Two-Dimensional Manifolds of Modified Chen System with Time Delay

Two-dimensional unstable manifolds of the modified Chen system are constructed at equilibrium solution by “expanding up” along the unstable eigen-direction, hence it is tangent to the unstable eigenspace. In general, unstable manifold expands to the attraction basin of the corresponding limit cycle or attractor. With the introduction of time delay, the two-dimensional unstable manifold of an unstable focus is simulated via expanding solution orbits with restriction condition on the associated foliations. The simulated unstable manifold coincides with the attraction basin of the limit cycle of the delay differential equations.

A Note on the Local Stability Theory for Caputo Fractional Planar System

In this manuscript a new approach into analyzing the local stability of equilibrium points of non-linear Caputo fractional planar systems is shown. It is shown that the equilibrium points of such systems can be a stable focus or unstable focus. In addition, it is proposed that previous results regarding the stability of equilibrium points have been incorrect, the results here attempt to correct such results. Lastly, it is proposed that a Caputo fractional planar system cannot undergo a Hopf bifurcation, contrary to previous results prior. Though, it is shown that such systems can undergo a Hopf bifurcation (topologically).

Focus of Attention in Children with ASD

The special way of understanding the world and the brain mechanisms of cognitive impairment in children with autism spectrum disorders (ASD) are becoming the core topics in the autism research. The article presents a study of the general characteristics of attention in children of primary school age with ASD using instruments aimed at detecting impairments of attention parameters: Schulte Tables, Go/No Go Task, Kraepelin’s Arithmetic Test, Stroop Test used to study the focus of attention in the context of the interference of verbal and object stimuli. Results of study of attention in children with ASD seemed to be ambivalent. Children with ASD are successful in performing the Schulte Tables; on the contrary, the Go/No Go Task shows the difficulties of performing the tasks and inhibition of dominant response. Different degrees of unstable focus of attention in tasks on visual perception also shown. The features of focusing attention in conditions of interference require further research. Preliminary results showed that the focus of attention in younger schoolchildren with ASD is associated with a preference for verbal rather than object characteristics of visual stimuli.

STABILITY OF THE REST POINTS FRACTIONAL OSCILLATOR FITZHUGH-NAGUMO

В работе с помощью качественного анализа были исследованы на устойчивость точки покоя дробного осциллятора ФитцХью-Нагумо в соизмеримом и несоизмеримом случаях. Для соответвующей точки покоя, с помощью численного метода теории конечно-разностных схем, была построена фазовая траектория. Показано, что точки покоя могут быть как асимптотически устойчивыми, что соответствуют устойчивым фокусам, так и являться асимптотически неустойчивыми (неустойчивыми фокусами), причем для них фазовые таректории, как правило, выходят на предельный цикл. In this paper, using the qualitative analysis, we studied the stability of the point of rest of the fractional oscillator FitzHugh-Nagumo in commensurate and incommensurable cases. For the corresponding point of rest, using the numerical method of the theory of finite difference schemes, phase trajectories were constructed. It is shown that quiescent points can be both asymptotically stable, which correspond to stable focus, and are asymptotically unstable (unstable focus), and for them the phase trajectories usually go to the limit cycle.

The Influence of Initial Values on Spatial Coherence Resonance in a Neuronal Network

Noise-induced single spatial coherence resonance (CR) and multiple spatial CRs simulated in a network have been reported independently in previous studies. In this paper, the relationship between the single and multiple spatial CRs is established by adjusting the initial values of the network composed of Morris–Lecar (ML) model neurons. The ML model manifests a saddle-node bifurcation on an invariant cycle through which a resting state is changed to a stable limit cycle corresponding to period-1 firing. Under resting state, a stable node, a saddle, and an unstable focus coexist. The membrane potential of the unstable focus is much higher than that of the stable node. When the initial value is closer to the unstable focus, the residence time of membrane potential on a high level is longer; correspondingly, the spatial CRs appear more frequently with respect to noise intensity and the coherence degree becomes stronger. The single spatial CR is induced by noise with high intensity. Multiple spatial CRs are induced by noise with high, middle, and even low noise intensities, respectively. When the initial values are closer to an unstable focus, the residence time of membrane potentials on a higher level is longer, which is important to the generation of multiple CRs, and builds a relationship between single and multiple spatial CRs.

Finite-Time Formation Control without Collisions for Multiagent Systems with Communication Graphs Composed of Cyclic Paths

This paper addresses the formation control problem without collisions for multiagent systems. A general solution is proposed for the case of any number of agents moving on a plane subject to communication graph composed of cyclic paths. The control law is designed attending separately the convergence to the desired formation and the noncollision problems. First, a normalized version of the directed cyclic pursuit algorithm is proposed. After this, the algorithm is generalized to a more general class of topologies, including all the balanced formation graphs. Once the finite-time convergence problem is solved we focus on the noncollision complementary requirement adding a repulsive vector field to the previous control law. The repulsive vector fields display an unstable focus structure suitably scaled and centered at the position of the rest of agents in a certain radius. The proposed control law ensures that the agents reach the desired geometric pattern in finite time and that they stay at a distance greater than or equal to some prescribed lower bound for all times. Moreover, the closed-loop system does not exhibit undesired equilibria. Numerical simulations and real-time experiments illustrate the good performance of the proposed solution.

Flow topology in compressible turbulent boundary layer

The flow topologies of compressible turbulent boundary layers at Mach 2 are investigated by means of direct numerical simulation (DNS) of the compressible Navier–Stokes equations, and statistical analysis of the invariants of the velocity gradient tensor. We identify a preference for an unstable focus/compressing topology in the inner layer and an unstable node/saddle/saddle (UN/S/S) topology in the outer layer. The dissipation and dissipation production originate mainly from this UN/S/S topology. The enstrophy depends mainly on an unstable focus/stretching (UFS) topology, and the enstrophy production relies on a UN/S/S topology in the inner layer and on a UFS topology in the outer layer. The compressibility effect on the statistical properties of the topologies is investigated in terms of the ‘incompressible’, compressed and expanding regions. It is found that the locally compressed region tends to be more stable and the locally expanding region tends to be more dissipative. The compressibility is mainly related to unstable focus/compressing and stable focus/stretching topologies. Moreover, the features of the average dissipation, enstrophy, dissipation production and enstrophy production of the various topologies are clarified in the locally compressed and expanding regions.