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This study introduces a brand-new, four-dimensional hyperchaotic system and provides a full analysis of its dynamical characteristics. Time series, phase portraits, Poincaré maps, bifurcation diagrams, estimating the Lyapunov exponents, and determining the Kaplan–Yorke dimension are all used to examine the system. Additionally, the system's equilibrium points are located and their stability is explored. Through the use of active, adaptive, and sliding-mode control approaches, we have also synchronized the newly introduced system. The Lyapunov stability theory and Vaidyanathan's theorem are the foundations upon which the controllers are built to guarantee synchronization between the systems. Finally, MATLAB-based numerical simulations are used to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability of quaternion-valued neural networks with time-varying delays is discussed. On the basis of the matrix measure method and Halanay inequality, some sufficient criteria of exponential stability for quaternion-valued neural with delays are given. Different from the existing methods, the criteria of global exponential stability are obtained without constructing Lyapunov function. Moreover, the activation function is no longer assumed to be differentiable, which makes the analysis much easier. Finally, the numerical simulations are used to prove the validity of the main results. [ABSTRACT FROM AUTHOR]

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to consider a continuous data assimilation algorithm for the three-dimensional planetary geostrophic model in the case that the observable measurements, obtained continuously in time, may be contaminated by systematic errors. In this paper, we will provide some suitable conditions on the nudging parameter and the spatial resolution, which are sufficient to show that the approximation solution of the proposed continuous data assimilation algorithm converges to the unique exact unknown reference solution of the original system at an exponential rate, asymptotically in time, under the assumption that the observed data is free of error. [ABSTRACT FROM AUTHOR]

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master–slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties. [ABSTRACT FROM AUTHOR]

This paper investigates finite-time synchronization of fractional-order quaternion-valued neural networks (F-QV-NNs), involving uncertainties in the system parameters. We adopt a combined approach of sliding mode control (SMC) and a non-separation strategy to achieve finite-time synchronization. Based on SMC theory, we construct a specific sliding surface and design a controller to ensure the occurrence of sliding motion. To achieve the desired sliding motion, we apply the fractional Lyapunov direct method to guide the system's states to the designed sliding surface. Moreover, the derived sufficient conditions ensure finite-time synchronization within the models. Lastly, we give a numerical example to validate the effectiveness of the acquired results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to consider a discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation in the case that the observable measurements, obtained discretely in time, may be contaminated by systematic errors, which works for a general class of observable measurements, such as low Fourier modes and local spatial averages over finite volume elements. We will provide some suitable conditions to establish asymptotic in time estimates of the difference between the approximating solution and the unknown exact (reference) solution in some appropriate norms for these two different kinds of interpolation operators, which also shows that the approximation solution of the proposed discrete data assimilation algorithm will convergent to the unique unknown exact (reference) solution of the original system at an exponential rate, asymptotically in time if the observational measurements are free of error. [ABSTRACT FROM AUTHOR]

The existence of synchronization is an important issue in complex dynamical networks. In this paper, we study the synchronization of impulsive coupled oscillator networks with the aid of rotating periodic solutions of impulsive system. The type of synchronization is closely related to the rotating matrix, which gives an insight for finding various types of synchronization in a united way. We transform the synchronization of impulsive coupled oscillators into the existence of rotating periodic solutions in a relevant impulsive system. Some existence theorems about rotating periodic solutions for a non-homogeneous linear impulsive system and a nonlinear perturbation system are established by topology degree theory. Finally, we give two examples to show synchronization behaviors in impulsive coupled oscillator networks. [ABSTRACT FROM AUTHOR]

We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally. [ABSTRACT FROM AUTHOR]

