Non-hyperbolic Point Research Articles

λ-lemma for nonhyperbolic point in intersection

The well known λ-Lemma has been proved by J. Palis for a hyperbolic fixed point of a C 1 -diffeomorphism. In this paper we show that the result is true for some cases of nonhyperbolic point.

Detailed qualitative dynamical analysis of a cosmological Higgs field

A scalar cosmological Higgs field is expected to exist in our universe in order to create inertial mass. Some results obtained at LHC suggest that this idea must be reconsidered. The cosmological effects of scalar fields have been proposed as a mechanism to drive the evolution of the universe in various scenarios. In this paper we investigate, from the dynamical systems perspective, the evolution of a Universe consisting of a matter component $$\rho $$ together with a scalar field $$\phi $$ exhibiting a quartic polynomial self-interacting potential. We consider an homogeneous and isotropic flat Friedmann–Robertson–Walker metric. Center Manifold Theory is employed to investigate the dynamics near a non-hyperbolic critical point. We prove that there are two possible late time attractors corresponding to stable de Sitter solutions.

Stabilization of a class of slow–fast control systems at non-hyperbolic points

In this document, we deal with the local asymptotic stabilization problem of a class of slow–fastsystems (or singularly perturbed Ordinary Differential Equations). The systems studied here have the following properties: (1) they have one fast and an arbitrary number of slow variables, and (2) they have a non-hyperbolic singularity at the origin of arbitrary degeneracy. Our goal is to stabilize such a point. The presence of the aforementioned singularity complicates the analysis and the controller design. In particular, the classical theory of singular perturbations cannot be used. We propose a novel design based on geometric desingularization, which allows the stabilization of a non-hyperbolic point of singularly perturbed control systems. Our results are exemplified on a didactic example and on an electric circuit.

Improving the Region of Attraction of a Non-Hyperbolic Point in Slow-Fast Systems With One Fast Direction

Through recent research combining the geometric desingularization or blow-up method and classical control tools, it has been possible to locally stabilize non-hyperbolic points of singularly perturbed control systems. In this letter we propose a simple method to enlarge the region of attraction of a non-hyperbolic point in the aforementioned setting by expanding the geometric analysis around the singularity. In this way, we can synthesize improved controllers that stabilize non-hyperbolic points within a large domain of attraction. Our theoretical results are showcased in a couple of numerical examples.

Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

In this letter, we investigate a class of slow-fast systems for which the classical model order reduction technique based on singular perturbations does not apply due to the lack of a normally hyperbolic critical manifold. We show, however, that there exists a class of slow-fast systems that after a well-defined change of coordinates have a normally hyperbolic critical manifold. This allows the use of model order reduction techniques and to qualitatively describe the dynamics from auxiliary reduced models even in the neighborhood of a non-hyperbolic point. As an important consequence of the model order reduction step, we show that it is possible to design composite controllers that stabilize the (non-hyperbolic) origin.

A remark on geometric desingularization of a non-hyperbolic point using hyperbolic space

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consider non-hyperbolic steady states, for example in the context of bifurcation theory. A geometric technique to desingularize non-hyperbolic points is the blow-up method. The classical case of the method is motivated by desingularization techniques arising in algebraic geometry. The idea is to blow up the steady state to a sphere or a cylinder. In the blown-up space, one is then often able to gain additional hyperbolicity at steady states. The method has also turned out to be a key tool to desingularize multiple time scale dynamical systems with singularities. In this paper, we discuss an explicit example of the blow-up method where we replace the sphere in the blow-up by hyperbolic space. It is shown that the calculations work in the hyperbolic space case as for the spherical case. This approach may be even slightly more convenient if one wants to work with directional charts. Hence, it is demonstrated that the sphere should be viewed as an auxiliary object in the blow-up construction. Other smooth manifolds are also natural candidates to be inserted at steady states. Furthermore, we conjecture several problems where replacing the sphere could be particularly useful, i.e., in the context of singularities of geometric flows, for avoiding compactification, and generating 'interior' steady states.

