Hyperbolic Points Research Articles

Stability of Periodic Points of a Diffeomorphism of a Plane in a Homoclinic Orbit

We considered a self-diffeomorphism of the plane with a fixed hyperbolic point at the origin and a non-transverse point homoclinic to it. Periodic points located in a sufficiently small neighborhood of the homoclinic point are divided into single-pass and multi-pass points depending on the location of the orbit of the periodic point with respect to the orbit of the homoclinic point. It follows from the works of W. Newhouse, L.P. Shil’nikov, B.F. Ivanov and other authors that for a certain method of tangency of the stable and unstable manifolds there can be an infinite set of stable periodic points in a neighborhood of a non-transverse homoclinic point, but at least one of the characteristic exponents of these points tends to zero with increasing period. Previous works of the author imply that for a different method of tangency of the stable and unstable manifolds there can be an infinite set of stable single-pass periodic points, the characteristic exponents of which are bounded away from zero in the neighborhood of a non-transverse homoclinic point. It is shown in this paper that under certain conditions imposed primarily on the method of tangency of the stable and unstable manifolds there can be a countable set of two-pass stable periodic points, the characteristic exponents of which are bounded away from zero in any neighborhood of a non-transverse homoclinic point.

Classification of rough transformations of a circle from a modern point of view

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

Ejection-Collision orbits in the symmetric collinear four–body problem

In this paper, we consider the collinear symmetric four-body problem, where four masses m3=α,m1=1,m2=1, and m4=α,α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejection-collision orbits, for any value of α. Some results from the explorations done for α=1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions.

Dynamical system analysis of quintom dark energy model

The present work deals with dynamical system analysis of a quintom model of dark energy. By suitable transformation of variables the Einstein field equations are converted to an autonomous system. The critical points are determined and stability of hyperbolic critical points are determined by Hartman–Grobman theorem. To analyze non-hyperbolic critical points different tools (notably center manifold theory) are used. Possible bifurcation scenarios have also been explained.

Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations

This paper is devoted to the numerical analysis of the abstract semilinear fractional problem $$D^\alpha u(t) = Au(t) + f(u(t)), u(0)=u^0,$$ in a Banach space E. We are developing a general approach to establish a semidiscrete approximation of stable manifolds. The phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to systems of initial value problems in the invariant subspaces corresponding to positive and negative real parts of the spectrum. We show that such a decomposition of the equation keeps the same structure under general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and can be verified for finite element as well as finite difference methods.

An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, $\{e^{mΔ}:\; m∈ \mathbb{N}\}$, where $Δ$ is the Laplace operator on the unbounded domain $\mathbb{R}^{n}\setminus \{0\}$.Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

Bi-Orbital States in Hyperbolic Traps

We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is the sum of a hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under the resonance 1 : (−1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, which define distinct classically allowed regions in the space of motion constants of the ideal trap. The corresponding stationary states of the trapped charge can form bi-orbital states, i.e., a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.

Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

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.

Assessment of a miniature four-roll mill and a cross-slot microchannel for high-strain-rate stagnation point flows

Stagnation point flows have been widely used to study the deformation and break-up of objects in two-dimensional pure straining flows. Here, we report a systematic study of the characterisation of stagnation point flows in two devices, a miniature Taylor’s four-roll mill and a cross-slot microchannel. The aim of the study is to find the best platform suitable for investigating the effect of strain rate on the mechanical properties of waterborne microorganisms. Using micro-PIV, the velocity field and the strain rates in both devices were measured at different flow rates and compared with an ideal hyperbolic stagnation point flow. The cross-slot microchannel was found to be a better experimental device than the miniature four-roll mill for the purpose of confining micron-sized objects in a controlled stagnation point flow. This is mainly due to the difficulty of maintaining a fixed location for the stagnation point within one micron in the miniature four-roll mill and achieving high strain rates beyond 10 . However, with no moving parts, the cross-slot microchannel was found to maintain a steady flow, with the stagnation point varying less than one micron at a cross-junction of , and was able to reach uniform strain rates up to 140 .

Cheap complex limit cycles

Consider a holomorphic foliation with singularities of a two-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have an infinite set of homologically independent complex limit cycles. In particular, if all leaves of a foliation are dense in the phase space, and it has a complex hyperbolic singular point, then it has an infinite set of homologically independent complex limit cycles.

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.

Finite time blow up in the hyperbolic Boussinesq system

In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

Statistical independence of the initial conditions in chaotic mixing.

Experimental evidence of the scalar convergence towards a global strange eigenmode independent of the scalar initial condition in chaotic mixing is provided. This convergence, underpinning the independent nature of chaotic mixing in any passive scalar, is presented by scalar fields with different initial conditions casting statistically similar shapes when advected by periodic unsteady flows. As the scalar patterns converge towards a global strange eigenmode, the scalar filaments, locally aligned with the direction of maximum stretching, as described by the Lagrangian stretching theory, stack together in an inhomogeneous pattern at distances smaller than their asymptotic minimum widths. The scalar variance decay becomes then exponential and independent of the scalar diffusivity or initial condition. In this work, mixing is achieved by advecting the scalar using a set of laminar flows with unsteady periodic topology. These flows, that resemble the tendril-whorl map, are obtained by morphing the forcing geometry in an electromagnetic free surface 2D mixing experiment. This forcing generates a velocity field which periodically switches between two concentric hyperbolic and elliptic stagnation points. In agreement with previous literature, the velocity fields obtained produce a chaotic mixer with two regions: a central mixing and an external extensional area. These two regions are interconnected through two pairs of fluid conduits which transfer clean and dyed fluid from the extensional area towards the mixing region and a homogenized mixture from the mixing area towards the extensional region.

Mixing and transport from combined stretching-and-folding and cutting-and-shuffling.

While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for nonmixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency. As a result, the maximum size of the nonmixing region occurs when the island is at the bifurcation point. We also demonstrate a bifurcation where periodic points are annihilated by the cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points that arise when invariant tori surrounding an elliptic point break up [according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they meet a cutting line. Consequently, the Poincaré index (a topological invariant of smooth systems) is not preserved.

Stability and Perturbations of Generalized Heteroclinic Loops in Piecewise Smooth Systems

We investigate a class of planar piecewise smooth systems with a generalized heteroclinic loop (a closed curve composed of hyperbolic saddle points, generalized singular points and regular orbits). We give conditions for the stability of the generalized heteroclinic loop and provide some sufficient conditions for the maximum number of limit cycles that bifurcate from the heteroclinic connection. The discussions rely on the approximation of the Poincare map, which is constructed near the generalized heteroclinic loop. To obtain it, we introduce the Dulac map and use Melnikov method. By analyzing the fixed point of the Poincare map, we get the number of limit cycles, which can be produced from the generalized heteroclinic loop. As applications to our theories, we give an example to show that two limit cycles can appear.

Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass

We establish the asymptotic completeness in the nonlinear Lamb system with nonzero mass for hyperbolic stationary states. For the proof, we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point, using the Banach space Inverse Function Theorem and a priori estimates. We give a counterexample showing that the hyperbolicity condition is essential.

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.