This paper investigates the leader and leaderless bipartite synchronization with the signed network utilizing the model of multiple memristor and coupled delayed neural network in an event-triggered pinning control. The usage of the descriptor method in fractional-order neural networks in case of a non-differentiable delay can be seen in this paper. Further, Lyapunov functional criteria, including Lur'e Postnikov Lyapunov functional, is established, and bipartite leader and leaderless synchronization are proved. The obtained numerical results can be seen as accurate to the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers the cluster synchronization of stochastic two-layer delayed neural networks via pinning impulsive control. First, the cluster synchronization of the first layer (leader-layer) network is explored by taking the average state of each subnet as the synchronization target. Then, a pinning impulsive controller is designed to synchronize the second layer (follower-layer) network to the leader-layer network in the mean square cluster sense. Based on the stochastic impulsive analysis and Lyapunov stability theory, some sufficient conditions for cluster synchronization are strictly obtained. At last, the correctness of the theoretical result is illustrated by a numerical example. Compared with previous works, the proposed network model is closer to the real networks and the control strategy is more energy efficient. [ABSTRACT FROM AUTHOR]

In this paper, a new scheme of anti difference multiswitching compound compound combination synchronization for achieving synchronization between seven chaotic systems has been proposed. This novel scheme of synchronization involves five drive systems and two response systems and its first of a kind. Due to multiswitching of signals, the security of signals transmitted through the systems enhances. Nonlinear controllers have been designed and synchronization has been achieved between the drive and response systems using Lyapunov stability criteria. Theoretical results have been given and numerical simulations have been performed using MATLAB to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings. [ABSTRACT FROM AUTHOR]

In this work we propose a new mathematical model of 2D cellular neural networks (CNN) in terms of the lattice FitzHugh-Nagumo equations with boundary feedback. The model features discrete Laplacian operators and periodic boundary feedback instead of the interior-clamped or mean-field feedback. We first prove the globally dissipative dynamics of the solutions through a priori uniform estimates. In the main result we show that the 2D cellular neural networks are exponentially synchronized if the computable threshold condition is satisfied by the synaptic gap signals of pairwise boundary cells of the grid and the system parameters with a linear feedback coupling coefficient tunable in applications. [ABSTRACT FROM AUTHOR]

In a coupled network cells can interact in several ways. There is a vast literature from the last 20 years that investigates this interacting dynamics under a graph theory formalism, namely as a graph endowed with an input-equivalence relation on the set of vertices that enables a characterisation of the admissible vector fields that rules the network dynamics. The present work goes in the direction of answering an inverse problem: for n ⩾ 2 , any mapping on R n can be realised as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) n. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to the appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realise couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators. [ABSTRACT FROM AUTHOR]

This research article mainly focuses on the development of a new six-dimensional dynamical system, principally based on the Lü system. The dynamic characteristics of this introduced system are evaluated and investigated by providing its 2D and 3D plots, time series for each state, discussing bifurcation scenarios and dissipation, the Poincaré map, and calculating its Kaplan–Yorke dimension. This system achieves hyper-chaos with two positive Lyapunov exponents. The generic stagnation points are computed, and their stability analysis is covered. The proposed system has also been studied via circuit implementation on Multisim. Towards the end of the manuscript, hybrid projective combination–combination synchronization among multiple drives and responses is attained through efficient adaptive control. Based on the principles of Lyapunov stability, several nonlinear controllers are created. All the acquired analytical results in the text are validated for their efficacy and viability through simulations on MATLAB. [ABSTRACT FROM AUTHOR]

The declaration of a nationwide lockdown in India led to millions of migrant workers, particularly from Uttar Pradesh (UP) and Bihar, returning to their home states without proper transportation and social distancing from cities such as Delhi, Mumbai, and Hyderabad. This unforeseen migration and social mixing accelerated the transmission of diseases across the country. To analyze the impact of reverse migration on disease progression, we have developed a disease transmission model for the neighboring Indian states of Delhi and UP. The model’s essential mathematical properties, including positivity, boundedness, equilibrium points (EPs), and their linear stability, as well as computation of the basic reproduction number (R0)\left({R}_{0}), are studied. The mathematical analysis reveals that the model with active reverse migration cannot reach a disease-free equilibrium, indicating that the failure of restrictive mobility intervention caused by reverse migration kept the disease propagation alive. Further, PRCC analysis highlights the need for effective home isolation, better disease detection techniques, and medical interventions to curb the spread. The study estimates a significantly shorter doubling time for exponential growth of the disease in both regions. In addition, the occurrence of synchronous patterns between epidemic trajectories of the Delhi and UP regions accentuates the severe implications of migrant plight on UP’s already fragile rural health infrastructure. By using COVID-19 incidence data, we quantify key epidemiological parameters, and our scenario analyses demonstrate how different lockdown plans might have impacted disease prevalence. Based on our observations, the transmission rate has the most significant impact on COVID-19 cases. This case study exemplifies the importance of carefully considering these issues before implementing lockdowns and social isolation throughout the country to combat future outbreaks.

