Global Stable Manifolds Research Articles

A generalized flux function for three-dimensional magnetic reconnection

The definition and measurement of magnetic reconnection in three-dimensional magnetic fields with multiple reconnection sites is a challenging problem, particularly in fields lacking null points. We propose a generalization of the familiar two-dimensional concept of a magnetic flux function to the case of a three-dimensional field connecting two planar boundaries. In this initial analysis, we require the normal magnetic field to have the same distribution on both boundaries. Using hyperbolic fixed points of the field line mapping, and their global stable and unstable manifolds, we define a unique flux partition of the magnetic field. This partition is more complicated than the corresponding (well-known) construction in a two-dimensional field, owing to the possibility of heteroclinic points and chaotic magnetic regions. Nevertheless, we show how the partition reconnection rate is readily measured with the generalized flux function. We relate our partition reconnection rate to the common definition of three-dimensional reconnection in terms of integrated parallel electric field. An analytical example demonstrates the theory and shows how the flux partition responds to an isolated reconnection event.

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

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.

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation x ¨ = f ( t , x ) be periodic in time, and let the equilibrium x ∗ ≡ 0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ( x , x ˙ ) ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C 2 equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C 1 global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.

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.

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).

Toward the Calculation of Higher-Dimensional Stable Manifolds and Stable Sets for Noninvertible and Piecewise-Smooth Maps

This paper presents a new numerical method for computing global stable manifolds and global stable sets of nonlinear discrete dynamical systems. For a given map f:ℝ d →ℝ d , the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region M⊂ℝ d . The algorithm divides the region M in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. In contrast to similar approaches, the method requires neither the system’s inverse nor its Jacobian. Hence, it can also be applied to noninvertible and piecewise-smooth maps. The successful application of the method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.

A note on unbounded solutions of a class of second order rational difference equations

We investigate the unbounded solutions of the second order difference equation where all parameters and C and initial conditions are nonnegative and such that for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC).

Competitive-exclusion versus competitive-coexistence for systems in the plane

We investigate global behavior of $x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E) where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive(monotone with respect to the south-east ordering) map on a set$\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assumethe existence of a unique fixed point $\overline{e}$ in theinterior of $\mathcal{ R}$. We give very general conditions whichare easily verifiable for (E) to exhibit eithercompetitive-exclusion or competitive-coexistence. More specifically,we obtain sufficient conditions for the interior fixed point$\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is arectangular region. We also show that when $T$ is strongly monotonein $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{R}$ is convex, the unique interior equilibrium $\overline{ e}$ isa saddle, and a technical condition is satisfied, the correspondingglobal stable and unstable manifolds are the graphs of monotonicfunctions, and the global stable manifold splits the domain into twoconnected regions, which under additional conditions on$\mathcal{R}$ and on $T$ are shown to be basins of attraction offixed points on the boundary of $\mathcal{R}$. Applications of themain results to specific difference equations are given.

Asymptotic behavior of a competitive system of linear fractional difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+x n )/(b+y n ), yn+1 = (d+y n )/(e+x n ), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

On the global stable manifold

We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms.

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Numerical approximations of strong (un)stable manifolds

The method of computing global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field is well known. Such manifolds consist only of two trajectories and arbitrarily large pieces can, for example, be generated using an initial point close to the equilibrium on the linear approximation of the manifold. The attraction properties (in forward or backward time) of the local manifolds ensure that the computational error, which depends on the arclength of the computed piece, remains bounded. This paper discusses how these error bounds change as the equilibrium loses its hyperbolicity, or when the one-dimensional, say, unstable manifold is, in fact, a ‘strong’ unstable manifold that is contained in a higher-dimensional unstable manifold. For these cases, the local manifolds are not locally attracting either in forward or in backward time and the standard error bound does not work. We illustrate the theoretical analysis with numerical computations, using an example for which the global manifolds can be found explicitly, as well as more general vector fields where the true manifolds are not known.

The Lorenz manifold as a collection of geodesic level sets

We demonstrate a method to compute a two-dimensional global stable or unstable manifold of a vector field as a sequence of approximate geodesic level sets. Specifically, we compute the Lorenz manifold—the two-dimensional stable manifold of the origin of the well-known Lorenz equations—which has emerged as a test example for manifold computations. The information given by the geodesic level sets can be used to visualize and understand the geometry of the Lorenz manifold, and one such visualization can be seen as the cover illustration.

Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields

Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and models of spiking neurons. The global dynamics of such a system is organized by the stable and unstable manifolds of the saddle points, of the saddle periodic orbits, and, more generally, of all compact invariant manifolds of saddle type. Except in very special circumstances the (un)stable manifolds are global objects that cannot be found analytically but need to be computed numerically. This is a nontrivial task when the dimension of the manifold is larger than one. In this paper we present an algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or a more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1< k < n. Stable manifolds are computed by considering the flow for negative time. The key idea is to view the unstable manifold as a purely geometric object, hence disregarding the dynamics on the manifold, and compute it as a list of approximate geodesic level sets, which are (topological) (k-1)-spheres. Starting from a (k-1)-sphere in the linear eigenspace of the equilibrium or periodic orbit, the next geodesic level set is found in a local (and changing) coordinate system given by hyperplanes perpendicular to the last geodesic level set. In this setup the mesh points defining the approximation of the next geodesic level set can be found by solving boundary value problems. By appropriately adding or removing mesh points it is ensured that the mesh that represents the computed manifold is of a prescribed quality. The algorithm is presently implemented to compute two-dimensional manifolds in a phase space of arbitrary dimension. In this case the geodesic level sets are topological circles and the manifold is represented as a list of bands between consecutive level sets. We use color to distinguish between consecutive bands or to indicate geodesic distance from the equilibrium or periodic orbit, and we also show how geodesic level sets change with increasing geodesic distance. This is very helpful when one wants to understand the often very complicated embeddings of two-dimensional (un)stable manifolds in phase space. The properties and performance of our method are illustrated with several examples, including the stable manifold of the origin of the Lorenz system, a two-dimensional stable manifold in a four-dimensional phase space arising in a problem in optimal control, and a stable manifold of a periodic orbit that is a Möbius strip. Each illustration is accompanied by an animation (supplied with this paper).

