Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to \(\tau \) with probability \(p_{\mathrm{delay}}\) and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling \(\tau \) and \(p_{\mathrm{delay}}\) under three cases. In these three cases, excitatory synapses are always considered to delayed with probability \(p_{\mathrm{delay}}\), while inhibitory synapses are considered to be without delays (case I), delayed with probability \(p_{\mathrm{delay}}\) (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small \(p_{\mathrm{delay}}\), while it could induce synchronization transitions at \(\tau \) around integer multiples of the period of individual neuron T when \(p_{\mathrm{delay}}\) is large enough, while in the case III, partial time delay could induce synchronization transitions at \(\tau \) being around odd integer multiples of T / 2 for small \(p_{\mathrm{delay}}\) and at \(\tau \) being around integer multiples of T for large \(p_{\mathrm{delay}}\). Most interesting observation is that partial time delay could induce frequent synchronization transitions at \(\tau \) being around integer multiples of T / 2 for intermediate \(p_{\mathrm{delay}}\). Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large \(p_{\mathrm{delay}}\) under the three cases.     

Periodically intermittent control strategies for $$\varvec{\alpha }$$-exponential stabilization of fractional-order complex-valued delayed neural networks

This paper studies the global \(\alpha \)-exponential stabilization of a kind of fractional-order neural networks with time delay in complex-valued domain. To end this, several useful fractional-order differential inequalities are set up, which generalize and improve the existing results. Then, a suitable periodically intermittent control scheme with time delay is put forward for the global \(\alpha \)-exponential stabilization of the addressed networks, which include feedback control as a special case. Utilizing these useful fractional-order differential inequalities and combining with the Lyapunov approach and other inequality techniques, some novel delay-independent criteria in terms of real-valued algebraic inequalities are obtained to ensure global \(\alpha \)-exponential stabilization of the discussed networks, which are very simple to implement in practice and avert to calculate the complex matrix inequalities. Finally, the availability of the theoretical criteria is verified by an illustrative example with simulations.     

$$H_\infty $$ switching synchronization for multiple time-delay chaotic systems subject to controller failure and its application to aperiodically intermittent control

This paper is concerned with the \(H_\infty \) synchronization control problem for a class of chaotic systems with multiple delays in the presence of controller temporary failure. Based on the idea of switching, the synchronization error systems with controller temporary failure are modeled as a class of switched synchronization error systems. Then, a switching condition that incorporates the controller failure time is constructed by using piecewise Lyapunov functional and average dwell-time methods, such that the switched synchronization error systems are exponentially stable and satisfy a weighted \(H_\infty \) performance level. In the meantime, a switching state feedback \(H_\infty \) controller is derived by solving a set of linear matrix inequalities. More incisively, the obtained results can also be applied to the issue of aperiodically intermittent control. Finally, three simulation examples are employed to illustrate the effectiveness and potential of the proposed method.     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.     

Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions

We study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible connection with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino, focusing close to the helimagnet/ferromagnet transition point corresponding to the critical value of the frustration parameter \(\alpha=4\). We reformulate this problem for any \(\alpha\geq0\) in the framework of surface energies in nonconvex discrete systems with nearest neighbours ferromagnetic and next-to-nearest neighbours antiferromagnetic interactions and we link it to the gradient theory of phase transitions, by showing a uniform equivalence by \(\varGamma \)-convergence on \([0,4]\) with Modica-Mortola type functionals.     

On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system

We consider the well-known Sprott A system, which depends on a single real parameter a and, for \(a=1\), was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for \(a=0\), the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For \(a\ne 0\), the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for \(a>0\) small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the \(\alpha \)- and \(\omega \)-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for \(a<1\). Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for \(a>0\), the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered.     

Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators

The experimental realization of a chaotic secure communication system is introduced herein taking advantage of Arduino’s open source integrated development environment along with its preset functions that reduce development time and complexity. The system uses multi-directional multi-scroll chaotic oscillators that are based on piecewise-linear functions. Experimental results are given for chaos generators in 1-direction (1-D), 2-D and 3-D of up to 20, \(20\times 20\) and \(20\times 20\times 20\) scrolls, respectively. Using two chaos generators, a chaotic secure communication system is implemented in a master–slave topology by applying generalized Hamiltonian forms and observer approach. Finally, we detail experimental results for the synchronization of \(6\times 6\) and \(20\times 20\) scroll attractors, which are used for transmitting a monochromatic image through the Arduino-based chaotic secure communication system.     

A novel approach for generating multi-direction multi-double-scroll attractors

Attention is focused in this work on quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies with a distance \(\omega _{d}\) apart, around a frequency \(\omega \) with \(\omega \gg \omega _{d}\) and its integer multiples, which are referred to as carrier frequencies. The ratio of the two frequencies \(\omega \) and \(\omega _{d}\) is an irrational number. A new method based on the traditional incremental harmonic balance (IHB) method with multiple timescales, referred to as Lau method, where two timescales, \(\tau _{1}=\omega t\) (a fast timescale) and \(\tau _{2}=\omega _{d}t\) (a slow timescale), are introduced, is presented to analyze quasiperiodic motion of nonlinear systems. An amplitude increment algorithm is adapted to deal with cases where the two frequencies \(\omega \) and \(\omega _{d}\) are    unknown a priori, in order to automatically trace frequency response of quasiperiodic motion of nonlinear systems and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasiperiodic free vibration of a hinged–clamped beam with internal resonance between two transverse modes are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration. The present IHB method is also used to analyze quasiperiodic free vibration of high-dimensional models of the hinged–clamped beam.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

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.     

