Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives

The link between the treatments of constrained systems with fractional derivatives by using both Hamiltonian and Lagrangian formulations is studied. It is shown that both treatments for systems with linear velocities are equivalent.     

Synchronization of nonlinear fractional order systems

This paper deals with the master-slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown state variables. For solving this problem we introduce a Fractional Algebraic Observability (FAO) property which is used as a main tool in the design of the master system. As numerical examples we consider a fractional order Rössler hyperchaotic system and a fractional order Lorenz chaotic system and by means of some simulations we show the effectiveness of the suggested approach.     

Ground state solution for a class fractional Hamiltonian systems

In this paper, we consider a class of Hamiltonian systems of the form $_tD_\infty^\alpha(_{-\infty} D_t^\alpha u(t))+L(t) u(t)-\nabla W(t,u(t))=0$ where $\alpha\in(\frac{1}{2},1)$, $_{-\infty}D_t^\alpha$ and $_{t}D_\infty^\alpha$ are left and right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole axis $R$ respectively. Under weaker superquadratic conditions on the nonlinearity and asymptotically periodic assumptions, ground state solution is obtained by mainly using Local Mountain Pass Theorem, Concentration-Compactness Principle and a new form of Lions Lemma respect to fractional differential equations.     

The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

Triple pendulum model involving fractional derivatives with different kernels

The aim of this work is to study the non-local dynamic behavior of triple pendulum-type systems. We use the Euler-Lagrange and the Hamiltonian formalisms to obtain the dynamic models, based on the Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivative definitions. In these representations, an auxiliary parameter σ is introduced, to define the equations in a fractal temporal geometry, which provides an entire new family of solutions for the dynamic behavior of the pendulum-type systems. The phase diagrams allow to visualize the effect of considering the fractional order approach, the classical behavior is recovered when the order of the fractional derivative is 1.     

Periodic solutions of Weyl fractional order integral systems†

In this paper, we show the existence and uniqueness results for periodic solutions of Weyl fractional order integral systems. A numerical example is given to illustrate our theoretical results. Our results show that periodic orbits can be obtained by putting the periodic conditions to some certain fractional order integral systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Integrable couplings of fractional L‐hierarchy and its Hamiltonian structures

Based on a general isospectral problem of fractional order, a fractional bilinear form variational identity, the new integrable coupling of fractional L‐hierarchy and the Hamiltonian structures of the integrable coupling of fractional L‐hierarchy are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Fault detection based on fractional order models: Application to diagnosis of thermal systems

The aim of this paper is to propose diagnosis methods based on fractional order models and to validate their efficiency to detect faults occurring in thermal systems. Indeed, it is first shown that fractional operator allows to derive in a straightforward way fractional models for thermal phenomena. In order to apply classical diagnosis methods, such models could be approximated by integer order models, but at the expense of much higher involved parameters and reduced precision. Thus, two diagnosis methods initially developed for integer order models are here extended to handle fractional order models. The first one is the generalized dynamic parity space method and the second one is the Luenberger diagnosis observer. Proposed methods are then applied to a single-input multi-output thermal testing bench and demonstrate the methods efficiency for detecting faults affecting thermal systems.     

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results.     

Stability by linear approximation and the relation between the stability of difference and differential fractional systems

The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.     

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ? ?² is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (?2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.     

Analysis of fractional order differential coupled systems

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B‐vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Recursive prediction error identification of fractional order models

This paper deals with time domain identification of fractional order systems. A new identification technique is developed providing recursive parameters estimation of fractional order models. The identification model is defined by a generalized ARX structure obtained by discretization of a continuous fractional order differential equation. The parameters are then estimated using the recursive least squares and the recursive instrumental variable algorithms extended to fractional order cases. Finally, the quality of the proposed technique is illustrated and compared through the identification of simulated fractional order systems.     

Nonlinear fractional cone systems with the Caputo derivative

Nonlinear fractional cone systems involving the Caputo fractional derivative are considered. We establish sufficient conditions for the existence of at least one cone solution to such systems. Sufficient conditions for the unique existence of the cone solution to a nonlinear fractional cone system are given.     

Quantum statistical systems in <Emphasis Type="Italic">D</Emphasis>-dimensional space using a fractional derivative

We investigate the thermodynamic properties of some quantum statistical systems with a fractional Hamiltonian in D-dimensional space. We calculate the partition function of the system of N fractional quantum oscillators and the thermodynamic quantities associated with it. We consider the thermal and critical properties of both Bose and Fermi gases in the context of the fractional energy and described by a fractional derivative.     

Modified projective synchronization of uncertain fractional order hyperchaotic systems

Base on the stability theory of fractional order system, this work mainly investigates modified projective synchronization of two fractional order hyperchaotic systems with unknown parameters. A controller is designed for synchronization of two different fractional order hyperchaotic systems. The method is successfully applied to modified projective synchronization between fractional order Rössler hyperchaotic system and fractional order Chen hyperchaotic system, and numerical simulations illustrate the effectiveness of the obtained results.     

Hamilton-Jacobi fractional mechanics

As a continuation of Rabei et al. work [Eqab M. Rabei, Khaled I. Nawafleh, Raed S. Hijjawi, Sami I. Muslih, Dumitru Baleanu, The Hamilton formalism with fractional derivatives, J. Math. Anal. Appl. 327 (2007) 891-897], the Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives. The Hamilton-Jacobi function in configuration space is obtained in a similar manner to the usual mechanics. Two problems are considered to demonstrate the application of the formalism. The result is found to be in exact agreement with Agrawal's formalism.     

Laplacian pretty good fractional revival

We develop the theory of pretty good fractional revival in quantum walks on graphs using their Laplacian matrices as the Hamiltonian. We classify the paths and the double stars that have Laplacian pretty good fractional revival.     

Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control

Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems.     

Synchronization of fractional order chaotic systems using active control method

In this article, the active control method is used for synchronization of two different pairs of fractional order systems with Lotka–Volterra chaotic system as the master system and the other two fractional order chaotic systems, viz., Newton–Leipnik and Lorenz systems as slave systems separately. The fractional derivative is described in Caputo sense. Numerical simulation results which are carried out using Adams–Bashforth–Moulton method show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order chaotic systems while it also allows both the systems to remain in chaotic states. A salient feature of this analysis is the revelation that the time for synchronization increases when the system-pair approaches the integer order from fractional order for Lotka–Volterra and Newton–Leipnik systems while it reduces for the other concerned pair.