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In this paper, we discuss the existence of a center stable manifold for planar fractional damped equations. By constructing a suitable LyapunovPerron operator via giving asymptotic behavior of MittagLeffler function, we obtain an interesting center stable manifold theorem. Finally, an example is provided to illustrate the result.

In this article we analyze n -dimensional slow–fast systems in a piecewise linear framework. In particular, we prove a Fenichel's-like Theorem where we give an explicit expression for the invariant slow manifold, that leads to the proof of the existence and location of maximal canards orbits. We show that these orbits perturb from singular orbits through contact points, of order greater than or equal to two, between the reduced flow and the fold manifold. In the particular case n = 3 , we show that the unique contact point is a visible two-fold singularity.

International audience; This paper considers near-equilibrium systems of ordinary differential equations with explicit separation of the slow and stable manifolds. Formal B-series like those previously used to analyze highly-oscillatory systems or to construct modified equations are employed here to construct expansions of the change of variables, the center invariant manifold and the reduced model. The new approach may be seen as a process of reduction to a normal form, with the main advantage, as compared to the standard view conveyed by the celebrated center manifold theorem, that it is possible to recover the complete solution at any time through an explicit change of variables.

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.

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the `physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors., Comment: 6 pages, 3 pdf figures, uses revtex4

In this paper, the homotopy perturbation method is applied to approximate the stable manifold of a nonlinear autonomous system of ordinary differential equations. The results reveal that the proposed method is very effective and simple.

A new algorithm is presented to compute both one dimensional stable and unstable manifolds of planar maps. The global manifold is grown from a local manifold and one point is added each step. It is proved that the gradient of the global manifold can be predicted by the known points on the manifold with a gradient prediction scheme and it can be used to locate the image or preimage of the new point quickly. A new accuracy criterion is derived from the gradient prediction scheme. The performance of the algorithm is tested with several well-known examples.

The new algorithm uses the idea of growing the manifold. The preimage of the new point is found quickly with a method called gradient prediction scheme and a new accuracy criterion is proposed. Furthermore, our algorithm is capable of computing both stable and unstable one dimensional manifold.

A new algorithm is presented for computing one dimensional unstable manifold of a map and Hénon map is taken as an example to test the performance of the algorithm. The unstable manifold is grown with new point added at each step and the distance between consecutive points is adjusted according to the local curvature. It is proved that the gradient of the manifold at the new point can be predicted by the known points on the manifold and in this way the preimage of the new point could be located immediately. During the simulation, it is found that the unstable manifold of Hénon map coincides with its direct iteration when canonical parameters are chosen which means order is obtained out of chaos. In the other several groups of parameters the two branches of the unstable manifolds are nearly symmetric, and they serve as the borderline of the Hénon map iteration sequence. We hope that this would contribute to the further exploration of Hénon map.

This study proposes an analytical method that determines the center manifold and identifies the reduced system on the center manifold. The proposed method expresses the center manifold through general equations containing only state variables, and not functions with respect to time. This is the so-called geometric structure of the center manifold. The location of periodic or quasi-periodic orbits is identified after the geometric structure of the center manifold is determined. The reduced system on the center manifold is described using ordinary differential equations, so that periodic or quasi-periodic orbits can be computed by numerically integrating the reduced system. The results indicate that the analytical method proposed in this study has higher precision compared with the Lindstedt-Poincare method of the same order.

In this paper we propose a numerical method for approximating connecting orbits on a manifold and its bifurcation parameters. First we extend the standard nondegeneracy condition to the connecting orbits on a manifold. Then we construct a well-posed system such that the nondegenerate connecting orbit pair on a manifold is its regular solution. We use a difference method to discretize the ODE part in this well-posed system and we find that the numerical solutions still remain on the same manifold. We also set up a modified projection boundary condition to truncate connecting orbits on a manifold onto a finite interval. Then we prove the existence of truncated approximate connecting orbit pairs and derive error estimates. Finally, we carry out some numerical experiments to illustrate the theoretical estimates.

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local. We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.

A method of greatly decreasing the activation time of a control method based on stable manifold information is proposed.

We show how to effectively link covering relations with cone conditions. We give a new, ‘geometric,’ proof of the stable manifold theorem for hyperbolic fixed point of a map.

We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.

We obtain a necessary as well as a sufficient condition for the existence of bifurcation points of a coincidence equation, and, in particular, of a parametrized fixed point problem. In both cases the trivial solutions are assumed to form a finite-dimensional submanifold of a Banach manifold. An application is given to a delay differential equation on a manifold: we detect periodic solutions that rotate close to an equilibrium point.