The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

We prove the complete and partial phase synchronization for the Kuramoto oscillators with distributed time-delays and inertia effect. Our results assert that the Kuramoto models incorporated with a small variation of distributed time-delays and inertia effect still exhibit synchronization. This shows the robustness of the synchronization phenomena of the original Kuramoto model in the perturbation of time-delay and inertia effects. We also present several numerical experiments supporting our main results and exhibiting interesting patterns. [ABSTRACT FROM AUTHOR]

The cluster synchronization problem in a class of two-layer nonlinear networks via sampling-based event-triggered control scheme is considered in this paper. First, through a sector-like condition and the linear matrix inequalities (LMIs), a cluster synchronization criterion of the first layer is strictly given. Then, taking the average state of the corresponding subset of nodes in the first layer as the target, an event-triggered control protocol based on the sampled information is designed to synchronize the second layer to the first layer. By applying the time-delay system method and the event-triggered control principle, the sufficient condition for cluster synchronization of the second layer is obtained. In this process, an advanced Lyapunov functional, generalized Jensen's inequality and convex combination method are exploited to reduce the conservation of the results. In addition, numerical example further verifies the feasibility and effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In many emerging scientific applications, the input of random initial time is often required for predicting and controlling the dynamics of real-order systems that include or does not include time-delays. In this work, we provide a new design for a class of random initial-time non-autonomous linear Caputo-type real-order time-delay systems that involves constant discrete delays. We first introduce a new comparison principle and a linear comparison lemma for Caputo derivative. By using the idea of comparison methodology and generalized Laplace transform, we develop some new elementary asymptotic theories and establish order-dependent and delay-independent conditions that give convergence of nontrivial solutions to such a class of system. One of the major challenges encountered in the investigation is discovering some simpler classes of autonomous linear real-order systems in bounding the coefficient matrices of a designed class of systems above by constant matrices in our assumptions. These assumptions including newly introduced matrices Λ , Θ k and G provide some fundamental key tools for an effective asymptotic analysis of newly designed class of systems. A typical theorem that we develop says that if all the diagonal entries of Λ become negative and every root of the characteristic equation associated with the matrix Λ + ∑ k = 1 m Θ k G lies in the open left-half complex plane, then it is possible to predict the limiting behavior of the designed class of systems. For a practical application of interest, we show how to synchronize the possible chaotic dynamics of the coupled nonlinear Duffing oscillators if the system starts at a negative initial time through some proposed results. [ABSTRACT FROM AUTHOR]

Nonlinear normal modes (NNMs) are defined as curved invariant structures (manifolds) in the phase-space of a nonlinear dynamical system, which confine the trajectories initiated on them to themselves and hence provide a reduced order subspace for the system to evolve. In this work, NNMs, obtained through graph style parameterization method, are used to study synchronization dynamics of two non-isochronous, frequency detuned limit cycle oscillators (LCOs) under displacement-based/reactive and velocity-based/dissipative coupling, also considering the effect of delayed interactions. The NNMs contain closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations. The motion initiated on these NNMs is found to evolve strictly on them, unless it encounters a fixed point or closed orbit with an out-of-plane unstable direction, in which case it may even transition to the other manifold. In all other cases, the motion settles onto a stable fixed point or orbit on either manifold if existent, or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to uncouple the governing equations of the system, directly giving the in-phase and out-of-phase synchronization frequencies and stability. Parametric study using averaged equations as well as the direct numerical integration-based evolution of the original system on the NNMs is used to identify various bifurcations. The properties of NNMs are tested for qualitatively different motions hence determined. NNMs for the delay coupled oscillators are obtained, by first expanding the delay terms in a Taylor series. It is found that even a small delay ( < 1 % of LCOs time period) brings a significant change in the behavior of the system. The quantitative predictions made by NNM in this case are found to be good for finite delay times ( ≈ 10 % of LCOs time period), while qualitative agreement was achieved for even larger delays. The results obtained here agree with prior work in the literature, at the same time extending them to hitherto unexplored parameter range and provide a new perspective on the same from the point of view of NNMs. [ABSTRACT FROM AUTHOR]

