scholarly journals An inclination lemma for normally hyperbolic manifolds with an application to diffusion

2014 ◽  
Vol 35 (7) ◽  
pp. 2269-2291 ◽  
Author(s):  
LARA SABBAGH
Keyword(s):  
Symplectic Manifold ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Hyperbolic Manifold ◽  
Finite Family ◽  
Hyperbolic Manifolds ◽  
Minimal Sets ◽  
Symplectic Diffeomorphism ◽  
Compact Submanifold ◽  
Normally Hyperbolic Manifolds

Let ($M$, ${\rm\Omega}$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^{l}$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^{1}$-submanifold ${\rm\Delta}$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_{N}M$. We suppose that ${\rm\Delta}$ transversely intersects the stable manifold of $N$. Then, we prove that for all ${\it\varepsilon}>0$, and for $n\in \mathbb{N}$ large enough, there exists $x_{n}\in N$ such that $f^{n}({\rm\Delta})$ is ${\it\varepsilon}$-close, in the $C^{1}$ topology, to the strongly unstable manifold of $x_{n}$. As an application of this ${\it\lambda}$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold’s example).

scholarly journals An intermittent control model of flexible human gait using a stable manifold of saddle-type unstable limit cycle dynamics

2014 ◽  
Vol 11 (101) ◽  
pp. 20140958 ◽  
Author(s):  
Chunjiang Fu ◽  
Yasuyuki Suzuki ◽  
Ken Kiyono ◽  
Pietro Morasso ◽  
Taishin Nomura
Keyword(s):  
Steady State ◽  
Limit Cycle ◽  
Unstable Manifold ◽  
Control Strategy ◽  
Stable Manifold ◽  
Joint Angle ◽  
Feedback Controller ◽  
Human Gait ◽  
Intermittent Control ◽  
Saddle Type

Stability of human gait is the ability to maintain upright posture during walking against external perturbations. It is a complex process determined by a number of cross-related factors, including gait trajectory, joint impedance and neural control strategies. Here, we consider a control strategy that can achieve stable steady-state periodic gait while maintaining joint flexibility with the lowest possible joint impedance. To this end, we carried out a simulation study of a heel-toe footed biped model with hip, knee and ankle joints and a heavy head-arms-trunk element, working in the sagittal plane. For simplicity, the model assumes a periodic desired joint angle trajectory and joint torques generated by a set of feed-forward and proportional-derivative feedback controllers, whereby the joint impedance is parametrized by the feedback gains. We could show that a desired steady-state gait accompanied by the desired joint angle trajectory can be established as a stable limit cycle (LC) for the feedback controller with an appropriate set of large feedback gains. Moreover, as the feedback gains are decreased for lowering the joint stiffness, stability of the LC is lost only in a few dimensions, while leaving the remaining large number of dimensions quite stable: this means that the LC becomes saddle-type, with a low-dimensional unstable manifold and a high-dimensional stable manifold. Remarkably, the unstable manifold remains of low dimensionality even when the feedback gains are decreased far below the instability point. We then developed an intermittent neural feedback controller that is activated only for short periods of time at an optimal phase of each gait stride. We characterized the robustness of this design by showing that it can better stabilize the unstable LC with small feedback gains, leading to a flexible gait, and in particular we demonstrated that such an intermittent controller performs better if it drives the state point to the stable manifold, rather than directly to the LC. The proposed intermittent control strategy might have a high affinity for the inverted pendulum analogy of biped gait, providing a dynamic view of how the step-to-step transition from one pendular stance to the next can be achieved stably in a robust manner by a well-timed neural intervention that exploits the stable modes embedded in the unstable dynamics.

scholarly journals Diastatic entropy and rigidity of complex hyperbolic manifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Roberto Mossa
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Continuous Map ◽  
The Hyperbolic Metric ◽  
Geometric Rigidity ◽  
Volume Entropy ◽  
Complex Hyperbolic Manifold ◽  
Real Analytic

AbstractLet f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we get that the minimal diastatic entropy is achieved if and only if g is isometric to the hyperbolic metric g0.

On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter

1991 ◽  
Vol 2 (2) ◽  
pp. 133-158 ◽  
Author(s):  
John G. Byatt-Smith
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Phase Plane ◽  
Careful Analysis ◽  
Heteroclinic Orbits ◽  
Perturbed System ◽  
Heteroclinic Connection ◽  
Small Dispersion

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

scholarly journals FLOER HOMOLOGY FOR SYMPLECTOMORPHISM

2009 ◽  
Vol 11 (06) ◽  
pp. 895-936 ◽  
Author(s):  
HAI-LONG HER
Keyword(s):  
Fixed Points ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Homology Groups ◽  
Symplectic Diffeomorphism ◽  
Novikov Ring ◽  
Compact Symplectic Manifold

Let (M,ω) be a compact symplectic manifold, and ϕ be a symplectic diffeomorphism on M, we define a Floer-type homology FH*(ϕ) which is a generalization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕ up to a Hamiltonian isotopy.

