Hyperbolic Equilibrium Points Research Articles

On the viability of Planck scale cosmology with quartessence

Growing evidence as the observations of the CMB (cosmic microwave background), galaxy clustering and high-redshift supernovae address a stable dynamically universe dominated by the dark components. In this paper, using a qualitative theory of dynamical systems, we study the stability of a unified dark matter-dark energy framework known as quartessence Chaplygin model (QCM) with three different equation-of-states within ultraviolet (UV) deformed Friedmann–Robertson–Walker (FRW) cosmologies without Big-Bang singularity. The UV deformation is inspired by the non-commutative (NC) Snyder spacetime approach in which by keeping the transformation groups and rotational symmetry there is a dimensionless, Planck scale characteristic parameter mu _0 with dual implications dependent on its sign that addresses the required invariant cutoffs for length and momentum in nature, in a separate manner. Our stability analysis is done in the (H,rho ) phase space at a finite domain concerning the hyperbolic critical points. According to our analysis, due to constraints imposed on the signs of mu _0 from the phenomenological parameters involved in quartessence models (Omega _m^*, c_s^2, rho _*), for an expanding and accelerating late universe, all three QCMs can be stable in the vicinity of the critical points. The requirement of stability for these quartessence models in case of admission of a minimum invariant length, can yield a flat as well as non-flat expanding and accelerating universe in which Big-Bang singularity is absent. This feedback also phenomenologically credits to braneworld-like framework versus loop quantum cosmology-like one as two possible scenarios which can be NC Snyder spacetime generators (correspond to mu _0<0 and mu _0>0, respectively). As a result, our analysis show that between quartessence models with Chaplygin gas equation-of-states and accelerating FRW backgrounds occupied by a minimum invariant length, there is a possibility of viability.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Dynamics and control of high area-to-mass ratio spacecraft and its application to geomagnetic exploration

This paper studies the dynamics and control of a spacecraft, whose area-to-mass ratio is increased by deploying a reflective orientable surface such as a solar sail or a solar panel. The dynamical system describing the motion of a non-zero attitude angle high area-to-mass ratio spacecraft under the effects of the Earth's oblateness and solar radiation pressure admits the existence of equilibrium points, whose number and the eccentricity values depend on the semi-major axis, the area-to-mass ratio and the attitude angle of the spacecraft together. When two out of three parameters are fixed, five different dynamical topologies successively occur through varying the third parameter. Two of these five topologies are critical cases characterized by the appearance of the bifurcation phenomena. A conventional Hamiltonian structure-preserving (HSP) controller and an improved HSP controller are both constructed to stabilize the hyperbolic equilibrium point. Through the use of a conventional HSP controller, a bounded trajectory around the hyperbolic equilibrium point is obtained, while an improved HSP controller allows the spacecraft to easily transfer to the hyperbolic equilibrium point and to follow varying equilibrium points. A bifurcation control using topologies and changes of behavior areas can also stabilize a spacecraft near a hyperbolic equilibrium point. Natural trajectories around stable equilibrium point and these stabilized trajectories around hyperbolic equilibrium point can all be applied to geomagnetic exploration.

Dynamics, implementation and stability of a chaotic system with coexistence of hyperbolic and non-hyperbolic equilibria

Hyperbolic-type equilibrium requires that all the real parts of the corresponding eigenvalues are nonzero. In this paper, a three-dimensional autonomous chaotic system is introduced, and interestingly we find that one non-hyperbolic equilibrium point and two hyperbolic equilibrium points coexist in this system, which, according to the information we know, has not been previously reported. We first reveal the basic dynamics of the system through analyzing phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent. Then, based on the idea of the improved modular technology, we build an analog circuit to realize the chaotic system, which further verifies the theoretical results. Finally, we design a simple feedback controller on account of Lyapunov asymptotic stability theory, to globally suppress the system to its equilibrium points.

Bifurcations of Heteroclinic Loop with Twisted Conditions

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

Boundary crisis for degenerate singular cycles

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)–(Q3) of Knobloch (2008 Nonlinearity 21 45–60). Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

ABSTRACTA complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.

Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold

This technical note addresses the structure preserving stabilization by output feedback of conservative input-output contact systems, a class of input-output Hamiltonian systems defined on contact manifolds. In the first instance, achievable contact forms in closed-loop and the associated Legendre submanifolds are analysed. In the second instance the stability properties of a hyperbolic equilibrium point of a strict contact vector field are analysed and it is shown that the stable and unstable manifolds are Legendre submanifolds. In the third instance the consequences for the design of stable structure preserving output feedback are derived: in closed-loop one may achieve stability only relatively to some invariant Legendre submanifold of the closed-loop contact form and furthermore this Legendre submanifold may be used as a control design parameter. The results are illustrated along the technical note on the example of heat transfer between two compartments and a controlled thermostat.

Subcritical Hopf Equilibrium Points in Boundary of the Stability Region

A complete characterization of the boundary of the stability region of a class of nonlinear autonomous dynamical systems is developed admitting the existence of Subcritical Hopf nonhyperbolic equilibrium points on the boundary of the stability region. The characterization of the stability region developed in this paper is an extension of the characterization already developed in the literature, which considers only hyperbolic equilibrium point. Under the transversality condition, it is shown the boundary of the stability region is comprised of the stable manifolds of all equilibrium points on the boundary of the stability region, including the stable manifolds of the subcritical Hopf equilibrium points of type k, with 0&lt;=k&lt;=(n-2), which belong to the boundary of the stability region.