A Shilnikov Phenomenon Due to State-Dependent Delay,by Means of the Fixed Point Index

The first part of this paper is a general approach towards chaotic dynamics for a continuous map \(f:X\supset M\rightarrow X\) which employs the fixed point index and continuation. The second part deals with the differential equation

$$\begin{aligned} x'(t)=-\alpha \,x(t-d_{{\varDelta }}(x_t)). \end{aligned}$$

with state-dependent delay. For a suitable parameter \(\alpha \) close to \(5\pi /2\) we construct a delay functional \(d_{{\varDelta }}\), constant near the origin, so that the previous equation has a homoclinic solution, \(h(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \), with certain regularity properties of the linearization of the semiflow along the flowline \(t\mapsto h_t\). The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.

     

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed \(c_0>0\). Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed \(c>0\). We prove that when \(c\ge c_0\), the species always dies out in the long-run, but when \(0<c<c_0\), the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form \(u_0(x)=\sigma \phi (x)\) with \(\phi \) fixed and \(\sigma >0\) a parameter, then there exists \(\sigma _0>0\) such that vanishing happens when \(\sigma \in (0,\sigma _0)\), borderline spreading happens when \(\sigma =\sigma _0\), and spreading happens when \(\sigma >\sigma _0\).     

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

In this paper we study the limit as \(\varepsilon \rightarrow 0\) of the singularly perturbed second order equation \(\varepsilon ^2 \ddot{u}_\varepsilon + \nabla _{\!x} V(t,u_\varepsilon (t))=0\), where V(t, x) is a potential. We assume that \(u_0(t)\) is one of its equilibrium points such that \(\nabla _{\!x}V(t,u_0(t))=0\) and \(\nabla _{\!x}^2V(t,u_0(t))>0\). We find that, under suitable initial data, the solutions \(u_\varepsilon \) converge uniformly to \(u_0\), by imposing mild hypotheses on V. A counterexample shows that they cannot be weakened.     

Topological Pressure of Generic Points for Amenable Group Actions

Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure \(\mu \) and any continuous map \(\varphi \),

$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$

where \(\{F_n\}\) is a Følner sequence, \(X_{\mu }\) is the set of generic points of \(\mu \) with respect to (w.r.t. for short) \(\{F_n\}\), \(P(X_{\mu },\varphi ,\{F_n\})\) is the topological pressure of \(X_{\mu }\) for \(\varphi \) w.r.t. \(\{F_n\}\) and \(h_{\mu }(X)\) is the measure-theoretic entropy.

     

Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).     

Pinning lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay

In this paper, a linear and adaptive feedback pinning strategy is used to study the lag synchronization of complex dynamical networks with known state time-delay and unknown channel time-delay. Firstly, based on the Lyapunov stability theory, a novel Lyapunov functional, which involves the estimated error \(\hat{e}_{i}(t)\) rather than the general synchronization error \(e_{i}(t)\), is constructed. Secondly, in view of the unknown information of the channel time-delay, two available pinning controllers are designed such that the considered networks achieve lag synchronization. Furthermore, by a proper adaptation mechanism, we estimate the unknown channel time-delay successfully under the case that the initial value of the estimated channel time-delay is larger than true channel time-delay, i.e., \(\hat{\tau }(0)>\tau \) and the another case \(\hat{\tau }(0)<\tau \). Finally, the effectiveness and correctness of the lag synchronization criteria are verified through two simulation experiments.     

Continuity and Invariance of the Sacker–Sell Spectrum

The Sacker–Sell (also called dichotomy or dynamical) spectrum \(\varSigma \) is a fundamental concept in the geometric, as well as for a developing bifurcation theory of nonautonomous dynamical systems. In general, it behaves merely upper-semicontinuously and a perturbation theory is therefore delicate. This paper explores an operator-theoretical approach to obtain invariance and continuity conditions for both \(\varSigma \) and its dynamically relevant subsets. Our criteria allow to avoid nonautonomous bifurcations due to collapsing spectral intervals and justify numerical approximation schemes for \(\varSigma \).     

Exact solution and invariant for fractional Cattaneo anomalous diffusion of cells in two-dimensional comb framework

This paper presents an investigation on anomalous diffusion of cells in a two-dimensional comb framework with effects of fractional Cattaneo flux. Formulated governing equation is an evolution equation with the coexisting characteristics of parabolic (diffusion) and hyperbolic (wave) for \(\alpha \) in (0, 1). Exact solution is obtained by the special fractional integral transformations, and a novel invariant is established, i.e., \(\left\langle {x^{2}\left( t \right) } \right\rangle \cdot \left\langle P \right\rangle = 0.5\) (the mean square displacement multiplied by the total number of cells along the x-axis = 0.5). Moreover, the characteristics of cells distribution, the total number and the mean square displacement of cells along the x-axis with different involved parameters, especially with the fractional parameter evolution, are shown graphically and analyzed in detail. For the cells distribution versus x, it turns from parabolic and hyperbolic with the decrease in t or the increase in \(\alpha \) or \(\xi \). It is monotonically decreasing for the cells distribution versus \(\alpha \) with different x, t and \(\xi \). For the distribution versus t with different \(\alpha \) and \(\xi \) or versus \(\alpha \) with different t, it is monotonically decreasing for the distribution of total number while monotonically increasing for the distribution of mean square displacement. It is remarkable that the anomalous subdiffusion happens along the x-axis for arbitrary parameters which is different from the classical Cattaneo diffusion.