scholarly journals On the global stable manifold

2006 ◽  
Vol 177 (2) ◽  
pp. 113-131 ◽  
Author(s):  
Alberto Abbondandolo ◽  
Pietro Majer
Keyword(s):  
Stable Manifold ◽  
Global Stable Manifold

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.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Local stable manifold theorem for fractional systems

2015 ◽  
Vol 83 (4) ◽  
pp. 2435-2452 ◽  
Author(s):  
Amey Deshpande ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Stable Manifold ◽  
Fractional Systems ◽  
Stable Manifold Theorem ◽  
Local Stable Manifold

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 Global center stable manifold for the defocusing energy critical wave equation with potential

2020 ◽  
Vol 142 (5) ◽  
pp. 1497-1557
Author(s):  
Hao Jia ◽  
Baoping Liu ◽  
Wilhelm Schlag ◽  
Guixiang Xu
Keyword(s):  
Wave Equation ◽  
Stable Manifold ◽  
Global Center
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