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional systems, even infinite dimensional systems in the case of a PDE with a low-dimensional attractor. The theory of global attractors approaches these issues analytically, and focuses on finding (depending on the question at hand), a slow-manifold, inertial manifold, or center manifold, on which a restricted dynamical system represents the interesting behavior of the dynamical system; the main issue depends on defining a stable invariant manifold in which the dynamical system is invariant. These approaches are analytical in nature, however, and are therefore not always appropriate for dynamical systems known only empirically through a dataset. Empirically, the collection of tools available are much more restricted, and are essentially linear in nature. Usually variants of Galerkin's method, project the dynamical system onto a function linear subspace spanned by modes of some chosen spanning set. Even the popular Karhunen–Loeve decomposition, or POD, method is exactly such a method. As such, it is forced to either make severe errors in the case that the invariant space is intrinsically a highly nonlinear manifold, or bypass low-dimensionality by retaining many modes in order to capture the manifold. In this work, we present a method of modeling a low-dimensional nonlinear manifold known only through the dataset. The manifold is modeled as a discrete graph structure. Intrinsic manifold coordinates will be found specifically through the ISOMAP algorithm recently developed in the Machine Learning community originally for purposes of image recognition.

An essentially nonlinear system of differential equations (i.e., a system without linear terms) is considered. It is proved that there exists a local-integral manifold of neutral type (central manifold) near an equilibrium point. Coefficient conditions for logarithmic norms are used.

A design of fuel-cell manifold to evenly distribute the reactants among the channels within bipolar plates is presented. Neglecting the friction term from the momentum equation, one-dimensional governing equations are manipulated to develop the manifold equation to achieve the required design. Some manifold shapes that satisfy the manifold equation, are suggested, together with the distribution of static pressure within the manifold itself.

We demonstrate that application of a generalized centre manifold is advantageous when low-dimensional reductions of continuous dynamical systems of hydrodynamic type are considered. The centre manifold theorem that we use differs from the standard one in the following way. We define a centre manifold as an invariant manifold, tangent to the eigenspace of the linearization of the mapping defining a dynamical system, where the eigenspace is associated with eigenvalues with small but not necessarily zero real parts (usually only eigenvalues with zero real parts are employed). As a result, several bifurcations following the onset of instability of the trivial steady state are reproduced in the eight-dimensional system that we obtain, whilst in the six-dimensional system constructed with application of the conventional centre manifold theorem only the first bifurcation of the steady state is reproduced correctly.

A dynamical system admitting an invariant manifold can be interpreted as a single element of an infinite class of dynamical systems that all exhibit the same behaviour on the invariant manifold. This observation is used in the context of autonomous ordinary differential equations to generalize a global stability result of Li and Muldowney. The new result is demonstrated on an epidemiological model.

The main result of our article is an analog of the local manifold theorem for a hyperbolic point. We give a set of sufficient conditions that ensure the existence of the global invariant manifold. We will prove the existence of invariant curves which lie in the quaternionic Julia set of the map f c ( X ) = X 2 + c .

If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results

Under area- and orientation-preserving maps, the images of the symmetry axes accumulate at the unstable manifold of a saddle fixed point P , while their inverse images accumulate at the stable manifold of P . Before or after the tangency of these stable and unstable manifolds, four types of generalized dynamical ordering are derived. They determine the order of the appearance of the symmetric periodic orbits. We derive these types of ordering for

We consider the Lagrangian , V 2π-periodic and δ bounded. The corresponding Euler–Lagrange equations have as the origin a saddle-centre stationary point whose (globally defined) centre manifold is foliated in periodic orbits. We prove that for ω small enough there exist multi-bump solutions, of large energy and heteroclinic to periodic orbits in the centre manifold.

First we prove a general spectral theorem for the linear Navier- Stokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H ' (' = 0,1,2,···). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits ( ! 0 + ) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-stable manifold of a steady state are O( p ), while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are O( ), as ! 0+. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete esti- mate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other two examples, partial estimates on the eigenvalues are obtained.

We consider dynamics generated by Hamiltonians with three degrees of freedom and symmetries. It is shown that locally, away from a possible saddle equilibrium, some codimension-1 invariant manifold may exist. They are stable/unstable manifolds of a codimension-2 hyperbolic invariant manifold. This structure appears when some periodic orbits constitutive of the Arnold web have bifurcated and become linearly unstable. This result generalizes the existence of normally hyperbolic invariant manifolds and their codimension-1 stable/unstable manifolds in the vicinity of an unstable ⊗ (stable)2 equilibrium point.

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.

This paper begins by providing a mathematical characterization of the Lorenz system. Then, we present a 1D algorithm for computing the global one-dimensional unstable manifold of a saddle point of Lorenz system. However, a higher-dimensional stable and unstable manifold of Lorenz system consist of infinitely many orbits, and a finite collection of orbits typically does not give an acceptable picture of the Lorenz system. We try to find a sophisticated algorithm for computing two-dimensional stable and unstable Lorenz manifolds of three-dimensional vector field. We are concerned with the technically challenging case of computing stable and unstable manifolds of Lorenz system in the three-dimensional vector fields in this paper.