ACTION AND PERIOD OF HOMOCLINIC AND PERIODIC ORBITS FOR THE UNFOLDING OF A SADDLE-CENTER BIFURCATION

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

Stability and chaotic dynamics of a rate gyro with feedback control under uncertain vehicle spin and acceleration

An analysis of stability and chaotic dynamics is presented by a single-axis rate gyro subjected to linear feedback control loops. This rate gyro is supposed to be mounted on a space vehicle which undergoes an uncertain angular velocity ω Z ( t) around its spin axis. And simultaneously acceleration ω ̇ X(t) occurs with respect to the output axis. The necessary and sufficient conditions of stability for the autonomous case, whose vehicle undergoes a steady rotation, were provided by Routh–Hurwitz theory. Also, the degeneracy conditions of the non-hyperbolic point were derived and the dynamics of the resulting system on the center manifold near the double-zero degenerate point by using center manifold and normal form methods were examined. The stability of the non-linear non-autonomous system was investigated by Liapunov stability and instability theorems. As the electrical time constant is much smaller than the mechanical time constant, the singularly perturbed system can be obtained by the singular perturbation theory. The Liapunov stability of this system by studying the reduced and boundary-layer systems was also analyzed. Numerical simulations were performed to verify the analytical results. The stable regions of the autonomous system were obtained in parametric diagrams. For the non-autonomous case in which ω Z ( t) oscillates near boundary of stability, periodic, quasiperiodic and chaotic motions were demonstrated by using time history, phase plane and Poincaré maps.

Viscous Profiles of Traveling Waves in Scalar Balance Laws: The Canard Case

The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold. In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in thesmall viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.

Approximation of attractors of semidynamical systems

The problem of approximation with prescribed accuracy of the attractors of semidynamical systems is considered. The problem is formulated in terms of the function of the rate of attraction. New results on the structure of unstable manifolds in the neighbourhood of a non-hyperbolic point are used. For a certain class of maps the unstable manifold is effectively constructed and an estimate of the rate of attraction to it is found.

The Local Solvability of a Hamilton--Jacobi--Bellman PDE around a Nonhyperbolic Critical Point

We show the existence of a local solution to a Hamilton--Jacobi--Bellman (HJB) PDE around a critical point where the corresponding Hamiltonian ODE is not hyperbolic, i.e., it has eigenvalues on the imaginary axis. Such problems arise in nonlinear regulation, disturbance rejection, gain scheduling, and linear parameter varying control. The proof is based on an extension of the center manifold theorem due to Aulbach, Flockerzi, and Knobloch. The method is easily extended to the Hamilton--Jacobi--Isaacs (HJI) PDE. Software is available on the web to compute local approximtate solutions of HJB and HJI PDEs.

Traveling Wave Phenomena in Some Degenerate Reaction-Diffusion Equations

In this paper we study the existence of travelling wave solutions (t.w.s.), u( x, t)=φ( x− ct) for the equation [formula]+ g( u), (*) where the reactive part g( u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion term D( u). Both functions D and g are defined on the interval [0, 1]. The existence problem is analysed in the following two cases. Case 1. D(0)=0, D( u)>0 ∀ u∈(0, 1], D and g∈ C 2 [0,1], D′(0)≠0 and D′′(0)≠0. We prove that if there exists a value of c, c*, for which the equation (*) possesses a travelling wave solution of sharp type, it must be unique. By using some continuity arguments we show that: for 0< c< c*, there are no t.w.s., while for c> c*, the equation (*) has a continuum of t.w.s. of front type. The proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations, which is also proved. Case 2. D(0)= D′(0)=0, D and g∈ C 2 [0,1], D′′(0)≠0. If, in addition, we impose D′′(0)>0 with D( u)>0 ∀ u∈(0, 1], We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if D′′(0)<0 with D( u)<0 ∀ u∈(0, 1] we analyse just one example ( D( u)=− u 2, and g( u)= u(1− u)) which has oscillatory t.w.s. for 0< c≤2 and t.w.s. of front type for c>2. In both the above cases we use higher order terms in the Taylor series and the Centre Manifold Theorem in order to get the local behaviour around a non-hyperbolic point of codimension one in the phase plane.