Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris–Lecar neuronal model

Adaptive infinite impulse response filters have received much attention due to its utilization in a wide range of real-world applications. The design of the IIR filters poses a typically nonlinear, non-differentiable and multimodal problem in the estimation of the coefficient parameters. The aim of the current study is the application of a novel hybrid optimization technique based on the combination of cellular particle swarm optimization and differential evolution called CPSO–DE for the optimal parameter estimation of IIR filters. DE is used as the evolution rule of the cellular part in CPSO to improve the performance of the original CPSO. Benchmark IIR systems commonly used in the specialized literature have been selected for tuning the parameters and demonstrating the effectiveness of the CPSO–DE method. The proposed CPSO–DE method is experimentally compared with two new design methods: the tissue-like membrane system (TMS), the hybrid particle swarm optimization and gravitational search algorithm (HPSO–GSA), the original CPSO-outer and CPSO-inner, and classical implementations of PSO, GSA and DE. Computational results and comparison of CPSO–DE with the other evolutionary and hybrid methods show satisfactory results. The hybridization of CPSO and DE demonstrates powerful estimation ability. In particular, to our knowledge, this hybridization has not yet been investigated for the IIR system identification.     

Stability of fractional-order systems with Prabhakar derivatives

Nonlinear Dynamics - Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields)...     

Existence of oscillatory solutions along the path of longitudinal flight equilibriums of an unmanned aircraft, when the automatic flight control system fails

The motion around the center of mass of a rigid unmanned aircraft, whose flight control system fails, in an “Aero Data Model In a Research Environment” is described, by a set of nine nonlinear ordinary differential equations. The longitudinal flight with constant forward velocity is described by a subset of three nonlinear differential equations, obtained from the general system. In this paper, the existence of oscillatory solutions of this system of three differential equations is proved by means of coincidence degree theory and Mawhin's continuation theorem.     

Multiple periodic solutions in impulsive hybrid neural networks with delays

The existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid Hopfield-type neural networks with both time-dependent and distributed delays, using the Leray-Schauder fixed point theorem and Lyapunov functionals. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses, providing flexibility for the analysis and design of delayed neural networks with impulse effects. Examples are given.     

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.     

Emergence of diverse dynamical responses in a fractional-order slow–fast pest–predator model

To explore the impact of pest-control strategy on integrated pest management, a three-dimensional (3D) fractional- order slow–fast prey–predator model is introduced in this article. The prey community (assumed as pest) represents fast dynamics and two predators exhibit slow dynamical variables in the three-species interacting prey–predator model. In addition, common enemies of that pest are assumed as predators of two different species. Pest community causes serious damage to the economy. Fractional-order systems can better describe the real scenarios than classical-order dynamical systems, as they show previous history-dependent properties. We establish the ability of a fractional-order model with Caputo’s fractional derivative to capture the dynamics of this prey–predator system and analyze its qualitative properties. To investigate the importance of fractional-order dynamics on the behavior of the pest, we perform the local stability analysis of possible equilibrium points, using certain assumptions for different sets of parameters and reveal that the fractional-order exponent has an impact on the stability and the existence of Hopf bifurcations in the prey–predator model. Next, we discuss the existence, uniqueness and boundedness of the fractional-order system. We also observe diverse oscillatory behavior of different amplitude modulations including mixed mode oscillations (MMOs) for the fractional-order prey–predator model. Higher amplitude pest periods are interspersed with the outbreaks of small pest concentration. With the decrease of fractional-order exponent, small pest concentration increases with decaying long pest periods. We further notice that the reduced-order model is biologically significant and sensitive to the fractional-order exponent. Additionally, the dynamics captures adaptation that occurs over multiple timescales and we find consistent differences in the characteristics of the model for various fractional exponents.

     

Numerical analysis of the oscillation susceptibility along the path of longitudinal flight equilibria of a reentry vehicle

In this paper, it is shown that when the automatic flight control system (AFCS) is decoupled, then on the path of longitudinal flight equilibria of the ALFLEX reentry vehicle, there exist saddle–node bifurcations resulting in oscillations: when the elevator angle exceeds the bifurcation value, the angle of attack and pitch rate oscillate with the same period, while the pitch angle increases or decreases infinitely. Hence, the orbit of the system is spiraling. It is also shown that a mild (non-catastrophic) stability loss occurs due to the bifurcations. Based on this analysis, an automatic flight control design is developed.     

Multistability in impulsive hybrid Hopfield neural networks with distributed delays

In this paper we investigate multistability of Hopfield-type neural networks with distributed delays and impulses, by using Lyapunov functionals, stability theory and control by impulses. Example and simulation results are given to illustrate the effectiveness of the results.     

Chaotic Dynamics of a Delayed Discrete-Time Hopfield Network of Two Nonidentical Neurons with no Self-Connections

A bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two delays, two internal decays and no self-connections, choosing the product of the interconnection coefficients as the characteristic parameter for the system. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified, and the existence of Fold/Cusp, Neimark–Sacker and Flip bifurcations is proved. All these bifurcations are analyzed by applying the center manifold theorem and the normal form theory. It is shown that the dynamics in a neighborhood of the null solution become more and more complex as the characteristic parameter grows in magnitude and passes through the bifurcation values. Under certain conditions, it is proved that if the magnitudes of the interconnection coefficients are large enough, the neural network exhibits Marotto’s chaotic behavior.