A new algorithm is presented to compute both one dimensional stable and unstable manifolds of planar maps. It is proved that the gradient of the global manifold can be predicted by the known points on the manifold with a gradient prediction scheme and it can be used to locate the image or preimage of the new point quickly. The performance of the algorithm is demonstrated with hyper chaotic Lorenz system. DOI : http://dx.doi.org/10.11591/telkomnika.v12i4.4740

A new algorithm is presented for computing one dimensional unstable manifold of a map and Henon map is taken as an example to test the performance of the algorithm. The unstable manifold is grown with new point added at each step and the distance between consecutive points is adjusted according to the local curvature. It is proved that the gradient of the manifold at the new point can be predicted by the known points on the manifold and in this way the preimage of the new point could be located immediately. During the simulation, it is found that the unstable manifold of Henon map coincides with its direct iteration when canonical parameters are chosen which means order is obtained out of chaos. In the other several groups of parameters the two branches of the unstable manifolds are nearly symmetric, and they serve as the borderline of the Henon map iteration sequence. We hope that this would contribute to the further exploration of Henon map. DOI : http://dx.doi.org/10.11591/telkomnika.v12i2.4328

In this paper we consider smooth differential 1-forms and smooth nonlinear control-affine systems with (n−1)-inputs evolving on an n-dimensional manifold with boundary. These systems are called hypersurface systems under the additional assumption that the drift vector field and control vector fields span the tangent space to the manifold. We locally classify all structurally stable differential 1-forms on a manifold with boundary. We give complete local classification of structurally stable hypersurface systems on a manifold with boundary under static state feedback defined by diffeomorphisms, which preserve the manifold together with its boundary.

We consider piecewise smooth, uniformly hyperbolic systems on a Riemannian manifold, where we allow the angle between the unstable direction and the singularity manifolds to vanish. Under natural assumptions we prove that such systems exhibit exponential decay of correlations and satisfy a central limit theorem with respect to a mixing Sinai-Ruelle-Bowen-measure. These results have been shown previously for systems in which the angle between the singularity manifold and the unstable direction is uniformly bounded away from zero.

The inertial manifold is a positive invariant set which exponentially attracts all the trajectories of a dissipative dynamic system. It was introduced for the purpose of studying the asymptotic behaviour of such systems. The initial infinitely dimensional dynamic system, generated by a partial differential evolution equation, can be projected on to it, in order to obtain the final system of ordinary differential equations (inertial equations). These equations then simulate the initial object. Although the inertial manifold is an object relatively simpler than the attractor (a very complicated set of non-integer dimension may be an attractor) it is more difficult to prove its existence than that of the attractor. The equation of a barotropic fluid on a rotating spherical surface is one of the examples of dissipative dynamic systems with an inertial manifold. This kindles the hope that also the equations of the dynamics of the real atmosphere will have an inertial manifold. The reduction of the sample system to this Lipschitz manifold of finite dimension thus justifies us in analysing the behaviour of the atmosphere on non-linear models of finite dimensions and few parameters, in a finite system of ordinary differential equations.

Consider the Henon's map Tb: (x→ 1-ax2+y, y→bx), the parameters (a, b) being such that |b|S(q1) of the saddle fixed point q1 belonging to the basin boundary of the attracting set A. A method of successive approximations of iterative type is used from the definition of a "generating approximation" g0(x, y)=0 (approximation of order zero). In the case of absence of homoclinic points to q1, a generating approximation is defined from the two parabolas constituting the degenerate stable manifold in the (x, y) plane when b=0. Formally the result is extended when homoclinic points to q1 are created for b=0, i.e. the degenerate stable manifold WS(q1) is made up of infinitely many parabolas.

The problem of finding bounds on the H ∞ -norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H ∞ -norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter,. The eigenvalues of the linearized system are complex for 0. Thus for 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation. One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for

In this paper we study the size of the set of equilibria on the pseudo-equilibrium manifold. It is shown that, with incomplete real asset markets, the set of equilibria is an open and dense subset of the pseudo-equilibrium manifold, whose complement is the disjoint union of a finite number of lower dimensional smooth submanifolds. When markets are generically complete, the pseudo-equilibrium manifold with fixed real asset structures is identical to the equilibrium manifold in the Arrow–Debreu model and the above result is still persistent. In addition, it is shown that the complement of the set of equilibria is also a trivial vector bundle over a semi-algebraic set.Journal of Economic LiteratureClassification Number: D52.

International audience; We develop a framework for the study of delayed neural fields equations and prove a center manifold theorem for these equations. Specific properties of delayed neural fields equations make it difficult to apply existing methods from the literature concerning center manifold results for functional differential equations. Our approach for the proof of the center manifold theorem uses the original combination of results from Vanderbauwhede et al. together with a theory of linear functional differential equations in a history space larger than the commonly used set of time-continuous functions

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.

For a system of nonlinear matrix differential equations with small parameter, we establish conditions of the existence of an invariant manifold in terms of the Green function.