This article explores a class of Clifford-valued neural networks with leakage delay and proportional delay. By using the non-decomposition method and Banach fixed point theorem, we obtain several sufficient conditions for the existence of weighted pseudo almost periodic solutions for Clifford-valued neural networks with leakage delay and proportional delay. By using the proof by contradiction, we get the global exponential synchronization for Clifford-valued neural networks with leakage delay and proportional delay. Finally, one illustrative example is given to illustrate the feasibility and effectiveness of the main results. [ABSTRACT FROM AUTHOR]

The Kuramoto model was developed to describe the coupling of oscillators, motivated by the natural synchronization phenomena. We are interested in modeling an epileptic seizure considering it as the synchronization of action potentials using and modifying this model. In this article, we propose to modify this model, changing the constant coupling force for a function with logistic growth to simulate the onset and epileptic seizure level in an adult male rat caused by the administration of lithium–pilocarpine. Later, we select some frequencies and their respective amplitude values using an algorithm based on the fast Fourier transform (FFT) from an electroencephalography signal when the rat is in basal conditions. Then, we take these values as the natural frequencies of the oscillators in the modified Kuramoto model, considering every oscillator as a single neuron to simulate the emergence of an epileptic seizure numerically by increasing the synchronization value in the coupling function. Finally, using Dynamic Time Warping algorithm, we compare the simulated signal by the Kuramoto model with an FFT approximation of the epileptic seizure. [ABSTRACT FROM AUTHOR]

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

This paper investigates chaotic and fractal dynamics of fractional coupled logistic maps constructed based on the Caputo fractional h-difference. The chaos of this map, affected by the memory and scale derived from the fractional operator, is examined through phase portrait, "0–1" test and Lyapunov exponent. Fractal synchronization is achieved by designing a coupled controller between Julia sets generated from two fractional coupled maps with different structures. Numerical simulations are presented to validate the main findings. [ABSTRACT FROM AUTHOR]

A novel 6D dissipative model with an unstable equilibrium point is introduced herein. Some of the dynamic characteristics of the proposed model were explored via analyses and numerical simulations including critical points, stability, Lyapunov exponents, time phase portraits, and circuit implementation. Also, anti-synchronization phenomena were implemented on the new system. Firstly, the error dynamics is found. Then, four different controllers are adopted to stabilize this error relying on the nonlinear control technique with two main ways: linearization and Lyapunov stability theory. In comparison with previous works, the present controllers realize anti-synchronization based on another method/linearization method. Finally, a comparison between the two ways was made. The simulation results show the effectiveness and accuracy of the first analytical strategy. [ABSTRACT FROM AUTHOR]

