Homoclinic Cycle Bifurcations in Planar Maps

2017 ◽  
Vol 27 (03) ◽  
pp. 1730012
Author(s):  
Kyohei Kamiyama ◽  
Motomasa Komuro ◽  
Kazuyuki Aihara
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Closed Curve ◽  
Bifurcation Structure ◽  
Stable And Unstable Manifolds ◽  
Planar Map ◽  
Unstable Manifolds ◽  
Planar Maps ◽  
Homoclinic Cycle

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.

PERTURBING TWO-DIMENSIONAL MAPS HAVING CRITICAL HOMOCLINIC ORBITS

1999 ◽  
Vol 09 (06) ◽  
pp. 1189-1195 ◽  
Author(s):  
FLAVIANO BATTELLI ◽  
CLAUDIO LAZZARI
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Hyperbolic Fixed Point ◽  
Unstable Manifolds ◽  
Planar Maps ◽  
Necessary And Sufficient ◽  
Critical Lines

Discrete planar maps such as xn+1=f(xn)+μ g(xn, μ), xn∈ℝ2, n∈ℤ, μ∈ℝ, are studied under the assumption that the unperturbed map xn+1=f(xn) has a critical homoclinic orbit to a hyperbolic fixed point. We give either necessary and sufficient conditions for a bifurcation from zero to two homoclinic orbits as the small parameter μ crosses zero. These conditions are stated in terms of geometrical objects such as critical lines, stable and unstable manifolds.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Detailed Explanation ◽  
Single Well ◽  
Unstable Manifolds ◽  
Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors

2005 ◽  
Author(s):  
M. Erol Ulucakli
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Fixed Points ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Rectangular Cell ◽  
Viscous Liquids ◽  
Unstable Manifolds ◽  
Time Periodic

The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.

A Gradient-Prediction Algorithm for Computing One Dimensional Stable and Unstable Manifolds of Maps

2012 ◽  
Vol 580 ◽  
pp. 78-81
Author(s):  
Meng Jia ◽  
Yi Ru ◽  
Jun Jie Xi ◽  
Qing Hua Ji ◽  
Zhong Wu
Keyword(s):  
Prediction Algorithm ◽  
Stable And Unstable Manifolds ◽  
One Dimensional ◽  
Prediction Scheme ◽  
Accuracy Criterion ◽  
Unstable Manifolds ◽  
Planar Maps

A new algorithm is presented to compute both one dimensional stable and unstable manifolds of planar maps. The global manifold is grown from a local manifold and one point is added each step. It is proved that the gradient of the global manifold can be predicted by the known points on the manifold with a gradient prediction scheme and it can be used to locate the image or preimage of the new point quickly. A new accuracy criterion is derived from the gradient prediction scheme. The performance of the algorithm is tested with several well-known examples.

A Numerical Scheme for Computing Stable and Unstable Manifolds in Nonautonomous Flows

2016 ◽  
Vol 26 (14) ◽  
pp. 1630041 ◽  
Author(s):  
Sanjeeva Balasuriya
Keyword(s):  
Lyapunov Exponent ◽  
High Resolution ◽  
Rossby Wave ◽  
Time Variation ◽  
Numerical Scheme ◽  
Numerical Approximation ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Finite Time Lyapunov Exponent ◽  
Unstable Manifolds

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

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.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

2021 ◽  
Vol 31 (10) ◽  
pp. 2150146
Author(s):  
Yuanyuan Si ◽  
Hongjun Liu ◽  
Yuehui Chen
Keyword(s):  
Phase Diagram ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Uniform Distribution ◽  
Bifurcation Diagram ◽  
Experimental Results ◽  
Kolmogorov Entropy ◽  
Symmetric Cryptography ◽  
Quadratic Map ◽  
Nonlinear Component

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

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