problem of embedding, in the flow of an autonomous vector field, a local diffeomorphism near a hyperbolic fixed point in a Banach space. To solve the problem, we first extend Floquet theory to Banach spaces, and then prove that two C∞ hyperbolic diffeomorphisms are formally equivalent if and only if they are C∞-equivalent. The latter result is a version, in the Banach space context, of a classical theorem by Chen.

Globalizing Two-Dimensional Unstable Manifolds of Maps

1998 ◽  
Vol 08 (03) ◽  
pp. 483-503 ◽  
Author(s):  
Bernd Krauskopf ◽  
Hinke Osinga
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Invariant Circle ◽  
Hyperbolic Fixed Point ◽  
Linear Foliation ◽  
Number Of Leaves ◽  
Unstable Manifolds ◽  
Global Stable Manifold

We present an algorithm for computing the global two-dimensional unstable manifold of a hyperbolic fixed point or a normally hyperbolic invariant circle of a three-dimensional map. The global stable manifold can be obtained by considering the inverse map. Our algorithm computes intersections of the unstable manifold with a finite number of leaves of a chosen linear foliation. In this way, we obtain a growing piece of the unstable manifold represented by a mesh of prescribed quality. The performance of the algorithm is demonstrated with several examples.

Nonlinear neutral integro-differential equations, stability by fixed point and inverses of delays

2017 ◽  
Vol 47 (1) ◽  
pp. 89-113
Author(s):  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi ◽  
Hocine Gabsi
Keyword(s):  
Fixed Point ◽  
Differential Equations

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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