Fatou maps in ℙn dynamics

We study the dynamics of a holomorphic self‐map f of complex projective space of degree d &gt; 1 by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics of f are a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified by f (and therefore any hyperbolic periodic point attracts a point of the critical set of f). We also show that Fatou components are hyperbolically embedded in ℙn and that a Fatou component which is attracted to a taut subset of itself is necessarily taut.

Globalizing Two-Dimensional Unstable Manifolds of Maps

We present an algorithm for computing the global two-dimensional unstable manifold of a hyperbolic fixed point or a normally hyperbolic invariant circle of a three-dimensional map. The global stable manifold can be obtained by considering the inverse map. Our algorithm computes intersections of the unstable manifold with a finite number of leaves of a chosen linear foliation. In this way, we obtain a growing piece of the unstable manifold represented by a mesh of prescribed quality. The performance of the algorithm is demonstrated with several examples.

Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system

This is a further study of the set of homoclinic solutions (i.e., nonzero solutions asymptotic to 0 as ¦x¦→∞) of the reversible Hamiltonian systemuiv +Pu″ +u−u2=0. The present contribution is in three parts. First, rigorously for P≤ −2, it is proved that there is a unique (up to translation) homoclinic solution of the above system, that solution is even, and on the zero-energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifolds. WhenP=−2 the origin is a node on its local stable and unstable manifolds, and whenP∈(−2,2) it is a focus. Therefore we can infer, rigorously, from the discovery by Devaney of a Smale horseshoe in the dynamics on the zero energy set, there are infinitely many distinct infinite families of homoclinic solutions forP∈(−2, −2+e) for somee>0. Buffoni has shown globally that there are infinitely many homoclinic solutions for allP∈(−2,0], based on a different approach due to Champneys and Toland. Second, numerically, the development of the set of symmetric homoclinic solutions is monitored asP increases fromP=−2. It is observed that two branches extend fromP=−2 toP=+2 where their amplitudes are found to converge to 0 asP ↗ 2. All other symmetric solution branches are in the form of closed loops with a turning point betweenP=−2 andP=+2. Numerically it is observed that each such turning point is accompanied by, though not coincident with, the bifurcation of a branch of nonsymmetrical homoclinic orbits, which can, in turn, be followed back toP=−2. Finally, heuristic explanations of the numerically observed phenomena are offered in the language of geometric dynamical systems theory. One idea involves a natural ordering of homoclinic orbits on the stable and unstable manifolds, given by the Horseshoe dynamics, and goes some way to accounting for the observed order (in terms ofP-values) of the occurrence of turning points. The near-coincidence of turning and asymmetric bifurcation points is explained in terms of the nontransversality of the intersection of the stable and unstable manifolds in the zero energy set on the one hand, and the nontransversality of the intersection of the same manifolds with the symmetric section in ℝ4 on the other. Some conjectures based on present understanding are recorded.

Codimension-one partitioning and phase space transport in multi-degree-of-freedom Hamiltonian systems with non-toroidal invariant manifold intersections

Invariant manifold formulations of phase-space transport theory offer a well-known means of prediction for the chaotic dynamics associated with 1 1 2 - and 2-degree-of-freedom systems. Whether such a means of prediction extends to systems with more than 2 degrees-of-freedom has thus far received little attention. We derive a multi-degree-of-freedom turnstile-based phase space transport theory for Hamiltonian flows with non-toroidal intersections of global stable and unstable manifolds of invariant normally hyperbolic sets of a Poincaré section on the energy surface. These non-toroidal manifold intersections are the typical case for flows with more than 2 degrees-of-freedom, and they disallow a partitioning of phase space via segments of stable and unstable manifolds alone. We show it is nevertheless possible for these intersection geometries to construct codimension-one partitions (dividing surfaces) of the Poincaré section that respect asymptotic behavior, and hence divide between regions of qualitatively different dynamics (bounded versus unbounded, rotation versus libration, and so on). These partitioning surfaces thus provide a geometrically exact criterion for capture into, and escape from, regions in the Poincaré section corresponding to qualitatively different dynamics. A turnstile-based phase space transport theory is then derived to describe transitions across the partitioning surfaces. We use Melnikov theory to study partition geometry, and to search through multi-dimensional phase space and parameter space to explore variation of O( ε) flux across the partition. Searches of this kind would be helpful in the control and optimization of any number of nonlinear physical processes associated with transitions across broken separatrices, such as molecular dissociation and atomic ionization. We observe significant variation in the effectiveness of separatrix crossing with differing phase and action values, and hence note the motivation to retain phase information, and the role of Arnold diffusion in sending points to regions of enhanced or diminished separatrix crossing, referred to as Arnold enhancement and Arnold suppression of transport across codimension-one partitions.