A Sun-Earth Stable Manifold-Based Method for Two-Impulse Earth-Moon Transfer Design

2021 ◽  
Author(s):  
Minghu Tan ◽  
Ke Zhang ◽  
Jingyu Wang
Keyword(s):  
Stable Manifold

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

Suboptimal Feedback Control of Nonlinear Distributed Parameter Systems by Stable Manifold Method

2014 ◽  
Vol 47 (3) ◽  
pp. 11357-11362 ◽  
Author(s):  
K. Hamaguchi ◽  
G. Nishida ◽  
N. Sakamoto
Keyword(s):  
Feedback Control ◽  
Stable Manifold ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Suboptimal Feedback Control

scholarly journals Local stable manifold of Langevin differential equations with two fractional derivatives

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
JinRong Wang ◽  
Shan Peng ◽  
D O’Regan
Keyword(s):  
Differential Equations ◽  
Stable Manifold ◽  
Fractional Derivatives ◽  
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

scholarly journals Determination of the flight dynamic envelope via stable manifold

2019 ◽  
Vol 52 (3-4) ◽  
pp. 244-251 ◽  
Author(s):  
Guoqiang Yuan ◽  
Yinghui Li
Keyword(s):  
Mesh Generation ◽  
Stable Manifold ◽  
Accurate Result ◽  
Flight Safety ◽  
Fast Method ◽  
Region Of Attraction ◽  
Actuator Failure ◽  
Advancing Front ◽  
Flight Envelope

The flight envelope plays an important role in flight safety. The concept of posing the flight envelope as a region of attraction is explored further, and it is investigated whether the stable manifold for the region of attraction computation is an efficient method for determining envelope. The stable manifold describes the flight dynamic envelope of an aircraft in an explicit representation, which means that the computation needs to be done only on the envelope, not the entire state space. In this paper, the stable manifold is computed by using a fast method which reduces the computation to solving a system of partial differential equation. Then, the stable manifold grows in the way of advancing front mesh generation framework. The stable manifold is then applied to the envelope determination of a nonlinear F-16 model. The result is compared to the results obtained with the level set method, demonstrating that the stable manifold provides a feasible and accurate result to the dynamic envelope. The proposed method is then used to investigate the effect of actuator failure on the flight safety. The proposed method can also be used as a safety assessing tool during the design phase of an aircraft.

scholarly journals Integrability of continuous bundles

2019 ◽  
Vol 2019 (752) ◽  
pp. 229-264 ◽  
Author(s):  
Stefano Luzzatto ◽  
Sina Tureli ◽  
Khadim War
Keyword(s):  
Dynamical Systems ◽  
Stable Manifold ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Classical Theorem ◽  
Uniqueness Of Solutions ◽  
Stable Manifold Theorem ◽  
New Criteria

Abstract We give new sufficient conditions for the integrability and unique integrability of continuous tangent subbundles on manifolds of arbitrary dimension, generalizing Frobenius’ classical theorem for {C^{1}} subbundles. Using these conditions, we derive new criteria for uniqueness of solutions to ODEs and PDEs and for the integrability of invariant bundles in dynamical systems. In particular, we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

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