Fractional Hamiltonian Systems Research Articles

Multiple Homoclinic Solutions for a Class of Fractional Hamiltonian Systems

Multiple Homoclinic Solutions for a Class of Fractional Hamiltonian Systems

Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems

In this paper, by the critical point theory, we consider the existence and multiplicity of solutions for the following fractional differential equation tD∞α(−∞Dtαu(t))+L(t)u(t)=∇W(t,u(t)),t∈R, where α∈(12,1], −∞Dtα and tD∞α are left and right Liouville–Weyl fractional derivatives of order α on the whole axis R respectively, u∈Rn, L(t) is positive definite symmetric matrix for all t∈R and W:R×Rn→R is a suitably chosen function.

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.

Solutions for perturbed fractional Hamiltonian systems without coercive conditions

In this paper we are concerned with the existence of solutions for the following perturbed fractional Hamiltonian systems: $-{}_{t}D^{\alpha}_{\infty}({}_{-\infty}D^{\alpha}_{t}u(t)) -L(t)u(t)+\nabla W(t,u(t))=f(t)$ , $u\in H^{\alpha}(\mathbb{R},\mathbb{R}^{n})$ (PFHS), where $\alpha\in(1/2,1)$ , $t\in\mathbb{R}$ , $u\in\mathbb{R}^{n}$ , $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$ , $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n},\mathbb{R})$ , and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at u, $f\in C(\mathbb{R},\mathbb{R}^{n})$ and belongs to $L^{2}(\mathbb{R},\mathbb{R}^{n})$ . The novelty of this paper is that, assuming $L(t)$ is bounded in the sense that there are constants $0<\tau_{1}<\tau_{2}<\infty$ such that $\tau_{1} |u|^{2}\leq(L(t)u,u)\leq\tau_{2} |u|^{2}$ for all $(t,u)\in\mathbb{R}\times\mathbb{R}^{n}$ and $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, $f(t)$ is sufficiently small in $L^{2}(\mathbb{R},\mathbb{R}^{n})$ , we show that (PFHS) possesses at least two nontrivial solutions. Recent results are generalized and significantly improved.

Multiplicity of Solutions for Fractional Hamiltonian Systems with Liouville-Weyl Fractional Derivatives

Abstract In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: tDα ∞(−∞Dα t u(t)) + L(t)u(t) = ∇W(t, u(t)), (0.1) u ∈ Hα(ℝ,ℝN), where α ∈ (1/2, 1), t ∈ ℝ, u ∈ ℝn, L ∈ C(ℝ,ℝn2 ) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C1(ℝ × ℝn,ℝ), and ∇W is the gradient of W at u. The novelty of this paper is that, assuming there exists l ∈ C(ℝ,ℝ) such that (L(t)u, u) ≥ l(t)|u|2 for all t ∈ ℝ, u ∈ ℝn and the following conditions on l: inf t ∈ ℝ l(t) &gt; 0 and there exists r0 &gt; 0 such that, for any M &gt;0 m({t ∈ (y − r0, y + r0)/ l(t) ≤ M}) → 0 as |y| →∞ are satisfied and W is of subquadratic growth as |u| → +∞, we show that (0.1) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in Z. Zhang and R. Yuan [24] are significantly improved.

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems urn:x-wiley:mma:media:mma3537:mma3537-math-0001 where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Multiple Solutions for a Class of Fractional Hamiltonian Systems

Abstract In this paper, we establish two existence theorems for multiple solutions for the following fractional Hamiltonian system { D ∞ α ( D ∞ α u ( t ) ) + L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) , u ∈ H ( ℝ , ℝ N ) , where α ∈ ( 1 / 2 , 1 ) , t ∈ ℝ , u = ( u 1 , … , u N ) T ∈ ℝ N , and L ∈ C ( ℝ , ℝ N 2 ) is a symmetric and positive definite matrix for all t ∈ ℝ , W ∈ C 1 ( ℝ × ℝ N × ℝ ) and ∇W is the gradient of W about u.

Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

Abstract In this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form t D ∞ α ( - ∞ D t α u ( t ) ) + L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) ${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$ , where α ∈ ( 1 2 , 1 ) ${\alpha \in (\frac{1}{2},1)}$ , t ∈ ℝ ${t\in \mathbb {R}}$ , u ∈ ℝ n ${u\in \mathbb {R}^n}$ , L ∈ C ( ℝ , ℝ n 2 ) ${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all t ∈ ℝ ${t\in \mathbb {R}}$ , W ∈ C 1 ( ℝ × ℝ n , ℝ ) ${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ∇ W ( t , u ) ${\nabla W(t,u)}$ is the gradient of W ( t , u ) ${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants 0 &lt; τ 1 &lt; τ 2 &lt; ∞ ${0&amp;lt;\tau _1&amp;lt;\tau _2&amp;lt; \infty }$ such that τ 1 | u | 2 ≤ ( L ( t ) u , u ) ≤ τ 2 | u | 2 ${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all ( t , u ) ∈ ℝ × ℝ n ${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and W ( t , u ) ${W(t,u)}$ is of the form ( a ( t ) / ( p + 1 ) ) | u | p + 1 ${({a(t)}/({p+1}))|u|^{p+1}}$ such that a ∈ L ∞ ( ℝ , ℝ ) ${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and 0 &lt; p &lt; 1 ${0&amp;lt;p&amp;lt;1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

Bifurcation results for a class of fractional Hamiltonian systems with Liouville–Weyl fractional derivatives

In this paper, by using variational methods, we prove the existence for a class of fractional Hamiltonian systems with Liouville–Weyl fractional derivatives { t D ∞ α ( - ∞ D t α u ( t ) ) + b ( t ) u ( t ) - λ u ( t ) = μ f ( t , u ( t ) ) , t ∈ ℝ u ∈ H α ( ℝ ) where and are the right and left inverse operators of the corresponding Liouville–Weyl fractional integrals of order α respectively, , are real parameters and is a function that satisfies some suitable conditions.

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015

Solutions for subquadratic fractional Hamiltonian systems without coercive conditions

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems urn:x-wiley:1704214:media:mma3031:mma3031-math-0001 where α ∈ (1 ∕ 2,1), , , is a symmetric and positive definite matrix for all , , and ∇ W is the gradient of W at u. The novelty of this paper is that, assuming L is bounded in the sense that there are constants 0 &lt; τ1 &lt; τ2 &lt; + ∞ such that τ1 | u | 2 ≤ (L(t)u,u) ≤ τ2 | u | 2 for all and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems, Math. Methods Appl. Sci., DOI:10.1002/mma.2941] are generalized and significantly improved. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Variational approach to solutions for a class of fractional Hamiltonian systems

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: urn:x-wiley:01704214:media:mma2941:mma2941-math-0001 where α ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ∇ W is the gradient of W at u. The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley &amp; Sons, Ltd.

About fractional Hamiltonian systems

We follow Stanislavsky's approach of Hamiltonian formalism for fractional systems, as a model problem for the study of chaotic Hamiltonian systems. We prove that his formalism can be retrieved from the fractional embedding theory. We deduce that the fractional Hamiltonian systems of Stanislavsky stem from a particular least action principle, said to be causal. In this case, the fractional embedding becomes coherent.

Fractional Hamiltonian analysis of irregular systems

The fractional Hamiltonian systems with linearly dependent constraints are investigated within fractional Riemann–Liouville derivatives. One example is analyzed in details and the consistency of fractional Euler–Lagrange and Hamilton equations is examined.

Fractional generalization of gradient and Hamiltonian systems

We consider a fractional generalization of Hamiltonian and gradient systems. We use differential forms and exterior derivatives of fractional orders. We derive fractional generalization of Helmholtz conditions for phase space. Examples of fractional gradient and Hamiltonian systems are considered. The stationary states for these systems are derived.

Fractional extensions of the classical isotropic oscillator and the Kepler problem

The class of fractional Hamiltonian systems that generalize the classical problem of the two-dimensional (2D) isotropic harmonic oscillator and the Kepler problem is considered. It is shown that, in the 4D space of structural parameters, the 2D isotropic harmonic oscillator can be extended along a line in such a way that the orbits remain closed and oscillations remain isochronous. Likewise, the Kepler problem can be extended along a line in such a way that the orbits remain closed for all finite motions and the third Kepler law remains valid. These curves lie on the 2D surfaces where any dynamical system is characterized by the same rotation number for all finite motions.