One-dimensional Invariant Manifold Research Articles

Breakdown of Heteroclinic Connections in the Analytic Hopf-Zero Singularity: Rigorous Computation of the Stokes Constant

Consider analytic generic unfoldings of the three- dimensional conservative Hopf-zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one-dimensional stable manifold and two-dimensional unstable manifold, whereas the other one has one- dimensional unstable manifold and two-dimensional stable manifold. Baldomá et al. (J Dyn Differ Equ 25(2):335–392, 2013) gave an asymptotic formula for the distance between the one-dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The nonvanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer-assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-zero singularity, obtaining a computer-assisted proof that the constant is nonzero.

Synchronization and Global Dynamics of a Cournot Model with Nonlinear Demand and R&D Spillovers

In this paper, a dynamical Cournot model with nonlinear demand and R&D spillovers is established. The system is symmetric when the duopoly firms have same economic environments, and it is proved that both the diagonal and the coordinate axes are the one-dimensional invariant manifolds of system. The results show that Milnor attractor of system can be found through calculating the transverse Lyapunov exponents. The synchronization phenomenon is verified through basins of attraction. The effects of adjusting speed and R&D spillovers on the dynamical behaviors of the system are discussed. The topological structures of basins of attraction are analyzed through critical curves, and the evolution process of “holes” in the feasible region is numerically simulated. In addition, various global bifurcation behaviors, such as two kinds of contact bifurcation and the blowout bifurcation, are shown.

Propagating vortices in ferrofluidic Couette flow under magnetic fields – Part II: Oblique orientated fields

This work provides a numerical investigations of propagating vortex flow states (pVs) for ferrofluidic Couette flow with small aspect ratio and fixed non-rotating end-walls. The system is subjected to oblique magnetic fields as superposition of axial and transverse orientated fields. Such field configuration breaks the basic system symmetries and renders all flow structures to be inherently three-dimensional with complexer flow dynamics. Under oblique field configuration pV states are not direct present at onset. Instead oscillating flow states (oVs) bifurcate out of the stationary state. Basically these oVs hold same symmetries as pVs, which eventually evolve in a smooth transformation out of the oVs. pVs under oblique fields appear periodic or quasi-periodic, which render them topological speaking to exist on either two- or one-dimensional invariant manifolds as 2-torus or limit cycle (1-torus) solutions, respectively. In detail, structural modifications and changes in spatial and temporal behavior for pV solutions are studied with changing the magnetic field strength of the applied magnetic field.

Global analysis of a generalized Nosé–Hoover oscillator

In this study, the generalized Nosé–Hoover oscillator is analyzed rigorously. We show that all trajectories not belonging to the two one-dimensional invariant manifolds (one is a straight line and the other is a unit circle) must traverse a two-dimensional unit disk infinitely many times transversally along the same direction in the state space. Consequently, all of the trajectories not located in the straight line invariant manifold have at least one ω-limiting point.

A dynamics approach to a low-order climate model

Energy Balance Models (EBM) are conceptual models which have proved useful in the study of planetary climate. The focus of EBM is placed on large scale climate components such as incoming solar radiation, albedo, outgoing longwave radiation and heat transport, and their interactions. Until recently, their study has centered on equilibrium solutions of an associated model equation, with no consideration of the dynamical nature of these solutions. In this paper we continue and expand upon recent efforts aimed at placing EBM in a more mathematical, dynamical systems context. In particular, the dynamical behavior of several variants of the Budyko-Sellers model, all but one of which involve the movement of glaciers, is shown to reduce to the study of the system on an attracting one-dimensional invariant manifold in an appropriately defined state space.

Invariant manifolds in the Clebsch–Tisserand–Brun problem

A series of new invariant manifolds of various dimensions is obtained in Clebsch–Tisserand–Brun problems using a certain modification of the Routh–Lyapunov procedure, and their stability, including their stability with respect to a part of the variables, is investigated. Cases of asymptotic stability of equilibrium positions on one-dimensional invariant manifolds are presented, and the consequences of this fact for the original system are indicated.

