Non-hyperbolic Equilibrium Point Research Articles

A Theorem for the Triple Zero bifurcation

Given a three-dimensional m-paremeterized vector field, which has a non-hyperbolic equilibrium point with a triple-zero eigenvalue with geometric multiplicity one, the called non-degeneracy and transversality conditions are given such that system undergoes the triple-zero bifurcation.

Basins of attraction of equilibrium points of second order difference equations

We investigate the basins of attraction of equilibrium points and period-two solutions of the difference equation of the form x n + 1 = f ( x n , x n − 1 ) , n = 0 , 1 , … , where f is decreasing in the first and increasing in the second variable. We show that the boundaries of the basins of attraction of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.

TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Non-Hyperbolic Equilibria in the Charged Collinear Three-Body Problem

Consider three charged masses moving along the line. For this model we study the solutions near total collision using blow up techniques obtaining that for given masses and charges the vector field on the collision manifold has a non-hyperbolic equilibrium point. To study this situation the vector field is written in normal form and the center manifold theory is used obtaining that all nonzero solutions near the origin escape to infinity.

Global behavior of a two-dimensional competitive system of difference equations with stocking

We investigate the global dynamics of solutions of competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of non-hyperbolic equilibrium points.

Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

Abstract We investigate global dynamics of the following systems of difference equations x n + 1 = α 1 + β 1 x n A 1 + y n y n + 1 = γ 2 y n A 2 + B 2 x n + y n , n = 0 , 1 , 2 , … where the parameters α 1, β 1, A 1, γ 2, A 2, B 2 are positive numbers, and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold. Mathematics Subject Classification (2000) Primary: 39A10, 39A11 Secondary: 37E99, 37D10

Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop

We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small time-periodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R 3 , whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff–Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called “ the chaos arising from infinity”, since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.

Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

We investigate global dynamics of the following systems of difference equations xn+1 = β1xn/(B1xn + yn), yn+1 = (α2 + γ2yn)/(A2 + xn), n = 0, 1, 2, …, where the parameters β1, B1, β2, α2, γ2, A2 are positive numbers, and initial conditions x0 and y0 are arbitrary nonnegative numbers such that x0 + y0 > 0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non‐hyperbolic equilibrium point.

Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of a type-zero saddle-node equilibrium point

Under the assumption that all equilibrium points are hyperbolic, the stability boundary of nonlinear autonomous dynamical systems is characterized as the union of the stable manifolds of equilibrium points on the stability boundary. The existing characterization of the stability boundary is extended in this paper to consider the existence of non-hyperbolic equilibrium points on the stability boundary. In particular, a complete characterization of the stability boundary is presented when the system possesses a type-zero saddle-node equilibrium point on the stability boundary. It is shown that the stability boundary consists of the stable manifolds of all hyperbolic equilibrium points on the stability boundary and of the stable manifold of the type-zero saddle-node equilibrium point.

Global Behavior of Four Competitive Rational Systems of Difference Equations in the Plane

We investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. (2009).

Global Dynamics of a Competitive System of Rational Difference Equations in the Plane

We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers such that . We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.

BIFURCATION AND STABILITY OF MARTENSITIC TRANSFORMATION DYNAMICS

In this paper, we have studied the mathematical properties and their physical implications of a system of nonlinear ordinary differential equations with two variables and six parameters, which was proposed to model the martensitic transformation of shape memory alloys. While the system could not have limit cycles, bifurcation to hysteresis was found in load and/or temperature controlled processes with large enough interfacial energies. This agrees qualitatively quite well with the experimental observation and theoretical understandings of the stress-strain-temperature hysteresis of the alloys. A three-dimensional bifurcation diagram was identified. Nonhyperbolic equilibrium points were found as saddle nodes and high order node points. The local behavior was studied and the phase portrait of the system was obtained for the load and temperature parameters. Accordingly, a stable node represents the stable martensitic or austenitic phase, a saddle stands for the unstable phase mixture, and a saddle node corresponds to the beginning or the end of the transformation. Therefore, for a thermal-mechanical loading path, the martensitic transformation process accords with the qualitative changes of the equilibrium points.

Singular perturbation approach to stability of a SIRS epidemic system

In this paper, a new method named multiple time scales is introduced to estimate the domain of attraction of the endemic equilibrium point of J. Mena-lorca and H. W. Hethcote’s SIRS model. Simultaneously, the recurrence formulae established by S. Balint, A. Balint and V. Negru are adopted to construct Lyapunov function for the reduced system. Furthermore, the stability for the non-hyperbolic equilibrium point of the reduced system is discussed. In the end, numerical simulations are carried out.

