Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition

2021 ◽  
pp. 1-24
Author(s):  
Wen Zhang ◽  
Gang Yang ◽  
Fangfang Liao
Keyword(s):  
Hamiltonian System ◽  
Growth Condition ◽  
Homoclinic Orbits ◽  
Quadratic Growth ◽  
First Order ◽  
Order Hamiltonian System ◽  
Quadratic Growth Condition

scholarly journals Adaptive restart of accelerated gradient methods under local quadratic growth condition

2019 ◽  
Vol 39 (4) ◽  
pp. 2069-2095 ◽  
Author(s):  
Olivier Fercoq ◽  
Zheng Qu
Keyword(s):  
Growth Condition ◽  
Error Bound ◽  
Gradient Methods ◽  
Quadratic Growth ◽  
Quadratic Error ◽  
Total Variation Denoising ◽  
Accelerated Gradient ◽  
Convergent Algorithm ◽  
Accelerated Gradient Methods ◽  
Quadratic Growth Condition

Abstract By analyzing accelerated proximal gradient methods under a local quadratic growth condition, we show that restarting these algorithms at any frequency gives a globally linearly convergent algorithm. This result was previously known only for long enough frequencies. Then as the rate of convergence depends on the match between the frequency and the quadratic error bound, we design a scheme to automatically adapt the frequency of restart from the observed decrease of the norm of the gradient mapping. Our algorithm has a better theoretical bound than previously proposed methods for the adaptation to the quadratic error bound of the objective. We illustrate the efficiency of the algorithm on Lasso, regularized logistic regression and total variation denoising problems.

scholarly journals Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Xiaoyan Lin ◽  
Qi-Ming Zhang ◽  
X. H. Tang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Satisfying Condition ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
First Order ◽  
Nonlinear Hamiltonian System

We give several sufficient conditions under which the first-order nonlinear Hamiltonian systemx'(t)=α(t)x(t)+f(t,y(t)),  y'(t)=-g(t,x(t))-α(t)y(t)has no solution(x(t),y(t))satisfying condition0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍,whereμ,ν>1and(1/μ)+(1/ν)=1,0≤xf(t,x)≤β(t)|x|μ,xg(t,x)≤γ0(t)|x|ν,β(t),γ0(t)≥0, andα(t)are locally Lebesgue integrable real-valued functions defined onℝ.

Homoclinic orbits for a class of Hamiltonian systems

1990 ◽  
Vol 114 (1-2) ◽  
pp. 33-38 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Mountain Pass Theorem ◽  
Local Maximum ◽  
Homoclinic Orbits ◽  
Order Hamiltonian System ◽  
Image Position ◽  
Second Order Hamiltonian System

SynopsisConsider the second order Hamiltonian system:where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

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