Stability of equilibrium points of demand-inventory model with stock-dependent demand

A model of demand and inventory of a product in one echelon of supply chain is considered. The model is formulated as a system of difference equations, in which every equilibrium point is nonhyperbolic. A positive invariant set of the system is constructed. An analysis of properties of equilibrium points of the system is based on the Lyapunov method or reducing it to the family of systems of difference equations with hyperbolic equilibrium points.

Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation

The present work deals with numerical methods for computing slow stableinvariant manifolds as well as their invariant stable and unstable normalbundles. The slow manifolds studied here are sub-manifolds of the stablemanifold of a hyperbolic equilibrium point. Our approach is based on studyingcertain partial differential equations equations whose solutions parameterizethe invariant manifolds/bundles. Formal solutions of the partial differentialequations are obtained via power series arguments, and truncating the formalseries provides an explicit polynomial representation for the desired chartmaps. The coefficients of the formal series are given by recursion relationswhich are amenable to computer calculations. The parameterizations conjugatethe dynamics on the invariant manifolds and bundles to a prescribed lineardynamical systems. Hence in addition to providing accurate representation ofthe invariant manifolds and bundles our methods describe the dynamics on theseobjects as well. Example computations are given for vector fields which ariseas Galerkin projections of a partial differential equation. As an application weillustrate the use of the parameterized slow manifolds and their linear bundlesin the computation of heteroclinic orbits.

A Geometric Approach to Stationary Defect Solutions in One Space Dimension

In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on $\mathbb{R}$. This problem corresponds to analyzing a discontinuous and non-autonomous $n$-dimensional system, $\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.$ under the assumption that the unperturbed system, i.e., the $\varepsilon \to 0$ limit system, possesses a heteroclinic orbit $\Gamma$ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit $\Gamma$ represents a localized structure in the PDE setting. We define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that $\lim_{\varepsilon \to 0} \Gamma_\varepsilon = \Gamma$ (as graphs). We distinguish between three types of defect solutions: trivial, local, and global defect solutions. The main goal of this manuscript is to develop a comprehensive and asymptotically explicit theory of the existence of local defect solutions. We find that both the dimension of the problem as well as the nature of the linearized system near the endpoints of the heteroclinic orbit $\Gamma$ have a remarkably rich impact on the existence of these local defect solutions. We first introduce the various concepts in the setting of planar systems $(n=2)$ and---for reasons of transparency of presentation---consider the three-dimensional problem in full detail. Then, we generalize our results to the $n$-dimensional problem, with special interest for the additional phenomena introduced by having $n \geq 4$. We complement the general approach by working out two explicit examples in full detail: (i) the existence of pinned local defect kink solutions in a heterogeneous Fisher--Kolmogorov equation ($n=4$) and (ii) the existence of pinned local defect front and pulse solutions in a heterogeneous generalized FitzHugh--Nagumo system $(n=6)$.

Local stable manifold theorem for fractional systems

In this paper we prove existence of local stable manifold around a hyperbolic equilibrium point of a fractional system. We illustrate this result with an example and plot corresponding local stable manifold.

Existence of nodal solutions for quasilinear elliptic problems in ℝN

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

Strange attractors with various equilibrium types

Of the eight types of hyperbolic equilibrium points in three-dimensional flows, one is overwhelmingly dominant in dissipative chaotic systems. This paper shows examples of chaotic systems for each of the eight types as well as one without any equilibrium and two that are nonhyperbolic. The systems are a generalized form of the Nose-Hoover oscillator with a single equilibrium point. Six of the eleven cases have hidden attractors, and six of them exhibit multistability for the chosen parameters.

Dynamics of the parabolic restricted three-body problem

The main purpose of the paper is the study of the motion of a massless body attracted, under the Newton’s law of gravitation, by two equal masses moving in parabolic orbits all over in the same plane, the planar parabolic restricted three-body problem. We consider the system relative to a rotating and pulsating frame where the equal masses (primaries) remain at rest. The system is gradient-like and has exactly ten hyperbolic equilibrium points lying on the boundary invariant manifolds corresponding to escape of the primaries in past and future time. The global flow of the system is described in terms of the final evolution (forwards and backwards in time) of the solutions. The invariant manifolds of the equilibrium points play a key role in the dynamics. We study the connections, restricted to the invariant boundaries, between the invariant manifolds associated to the equilibrium points. Finally we study numerically the connections in the whole phase space, paying special attention to capture and escape orbits.

Successive approximation: A survey on stable manifold of fractional differential systems

Abstract This paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

Expansive flows of the three-sphere

In this article we show that the three-dimensional sphere admits transitive expansive flows in the sense of Komuro with hyperbolic equilibrium points. The result is based on a construction that allows us to see the geodesic flow of a hyperbolic three-punctured two-dimensional sphere as the flow of a smooth vector field on the three-dimensional sphere.

Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u 0, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10, 11]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.