The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Li Yan-mei - One of the best experts on this subject based on the ideXlab platform.
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Phase Portraits of Some Planar Quintic Hamiltonian Vector Field with Z_2-Equivariant Property
Journal of Chuxiong Normal University, 2011Co-Authors: Li Yan-meiAbstract:In this paper,a quintic Hamiltonian Vector Field withZ2-Equivariant property is studied and the phase portraits are classified.
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The classification of phase portraits about some Hamiltonian Vector Field with Z_3-equivariant property
Journal of Yunnan Normal University, 2003Co-Authors: Li Yan-meiAbstract:In this paper, we consider some planar quintic Hamiltonian Vector Field with Z3-equivariant property, get its global phase portraits, and classify its parameter space.
Li Yanme - One of the best experts on this subject based on the ideXlab platform.
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The global phase portrait of Hamiltonian Vector Field with Z_2-equivalent property
Journal of Chuxiong Teacher's College, 2004Co-Authors: Li YanmeAbstract:The presentpaper probes into the global property of some planar quintic Hamiltonian Vector Field with Z2-equivalent property. As a result, its global phase portrait is obtained, and its parameter space is classified.
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Classification of phase portrait about planar quintic Hamiltonian Vector Field
Journal of Chuxiong Teacher's College, 2002Co-Authors: Li YanmeAbstract:The article gives an analysis to planar quintic Z 4 Hamiltonian equivariant Vector Field, finding out its whole phase portrait. Meanwhile, the present author makes a classification of the parameter space.
Li Yan - One of the best experts on this subject based on the ideXlab platform.
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Classification of Phase Portraits of Z_2- Equivariant Planar Hamiltonian Vector Fields of Degree 7(III)
Journal of Chuxiong Normal University, 2013Co-Authors: Li YanAbstract:In this paper,applying the method of qualitative analysis of differential equations,we study the phase portraits of a new Z2- equivariant planar Hamiltonian Vector Field of degree 7 and get some new phase portraits. In the meanwhile,the parameter space of the system is classified.
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the global properties of some planar quintic Hamiltonian Vector Field with z_2 equivariant property
Journal of Yunnan University, 2001Co-Authors: Li YanAbstract:A class of quintic Hamiltonian system with Z 2-symmetry is considered.Using the methods of qualitative analysis,the global phase portraits are given.
Lubomir Gavrilov - One of the best experts on this subject based on the ideXlab platform.
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Cubic perturbations of elliptic Hamiltonian Vector Fields of degree three
Journal of Differential Equations, 2016Co-Authors: Lubomir Gavrilov, Iliya D. IlievAbstract:Abstract The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian Vector Field X e X e : { x ˙ = H y + e f ( x , y ) y ˙ = − H x + e g ( x , y ) , H = 1 2 y 2 + U ( x ) which bifurcate from the period annuli of X 0 for sufficiently small e . Here U is a univariate polynomial of degree four without symmetry, and f , g are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map d ( h , e ) by the Hamiltonian value h and by the small parameter e . Let M k ( h ) be the k -th coefficient in its expansion with respect to e . We establish the general form of M k and study its zeroes. We deduce that the period annuli of X 0 can produce for sufficiently small e , at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian Fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of X 0 of higher degrees are also studied.
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Cubic perturbations of elliptic Hamiltonian Vector Fields of degree three
arXiv: Dynamical Systems, 2014Co-Authors: Lubomir Gavrilov, Iliya D. IlievAbstract:The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian Vector Field $X_\varepsilon$ $$ X_\varepsilon : \left\{ \begin{array}{llr} \dot{x}=\;\; H_y+\varepsilon f(x,y)\\ \dot{y}=-H_x+\varepsilon g(x,y), \end{array} \;\;\;\;\; H~=\frac{1}{2} y^2~+U(x) \right. $$ which bifurcate from the period annuli of $X_0$ for sufficiently small $\varepsilon$. Here $U$ is a univariate polynomial of degree four without symmetry, and $f, g$ are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map $d(h,\varepsilon)$ by the Hamiltonian value $h$ and by the small parameter $\varepsilon$. Let $M_k(h)$ be the $k$-th coefficient in its expansion with respect to $\varepsilon$. We establish the general form of $M_k$ and study its zeroes. We deduce that the period annuli of $X_0$ can produce for sufficiently small $\varepsilon$, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian Fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of $X_0$ of higher degrees are also studied.
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The infinitesimal 16th Hilbert problem in the quadratic case
Inventiones mathematicae, 2001Co-Authors: Lubomir GavrilovAbstract:Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let ${X_H} = {H_y}\frac{\partial }{{\partial x}} - {H_x}\frac{\partial }{{\partial y}}$ be the corresponding Hamiltonian Vector Field. We show that there is a neighborhood ? of X _ H in the space of all quadratic plane Vector Fields, such that any X ∈? has at most two limit cycles.
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Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems
Ergodic Theory and Dynamical Systems, 2000Co-Authors: Lubomir Gavrilov, Iliya D. IlievAbstract:We study degree n polynomial perturbations of quadratic reversible Hamiltonian Vector Fields with one center and one saddle point. It was recently proved that if the first Poincare-Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is n 1. In the present paper we prove that if the first Poincare-Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is 2 .n 1/. In the case when the perturbation is quadratic (n D 2) we obtain a complete result—there is a neighborhood of the initial Hamiltonian Vector Field in the space of all quadratic Vector Fields, in which any Vector Field has at most two limit cycles.
A.j. Van Der Schaft - One of the best experts on this subject based on the ideXlab platform.
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On a state space approach to nonlinear H ∞ control
Systems & Control Letters, 1991Co-Authors: A.j. Van Der SchaftAbstract:We study the standard H∞ optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the L2-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant γ > 0 if the corresponding H∞ norm for the system linearized at the equilibrium can be made smaller than γ by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known, e.g. from the state space approach to linear H∞ optimal control. Our approach to the nonlinear H∞ optimal control problem generalizes the state space approach to the linear H∞ problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian Vector Field together with a Hamiltonian-Jacobi equation corresponding to its stable invariant manifold.
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on a state space approach to nonlinear h control
Systems & Control Letters, 1991Co-Authors: A.j. Van Der SchaftAbstract:We study the standard H∞ optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the L2-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant γ > 0 if the corresponding H∞ norm for the system linearized at the equilibrium can be made smaller than γ by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known, e.g. from the state space approach to linear H∞ optimal control. Our approach to the nonlinear H∞ optimal control problem generalizes the state space approach to the linear H∞ problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian Vector Field together with a Hamiltonian-Jacobi equation corresponding to its stable invariant manifold.