Invariant Manifolds of Binomial-Like Nonautonomous Master Equations

It has recently been shown that certain finite-state jump processes used to model ion channel kinetics give rise to nonautonomous master equations which possess low-dimensional invariant manifolds. In particular, for the model of an ion channel with many identical and independent subunits existing in either an open or closed state, the one-dimensional manifold of binomial distributions, whose parameter is the probability of a single subunit being in an open state, is an invariant manifold of the associated master equation, and it is the only such one-dimensional manifold. In this paper, we make explicit how the invariance of this manifold arises from the structure of the transition rate matrix of the master equation. We also show that, despite not having any apparent underlying structure or symmetry akin to the ion channel example, there exists a two-parameter family of master equations that shares the essential features of this example and therefore possesses unique, one-dimensional invariant manifolds similar to the manifold of binomial distributions. The mere existence of such a family suggests that there may be other nonautonomous master equations that are devoid of any obvious structure but nevertheless possess invariant manifolds of low dimension.

Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $\partial_t u - \Delta u + |\nabla u|^q = 0$ is investigated for the critical exponent q=(N+2)/(N+1). Convergence towards a rescaled self-similar solution to the linear heat equation is shown, the rescaling factor being $(\ln{t})^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.

Some topological invariants for three-dimensional flows.

We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. (c) 2001 American Institute of Physics.

Appearance of an Invariant Manifold and On-Off Intermittency in Coupled-Maps: Application to a 3-Valued Neuron

Yamada and Fujisaka's three· logistic-maps model, whose maps are slightly different from each other, is studied theoretically and numerically_ It is found that motion of the model exhibits a nearly-synchronized state whose time series manifests on-off intermittency, and a one-dimensional invariant manifold appears at a critical coupling constant. Association in a network of 3-valued neurons, where the model acts the role of the neuron, is also studied. 387 Recently, a considerable number of studies have been made of chaos synchroniza­ tion 1 )-20) and on-off intermittency8),21)-29) in chaotic systems. However, only a few such attempts 10 ),14),30) have so far been made at nearly-synchronized states which can be observed in a coupled-maps system whose maps are slightly different from each other. The nearly-synchronized state has been treated as synchronization with noise, and with coherency inherent in the noise. In a previous paper,31) we found a one­ dimensional invariant manifold of a coupled two-logistic-maps model (Yamada and Fujisaka's2) model of two maps). In the present paper, we study a nearly­ synchronized state inYamada and Fujisaka's three-logistic-maps model. The coupled system has been utilized for a model neuron, 32 1 where synchronous, partially synchronous and asynchronous are represented by 1, 0 and -1, respectively. A network of the neurons can act as Nakano's33) associatron. The equations of motion of the Yamada-Fujisaka model are expressed as

Smooth invariant curves of singularities of vector fields on [formula omitted]

We describe singularities of C∞ vector fields, mainly in ℝ3, in the neighbourhood of a given « generalized » direction (this is: the image of a C∞ germ γ:(ℝ+,0)→(ℝ3,0) with γ′(0) ≠ 0). It is a local study. One of the major results is: if X is a germ in 0∈ℝ3 of a C∞ vector field and if X is not infinitely flat along a direction D then there exists a cone of finite contact around D in which four specific situations can occur. In three of these situations D is formally invariant under X (with formally we mean: up to the level of formal Taylor series) and there exists a C∞ one-dimensional invariant manifold having infinite contact with D. In particular, we obtain that the existence of a formally invariant direction D always implies the existence of a « real life » invariant direction having infinite contact with D, provided that X is not infinitely flat along D. Using the blowing up method for singularities of vector fields we reduce a singularity always to either a « flow box » or to a singularity with nonzero 1-jet and with a formally invariant direction. Finally we give topological models for the obtained situations.I would like to thank Freddy Dumortier for suggesting me the problem and for his valuable help.

Invariant manifolds in the measure-preserving mappings with three-dimensions

In this paper, we study the following three-dimensional mappings $$T:\left\{ \begin{gathered} x_{n + 1} = x_n + y_n + B sin z_n , \hfill \\ y_{n + 1} = y_n + A sin x_{n + 1} , \hfill \\ z_{n + 1} = z_n + C sin y_{n + 1} + D, \hfill \\ \end{gathered} \right.\left( {\bmod 2\Pi } \right)$$ where A, B, C, D are parameters. When D>BC and 2π/D is an irrational number, we find numerically-two-dimensional and one-dimensional invariant manifolds, but when D BC and 2π/D is a rational number we find numerically one-dimensional manifolds and the fixed points for some cycles.