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.

Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator

2003 ◽  
Vol 9 (3-4) ◽  
pp. 281-315 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Finite Number ◽  
Control Method ◽  
Mathematical Problem ◽  
Melnikov Method ◽  
Excitation Amplitude ◽  
Homoclinic Bifurcation ◽  
Theoretical Treatment ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Basin Erosion

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

CHAOS AND CHAOTIC TRANSIENTS IN A FORCED MODEL OF THE ECONOMIC LONG WAVE: THE ROLE OF HOMOCLINIC BIFURCATION TO INFINITY

1995 ◽  
Vol 05 (03) ◽  
pp. 741-749 ◽  
Author(s):  
JEPPE STURIS ◽  
MORTEN BRØNS
Keyword(s):  
Limit Cycle ◽  
Basin Of Attraction ◽  
Homoclinic Bifurcation ◽  
Periodic Forcing ◽  
Stable And Unstable Manifolds ◽  
Long Wave ◽  
Unstable Manifolds ◽  
Chaotic Transients ◽  
The Basin Of Attraction

When an autonomous system of ordinary differential equations exhibits limit cycle behavior but is close in parameter space to a homoclinic bifurcation to infinity in which the limit cycle blows up to infinite amplitude and disappears, periodic forcing of the system may result in the appearance of both chaos and chaotic transients. In this paper, we use numerical techniques to map out Arnol’d tongues of a forced model of the economic long wave and illustrate how the system becomes chaotic and also exhibits chaotic transients for certain parameter combinations. Based on linearizations at infinity, we argue that infinity acts like a saddle with stable and unstable manifolds. By numerical computation, we show that chaotic transients occur when the manifolds intersect. Depending on parameters, two types of bifurcations have been identified: A chaotic attractor blows up to infinite size and disappears or the boundary of the basin of attraction of a periodic solution becomes fractal.

HOMOCLINIC BIFURCATION TO INFINITY IN A MODEL OF THE ECONOMIC LONG WAVE

1992 ◽  
Vol 02 (01) ◽  
pp. 129-136 ◽  
Author(s):  
MORTEN BRØNS ◽  
JEPPE STURIS
Keyword(s):  
Numerical Method ◽  
Limit Cycle ◽  
Consumer Goods ◽  
Formal Proof ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Economic Implications ◽  
New Model ◽  
Long Wave ◽  
Unstable Manifolds

In a homoclinic bifurcation to infinity, a limit cycle blows up to infinite amplitude and disappears. Based on linearizations at infinity, we argue that this bifurcation occurs in models of the Kondratieff economic long wave. A formal proof of the existence of stable and unstable manifolds at infinity is given. A numerical method is proposed and applied to a new model in which regulation of consumer goods production is taken into account. The economic implications of the analysis are discussed.

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.

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

Smooth conjugacy of centre manifolds

1992 ◽  
Vol 120 (1-2) ◽  
pp. 61-77 ◽  
Author(s):  
Almut Burchard ◽  
Bo Deng ◽  
Kening Lu
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Geometric Theory ◽  
Unstable Manifolds ◽  
Centre Manifolds ◽  
Smooth Conjugacy

SynopsisIn this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

Dynamics of an ecological system

2019 ◽  
Vol 10 (4) ◽  
pp. 355-376
Author(s):  
Shashi Kant
Keyword(s):  
Saddle Point ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Local Stability ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Trivial Equilibrium ◽  
Stability Results ◽  
Predator System

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

Sign in / Sign up

Export Citation Format

Share Document