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.     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

Dissipative and generative fractional electric elements in modeling $${varvec{RC}}$$ RC and $${varvec{RL}}$$ RL circuits

Nonlinear Dynamics - Two types of constitutive equations consisting of instantaneous and power type hereditary contributions are proposed in order to model generalized capacitor (inductor). The...     

Robust {\varvec{L}}_{\varvec{2}}-gain control for a class of cascade switched nonlinear systems

In this paper, we study the robust finite \(L_2 \) -gain control for a class of cascade switched nonlinear systems with parameter uncertainty. Each subsystem of the switched system under consideration is composed of a zero-input asymptotically stable nonlinear part which is a lower dimension switched system, and of a linearizable part. The uncertainty appears in the control channel of each subsystem. We give sufficient conditions under which the nonlinear feedback controllers are derived to guarantee that the \(L_2 \) -gain of the closed-loop switched system is less than a prespecified value for all admissible uncertainty under arbitrary switching. Moreover, we also develop the \(L_2\) -gain controllers for the switched systems with nonminimum phase case.     

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.     

Isochronicity for a $$\varvec{Z_{2}}$$ Z 2 -equivariant cubic system

Nonlinear Dynamics - This paper is concerned with the bi-isochronous centers problem for a cubic systems in $${Z}_2$$ -equivariant vector field. Being based on bi-centers condition, we compute the...     

Global exponential stability of stochastic memristor-based complex-valued neural networks with time delays

In recent years, the dynamic behaviors of complex-valued neural networks have been extensively investigated in a variety of areas. This paper focuses on the stability of stochastic memristor-based complex-valued neural networks with time delays. By using the Lyapunov stability theory, Halanay inequality and Itô formula, new sufficient conditions are obtained for ensuring the global exponential stability of the considered system. Moreover, the obtained results not only generalize the previously published corresponding results as special cases for our results, but also can be checked with the parameters of system itself. Finally, simulation results in three numerical examples are discussed to illustrate the theoretical results.     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Intermittent control on switched networks via {\varvec{\omega }}-matrix measure method

This paper is concerned with global exponential synchronization problem for a class of switched delay networks with interval parameters uncertainty, different from the most existing results, without constructing complex Lyapunov–Krasovskii functions; \(\omega \) -matrix measure method is firstly introduced to switched interval networks, combining Halanay inequality technique, designing proper intermittent and non-intermittent control strategy; some easy-to-verify synchronization criteria are given to ensure the global exponential synchronization of switched interval networks under arbitrary switching rule and for admissible interval uncertainties. Moreover, as an application, the proposed scheme can be applied to chaotic neural networks. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results and show the obtained results via employing \(\omega \) -measure are superior to previous results by using \(1\) -measure.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Effect of perforation on flow past a conic cylinder at $${\varvec{Re}}~=~100$$: wavy vortex and sign laws

In order to find the intrinsic physical mechanism of the original Kármán vortex wavily distorted across the span due to the introduction of three-dimensional (3-D) geometric disturbances, a flow past a peak-perforated conic shroud is numerically simulated at a Reynolds number of 100. Based on previous work by Meiburg and Lasheras (1988), the streamwise and vertical interactions with spanwise vortices are introduced and analyzed. Then vortex-shedding patterns in the near wake for different flow regimes are reinspected and illustrated from the view of these two interactions. Generally, in regime I, spanwise vortices are a little distorted due to the weak interaction. Then in regime II, spanwise vortices, even though curved obviously, are still shed synchronously with moderate streamwise and vertical interactions. But in regime III, violently wavy spanwise vortices in some vortex-shedding patterns, typically an \(\Omega \)-type vortex, are mainly attributed to the strong vertical interactions, while other cases, such as multiple vortex-shedding patterns in sub-regime III-D, are resulted from complex streamwise and vertical interactions. A special phenomenon, spacial distribution of streamwise and vertical components of vorticity with specific signs in the near wake, is analyzed based on two models of streamwise and vertical vortices in explaining physical reasons of top and bottom shear layers wavily varied across the span. Then these two models and above two interactions are unified. Finally two sign laws are summarized: the first sign law for streamwise and vertical components of vorticity is positive in the upper shear layer, but negative in the lower shear layer, while the second sign law for three vorticity components is always negative in the wake.     

Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on $${\mathbb {S}}^1$$ and $${\mathbb {S}}^2$$S2

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to \({\mathbb {S}}^1\), but unstable if the bodies are considered in \({\mathbb {S}}^2\).     

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in $${\mathbb{R}^{n}}$$

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step. Research partially supported by the INTAS project Attractors for Equations of Mathematical Physics, by CRDF and by the Alexander von Humboldt–Stiftung.     

The Stationary Oseen Equations in $${\mathbb{R}}^{3}$$ . An Approach in Weighted Sobolev Spaces

In this paper we solve the stationary Oseen equations in . The behavior of the solutions at infinity is described by setting the problem in weighted Sobolev spaces including anisotropic weights. The study is based on a L^p theory for 1 < p < ∞.     

Globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching

The problem of globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is studied in this paper. By using the Lyapunov?CKrasovskii method and the stochastic analysis approach, a sufficient condition to ensure globally exponential stability for the stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching is derived. Finally, a numerical example is given to illustrate the effectiveness of the result proposed in this paper.     

Adaptive stabilization for a class of uncertain $${textit{p}}$$ p -normal nonlinear systems via a generalized homogeneous domination technique

Nonlinear Dynamics - This paper investigates a generalized homogeneous adaptive stabilization method for a class of high-order nonlinear systems without controllable/observable linearizations....     

Nonparabolic Asymptotic Limits of Solutions of the Heat Equation on $${\mathbb{R}}^N$$

In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.      

Switching stabilization and { H}_{\varvec{\infty }} performance of a class of discrete switched LPV system with unstable subsystems

Stability and $H_\infty $ performance are analyzed in this paper for a class of discrete switched linear parameter-varying (LPV) systems in which all subsystems’ state-space matrices are parametrically affine, and any subsystem is not stable for parameters varying in a convex set. A switching law is designed to stabilize and satisfy the $H_\infty $ performance of the switched LPV system. By means of the multiple Lyapunov functions method, linear matrix inequality (LMI) conditions for the existence of parameter-dependent Lyapunov functions are proposed. An example shows the effectiveness of the proposed methods.