Chaotic behavior of a class of discontinuous dynamical systems of fractional-order

In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that end, the initial value problem is first transformed, by using the Filippov regularization (Filippov in Differential Equations with Discontinuous Right-Hand Sides, 1988), into a set-valued problem of fractional-order, then by Cellina’s approximate selection theorem (Aubin and Cellina in Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska in Set-valued Analysis, 1990). The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical scheme proposed by Diethelm et al. (Nonlinear Dyn. 29:3–22, 2002). Two typical examples of systems belonging to this class are analyzed and simulated.     

Fractional conservation laws in optimal control theory

Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler–Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative of the state variable. Partially presented at FDA ’06—2nd IFAC Workshop on Fractional Differentiation and its Applications, 19–21 July 2006, Porto, Portugal.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Synchronization of piecewise continuous systems of fractional order

This paper proves analytically that synchronization of a class of piecewise continuous fractional-order systems can be achieved. Since there are no dedicated numerical methods to integrate differential equations with discontinuous right-hand sides for fractional-order models, Filippov’s regularization (Filippov, Differential Equations with Discontinuous Right-Hand Sides, 1988) is applied, and Cellina’s Theorem (Aubin and Cellina, Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska, Set-valued Analysis, 1990) is used. It is proved that the corresponding initial value problem can be converted to a continuous problem of fractional-order systems, to which numerical methods can be applied. In this way, the synchronization problem is transformed into a standard problem for continuous fractional-order systems. Three examples are presented: the Sprott’s system, Chen’s system, and Shimizu–Morioka’s system.     

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

A historical perspective of Menabrea’s theorem in elasticity

This paper presents the theorem proposed by Luigi Federico Menabrea to study linear elastic redundant systems. Some of Menabrea’s papers on the subject are examined, as well as the criticism and the corrections brought to his first proof. We consider Menabrea’s work in the frame of the studies of his contemporaries; we try to provide a historical and epistemological background for Menabrea’s theorem and for its consequences in modern mechanics.     

On the instability of steady motion

This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov’s generalization of the first Lyapunov’s method, as well as Rout’s method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin’s series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938–944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov’s generalizations of the first Lyapunov’s method are formulated in a form including one variable less than was the case in existing literature.     

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes’ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes.     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

Lyapunov’s first and second instability theorems for Caputo fractional-order systems

As well as proving stability, Lyapunov functions can also be used to prove instability, for which there are two well-known theorems: Lyapunov’s first and second instability theorems, in the integer-order (ordinary differential equation) case. However, these instability theorems for Caputo fractional-order systems remain blank, due to the long lack of general results on continuation of solution and Caputo fractional derivative of Lyapunov functions along trajectories. In this paper, based on recent advances in these two aspects, the Lyapunov’s first and second instability theorems for Caputo fractional-order systems are presented with proofs and then illustrated by examples.

     

Knudsen’s Permeability Correction for Tight Porous Media

Various flow regimes including Knudsen, transition, slip and viscous flows (Darcy’s law), as applied to flow of natural gas through porous conventional rocks, tight formations and shale systems, are investigated. Data from the Mesaverde formation in the United States are used to demonstrate that the permeability correction factors range generally between 1 and 10. However, there are instances where the corrections can be between 10 and 100 for gas flow with high Knudsen number in the transition flow regime, and especially in the Knudsen’s flow regime. The results are of practical interest as gas permeability in porous media can be more complex than that of liquid. The gas permeability is influenced by slippage of gas, which is a pressure-dependent parameter, commonly referred to as Klinkenberg’s effect. This phenomenon plays a substantial role in gas flow through porous media, especially in unconventional reservoirs with low permeability, such as tight sands, coal seams, and shale formations. A higher-order permeability correlation for gas flow called Knudsen’s permeability is studied. As opposed to Klinkenberg’s correlation, which is a first-order equation, Knudsen’s correlation is a second-order approximation. Even higher-order equations can be derived based on the concept used in developing this model. A plot of permeability correction factor versus Knudsen number gives a typecurve. This typecurve can be used to generalize the permeability correction in tight porous media. We conclude that Knudsen’s permeability correlation is more accurate than Klinkenberg’s model especially for extremely tight porous media with transition and free molecular flow regimes. The results from this study indicate that Klinkenberg’s model and various extensions developed throughout the past years underestimate the permeability correction especially for the case of fluid flow with the high Knudsen number.     

A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control

In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincaré maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

Dissipation-induced instabilities and symmetry

The paradox of destabilization of a conservative or non-conservative system by small dissipation,or Ziegler’s paradox(1952),has stimulated a growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations.Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary,associated with Whitney’s umbrella.The first explanation of Ziegler’s paradox was given(much earlier)by Oene Bottema in 1956.The aspects of the mechanics and geometry of dissipation-induced instabilities with an application to rotor dynamics are discussed.     

Stability and stabilization of a class of fractional-order nonlinear systems for $$\varvec{0<}\,{\varvec{\alpha }} \,\varvec{< 2}$$

This paper investigates the stability and stabilization problem of fractional-order nonlinear systems for \(0<\alpha <2\). Based on the fractional-order Lyapunov stability theorem, S-procedure and Mittag–Leffler function, the stability conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with \(0<\alpha <2\) are proposed. Finally, typical instances, including the fractional-order nonlinear Chen system and the fractional-order nonlinear Lorenz system, are implemented to demonstrate the feasibility and validity of the proposed method.     

Impulsive stabilization of mechanical systems in Takagi–Sugeno models

Takagi–Sugeno fuzzy impulsive systems are analyzed for Lyapunov stability. Lyapunov’s second method is used to establish sufficient stability conditions for such systems. It is shown that these conditions are expressed by a system of matrix inequalities. Impulsive fuzzy control of two coupled pendulums is considered as an example     

Forced convection with laminar pulsating flow in a tube

The heat transfer model of laminar pulsating flow in a tube in rolling motion is established. The correlations of velocity, temperature and Nusselt number are obtained. The effects of several parameters on Nusselt number are investigated. The theoretical results are consistent with experimental data. Then the results are evaluated with Nield and Kuznetsov’s results. It is found that Nield and Kuznetsov’s results are not applicable for the laminar pulsating flow in nuclear power systems in ocean environments.     

Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.