Analysis of a bacteria–immunity model with delay quorum sensing

A bacteria–immunity model with bacterial quorum sensing is formulated, which describes the competition between bacteria and immune cells. A distributed delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In this paper, we focus on a subsystem of the bacteria–immunity model, analyze the stability of the equilibrium points, discuss the existence and stability of periodic solutions bifurcated from the positive equilibrium point, and finally investigate the stability of the nonhyperbolic equilibrium point by the center manifold theorem.

The structure of slow invariant manifolds and their bifurcational routes in chemical kinetic models

This article analyzes the global geometric properties of slow invariant manifolds in two-dimensional chemical kinetic models. By enforcing the concept of Lyapunov-type numbers, a classification of slow manifolds into global and generalized structures is obtained, and applied to explain the occurrence of different dynamic phenomena. Several related concepts such as stretching heterogeneity and α – ω inversion are introduced and commented by taking the Semenov system as a paradigmatic example. We show that the existence of a global slow manifold along with its properties are controlled by a transcritical bifurcation of the points at infinity, that can be readily identified by analyzing the Poincaré projected (Pp) system. The information that can be obtained from the analysis of the Pp-system, and specifically the presence of saddle-points on the Poincaré circle, are extremely helpful in the construction of a complete picture of the structure and properties of slow invariant manifolds even when the system exhibits non-hyperbolic equilibrium points or stable limit cycles.

Invariant closed surface and stability of non-hyperbolic equilibrium point for polynomial differential systems in [formula omitted

In this paper, by algebraic method and Lyapunov function, we discuss the stability of non-hyperbolic equilibrium point in R 3 , that the coefficient matrix of linearized system have a pair purely imaginary eigenvalues and a zero eigenvalue, with the perturbations of 3th-degree homogeneous and 3th-degree and 5th-degree homogeneous. We shall give the sufficiently conditions which can immediately distinguish that the equilibrium point is asymptotically stable or unstable and a ball-center by the coefficients of perturbed terms, meantime, we discuss the condition which produce invariant closed surface by changing the stability of equilibrium point with perturbation.

Hydrodynamic limit for perturbation of a hyperbolic equilibrium point in two-component systems

We consider one-dimensional, locally finite interacting particle systems with two conservation laws. The models have a family of stationary measures with product structure and we assume the existence of a uniform bound on the inverse of the spectral gap which is quadratic in the size of the system. Under Eulerian scaling the hydrodynamic limit for the macroscopic density profiles leads to a two-component system of conservation laws. The resulting pde is hyperbolic inside the physical domain of the macroscopic densities, with possible loss of hyperbolicity at the boundary. We investigate the propagation of small perturbations around a hyperbolic equilibrium point. We prove that the perturbations essentially evolve according to two decoupled Burgers equations. The scaling is not Eulerian: if the lattice constant is n −1 , the perturbations are of order n − β then time is speeded up by n 1 + β . Our derivation holds for 0 < β < 1 5 . The proof relies on Yau's relative entropy method, thus it applies only in the regime of smooth solutions. This result is an extension of [T. Seppäläinen, Perturbation of the equilibrium for a totally asymmetric stick process in one dimension, Ann. Probab. 29 (2001) 176–204] and [B. Tóth, B. Valkó, Between equilibrium fluctuations and Eulerian scaling. Perturbation of equilibrium for a class of deposition models, J. Statist. Phys. 109 (2002) 177–205] where the analogue result was proved for systems with one conservation law. It also complements [B. Tóth, B. Valkó, Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit, Commun. Math. Phys. 256 (2005) 111–157] where it was shown that perturbations around a nonhyperbolic boundary equilibrium point are driven by a universal two-by-two system of conservation laws.

Homoclinic Bifurcations with Nonhyperbolic Equilibria

A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong $\lambda $-lemma, and Lyapunov–Schmidt reduction for the Poincare maps under Sil’nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.

Center manifolds and contractions on a scale of Banach spaces

We give a new proof for the existence of a C k -center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vector field of class C k . The problem is reduced to a fixed point problem on a scale of Banach spaces; these Banach spaces consist of mappings with a certain maximal exponential growth at infinity. We give conditions under which there is a unique fixed point depending differentiably on the parameters; the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of Banach spaces.