Chains that realize the Gromov invariant of hyperbolic manifolds

1997 ◽  
Vol 17 (3) ◽  
pp. 643-648 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Hyperbolic Manifold ◽  
Homology Class ◽  
Hyperbolic Manifolds ◽  
Uniform Measure ◽  
Geodesic Boundary ◽  
Maximal Volume ◽  
Totally Geodesic ◽  
Gromov Norm

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

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

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Field ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Lorenz System ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
The Lorenz System ◽  
Global Invariant

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.

Global Stability Analysis on Combination Self-Excited Vibration of BTA Deep-Hole Boring Rotor Materials

2015 ◽  
Vol 667 ◽  
pp. 75-81 ◽  
Author(s):  
Wu Zhao ◽  
Zhan Qi Hu ◽  
Dan Huang
Keyword(s):  
Plastic Deformation ◽  
Phase Transformation ◽  
Cutting Force ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Processing System ◽  
Martensite Phase ◽  
Deep Hole ◽  
Excited Vibration ◽  
Cutting Chatter

Cr, Ni, Mo, V exist in materials of high, medium and low pressure turbine rotor. Its composition contains higher carbon content, so leads to poor hardenability. Shear characteristics of martensite phase transformation is caused by a large number of micro defects inside crystal of microstructure in rotor during the service status. Martensite phase transformation is by strengthening because of plastic deformation in cutting process. Mechanical properties and phase transformation plasticity in cutting process show that rotor is hard machining material. Plastic deformation aggravates tool wear, and leads to an increased cutting force; hard points of uneven material internal organization lead to fluctuation of cutting force, and induced cutting chatter. During the course of machining critical or ultra-supercritical rotor, boring bar becomes a flexible system surrounded by rotating stirred vortex cutting fluid inside and outside it. BTA processing system is in a complicated hydraulic-solid coupling environment. Whether or not BTA deep hole boring processing system is stable, the essence is converted into estimating dynamic stability of BTA boring bar coupled fluid--solid with multi-physics environment. The system will be triggered cutting chatter, fluid flutter and self-excited vibration combination. This paper establishes nonlinear mathematical physics model on BTA deep hole processing system. The perturbation equation and Lyapunov function of BTA system are obtained, through introducing the stable Rumjantsev part variables analysis theory. Based on stability characteristic under the significance of Lyapunov, the constraint condition undisturbed stability is gotten and the global bifurcation analysis is finished. Hyperbolic periodic orbit and heteroclinic connection are found throughout the entire phase space of Hamiltonian system. The bifurcation condition on quadratic heteroclinic is acquired, by analyzing intersection relationship between hyperbolic stable manifold and its unstable manifold of Poincare mapping. The intersection critical conditions are shown on hyperbolic fixed stable manifold and its unstable manifold, by using computer numerical simulation to draw 3-D surface relationship among resonant frequency, cutting fluids flow velocity and the radius of free surface. The result could provide a reference on multi-machining parameters optimization for BTA deep hole boring process practical production.

scholarly journals Showcase of Blue Sky Catastrophes

2014 ◽  
Vol 24 (08) ◽  
pp. 1440003 ◽  
Author(s):  
Leonid Pavlovich Shilnikov ◽  
Andrey L. Shilnikov ◽  
Dmitry V. Turaev
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Invariant Torus ◽  
Complex Dynamics ◽  
Klein Bottle ◽  
System Of Differential Equations ◽  
Homoclinic Connection ◽  
Global Stable Manifold

Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

DETERMINATION OF CRISIS PARAMETER VALUES BY DIRECT OBSERVATION OF MANIFOLD TANGENCIES

1992 ◽  
Vol 02 (02) ◽  
pp. 383-396 ◽  
Author(s):  
JOHN C. SOMMERER ◽  
CELSO GREBOGI
Keyword(s):  
Periodic Point ◽  
Chaotic System ◽  
Direct Observation ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Critical Parameter ◽  
Scaling Relation ◽  
Chaotic Transients ◽  
Parameter Values

We discuss an algorithm to find the parameter value at which a nonlinear, dissipative, chaotic system undergoes crisis. The algorithm is based on the observation that, at crisis, the unstable manifold of an unstable periodic point becomes tangent to the stable manifold of the same or another, related unstable periodic point. This geometric algorithm uses much less computation (or data) than estimating the critical parameter value by using the scaling relation for chaotic transients, τ~(p−pc)−γ. We demonstrate the algorithm in both numerical and experimental contexts.

scholarly journals Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

2021 ◽  
Author(s):  
Werner Müller
Keyword(s):  
Fundamental Group ◽  
General Result ◽  
Hyperbolic Manifold ◽  
Zeta Functions ◽  
Hyperbolic Manifolds ◽  
Reidemeister Torsion ◽  
Finite Dimensional ◽  
Orthogonal Representations ◽  
Ruelle Zeta Function ◽  
Complex Valued

AbstractThis paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $$s=0$$ s = 0 and its value at $$s=0$$ s = 0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

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