• 충청수학회지
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• 제19권2호
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• pp.141-151
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• 2006

We investigate the multiplicity of $2{\pi}$-periodic solutions of the nonlinear Hamiltonian system with almost polynomial and exponential potentials, $\dot{z}=J(G^{\prime}(z)+h(t))$, where $z:R{\rightarrow}R^{2n}$, $\dot{z}=\frac{dz}{dt}$, $J=\(\array{0&-I\\I&o}\)$, I is the identity matrix on $R^n$, $H:R^{2n}{\rightarrow}R$, and $H_z$ is the gradient of H. We look for the weak solutions $z=(p,q){\in}E$ of the nonlinear Hamiltonian system.

• 대한수학회보
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• 제53권3호
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• pp.667-680
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• 2016

We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the $S^1$-invariant functions and the $S^1$-invariant linear subspaces.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.383-388
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• 2007

A (g, f)-factor F of a graph G is Called a Hamiltonian (g, f)-factor if F contains a Hamiltonian cycle. The binding number of G is defined by $bind(G)\;=\;{min}\;\{\;{\frac{{\mid}N_GX{\mid}}{{\mid}X{\mid}}}\;{\mid}\;{\emptyset}\;{\neq}\;X\;{\subset}\;V(G)},\;{N_G(X)\;{\neq}\;V(G)}\;\}$. Let G be a connected graph, and let a and b be integers such that $4\;{\leq}\;a\;<\;b$. Let g, f be positive integer-valued functions defined on V(G) such that $a\;{\leq}\;g(x)\;<\;f(x)\;{\leq}\;b$ for every $x\;{\in}\;V(G)$. In this paper, it is proved that if $bind(G)\;{\geq}\;{\frac{(a+b-5)(n-1)}{(a-2)n-3(a+b-5)},}\;{\nu}(G)\;{\geq}\;{\frac{(a+b-5)^2}{a-2}}$ and for any nonempty independent subset X of V(G), ${\mid}\;N_{G}(X)\;{\mid}\;{\geq}\;{\frac{(b-3)n+(2a+2b-9){\mid}X{\mid}}{a+b-5}}$, then G has a Hamiltonian (g, f)-factor.

• 대한수학회지
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• 제48권6호
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• pp.1143-1152
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• 2011

The existence of periodic solutions for the planar Hamiltonian systems with positively homogeneous Hamiltonian is discussed. The asymptotic expansion of the Poincar$\acute{e}$ map is calculated up to higher order and some sufficient conditions for the existence of periodic solutions are given in the case when the first order term of the Poincar$\acute{e}$ map is identically zero.

• Bulletin of the Korean Chemical Society
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• 제24권6호
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• pp.744-750
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• 2003

In quantum dynamics of many-body systems, to identify the Hamiltonian becomes more difficult very rapidly as the number of degrees of freedom increases. In order to simplify the dynamics and to deduce dynamically relevant Hamiltonian information, it is desirable to control the dynamics to lie within a reduced space. With a judicious choice for the cost functional, the closed loop optimal control experiments can be manipulated efficiently to steer the dynamics to lie within a subspace of the system eigenstates without requiring any prior detailed knowledge about the system Hamiltonian. The procedure is simulated for optimally controlled population transfer experiments in the system of two degrees of freedom. To show the feasibility of steering the dynamics to lie in a specified subspace, the learning algorithms guiding the dynamics are presented along with frequency filtering. The results demonstrate that the optimal control fields derive the system to the desired target state through the desired subspace.

• 대한수학회논문집
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• 제38권3호
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• pp.741-754
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• 2023

In this paper, we study the existence and nonexistence of solutions for a class of Hamiltonian strongly degenerate elliptic system with subcritical growth $$\left{\array{-{\Delta}_{\lambda}u-{\mu}v={\mid}v{\mid}^{p-1}v&&\text{in }{\Omega},\\-{\Delta}_{\lambda}v-{\mu}u={\mid}u{\mid}^{q-1}u&&\text{in }{\Omega},\\u=v=0&&\text{ on }{\partial}{\Omega},}$$ where p, q > 1 and Ω is a smooth bounded domain in ℝ^N, N ≥ 3. Here Δ_λ is the strongly degenerate elliptic operator. The existence of at least a nontrivial solution is obtained by variational methods while the nonexistence of positive solutions are proven by a contradiction argument.

• 대한기계학회논문집
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• 제10권1호
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• pp.170-180
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• 1986

본 연구에서는 2자유도 Hamiltonian 운동계에서 비선형 정규모우드(normal mode)들의 안정성을 예측하기 위한 제3의 운동상수를 선형계의 진동수비가 1:1이고 포텐셜이 4차항까지 우함수인 일반계에 적용하여 발전시켰다. 이는 Hamiltonian을 정규모우드로 바꾸는 B-G변환과 함수들을 부호처리함과 Poincare map을 이용하다. 비선형계에서 비선형상수에 따라 모우드가 bifurcate되며, 각각의 모우드 안정성은 제3의 운동상수와 Poincare map으로 정확히 판정할 수 있다는 결론을 얻었다.

• 대한수학회보
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• 제32권1호
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• pp.1-11
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• 1995

Let $p, q \in R^2 and H : R^{2n} \to R^n$ be differentiable. An autonomous Hamiltonian system has the form $$ (0.1) \dot{p} = -\frac{\partial q}{\partial H}(p, q), \dot{q} = \frac{\partial p}{\partial H}(p, q) $$.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2004년도 ICCAS
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• pp.1623-1628
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• 2004

In the 3-dimensional cyclic Lotka-Volterra equations, we show the solution on the invariant hyperplane. In addition, we show the existence of the invariant hyperplane by the center manifold theorem under the some conditions. With this result, we can lead the hyperplane of the n-dimensional cyclic Lotka-Volterra equaions. In other section, we study the 3- or 4-dimensional Hamiltonian Lotka-Volterra equations which satisfy the Jacobi identity. We analyze the solution of the Hamiltonian Lotka- Volterra equations with the functions called the split Liapunov functions by [4], [5] since they provide the Liapunov functions for each region separated by the invariant hyperplane. In the cyclic Lotka-Volterra equations, the role of the Liapunov functions is the same in the odd and even dimension. However, in the Hamiltonian Lotka-Volterra equations, we can show the difference of the role of the Liapunov function between the odd and the even dimension by the numerical calculation. In this paper, we regard the invariant hyperplane as the important item to analyze the motion of Lotka-Volterra equations and occur the chaotic orbit. Furtheremore, an example of the asymptoticaly stable and stable solution of the 3-dimensional cyclic Lotka-Volterra equations, 3- and 4-dimensional Hamiltonian equations are shown.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 1993년도 추계학술대회논문집; 반도아카데미, 26 Nov. 1993
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• pp.77-83
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• 1993

In this paper, the dynamics of a 2-DOF not 1:1 resonant Hamiltonian system are studied. In the first part of the work, the behaviors of special periodic orbits called normal modes are examined by means of the harmonic balance method and their approximate stability ar analyzed by using the Synge's concept named stability in the kinematico-statical sense. Secondly, the global dynamics of the system for low and high energy are studied in terms of a perturbation analysis and Poincare' maps. In this part, one can see that the unstable normal mode generates chaotic motions resulting from the transverse intersections of the stable and unstable manifolds. Although there exist analytic methods for proving the existence of infinitely many periodic orbits, chaos, they cannot be applied in our case and thus, the Poincare' maps constructed by direct numerical integrations are utilized fot detecting chaotic motions. In the last part of the work, the existence of arbitrarily many periodic orbits of the system are proved by using a subharmonic Melnikov's method. We also study the possibility of the breakdown of invariant KAM tori only when h>h$_{0}$ (h$_{0}$:bifurcating energy) and investigate the generality of the destruction phenomena of the rational tori in the systems perturbed by stiffness and inertial coupling.