In this paper, the exponential synchronization of fractional-order multi-links complex dynamical networks (CDNs) is studied based on non-periodically intermittent control. By means of the Lyapunov method and graph-theoretic approach, a Lyapunov-type theorem is provided based on the existence of vertex-Lyapunov functions. Then by giving the specific vertex-Lyapunov functions, a coefficients-type theorem is presented where the conditions of it are based on the coefficients of system. Moreover, to show the practicality of theoretical results, we give two applications to fractional-order chaotic CDNs with multiple links and fractional-order Hindmarsh–Rose neuron systems with multiple links, respectively. Meanwhile, two numerical examples are given to demonstrate the validity and feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the temporal asymptotic behavior of the all-to-all coupled Kuramoto model with inertia and time-periodic natural frequencies. Due to the inertial effect, there are three cases of the dynamical ensemble with respect to the coupling strength; large coupling, near boundary, and small coupling. For each case, we present the asymptotic behavior of the solution to the inertial Kuramoto model with periodic natural frequencies: the solutions commonly consist of a macroscopic phase, a mean-centered-periodic solution, and an exponential decay term. The macroscopic phase is a drift-type term determined by initial data and natural frequencies, and the mean-centered-periodic solution is a standing wave independent of initial data. We provide sufficient conditions for the existence of a mean-centered-periodic solution with a time-periodic phase difference between nodes for each case and its exponential stability. We also provide several simulations to confirm our mathematical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotical stability and synchronization of fractional neural networks. Multiple time-varying delays and distributed delays are taken into consideration simultaneously. First, by applying the Banach's fixed point theorem, the existence and uniqueness of fractional delayed neural networks are proposed. Then, to guarantee the asymptotical stability of the demonstrated system, two sufficient conditions are derived by integral-order Lyapunov direct method. Furthermore, two synchronization criteria are presented based on the adaptive controller. The above results significantly generalize the existed conclusions in the previous works. At last, numerical simulations are taken to check the validity and feasibility of the achieved methods. [ABSTRACT FROM AUTHOR]

Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with electrical coupling show a diversity of post-spike subthreshold fluctuations, often linked to subthreshold resonance, which are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with damped subthreshold oscillations and a range of post-spike voltage dynamics. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuous post-spike reset rule. We find that both spikes and subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau potential. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs.

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

This paper discusses a class of delayed Clifford-valued fuzzy cellular neural networks. First, although the multiplication of Clifford algebras does not satisfy the commutativity, without separating the Clifford-valued systems into real-valued systems. Second, we can obtain several sufficient conditions for the existence of anti-periodic solutions for Clifford-valued fuzzy cellular neural networks by using the non-decomposition method, and Schauder fixed point theorem. Third, we can obtain the global exponential stability of anti-periodic solutions for Clifford-valued fuzzy cellular neural networks by using the the proof by contradiction. Finally, we give one example to illustrate the feasibility and effectiveness of the main results. [ABSTRACT FROM AUTHOR]

In this paper, the quasi-synchronization and global asymptotic synchronization of memristor-based complex-valued neural networks with time-varying delays are studied respectively. Two suitable controllers are designed to achieve the control objective. To get the corresponding conclusion, the method of vector ordering is employed, which can be used to compare the size of two different complex numbers, as a result, the closed convex hull derived by the complex-valued connections can be derived. Finally, the effectiveness of the proposed method is verified by two simulation examples. [ABSTRACT FROM AUTHOR]

This paper focuses on the global Mittag–Leffler (M–L) synchronization for fractional-order inertial neural networks (FOINNs) including the Caputo derivative and delay. By choosing an appropriate variable transformation, the considered system including the inertial term is transformed into two fractional-order nonlinearly coupled interconnected subsystems. Combining with the delayed-feedback controller, fractional Lyapunov functional approach and inequality analysis technique, multi sets of novel algebraic criteria on the global M–L synchronization between derive-response systems are established. The main prominent highlight of this paper is that the proposed results are characterized by the algebraic inequalities form, which can simplify the calculation and facilitate the test. Two numerical simulation examples validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considered Generalized Function Projective Synchronisation (GFPS) between identical hyperchaotic systems by utilizing two different matrices with the new class of scaling functions. The nonlinear control approach is applied for chaos synchronization. The stability analysis of the synchronization error is carried out through Lyapunov's second method. Two different controllers are introduced and a comparison between the two methods is given. Finally, the results were discussed and the effectiveness of this was achieved through numerical theory and simulation. [ABSTRACT FROM AUTHOR]

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring. [ABSTRACT FROM AUTHOR]

This paper presents a novel simple 5D chaotic system with no critical point having hidden attractors which are derived via state feedback control strategy from the known 3D Sprott-E system. The proposed system is different from some well-known hidden chaotic systems, it exhibits chaotic 2-torus and consists of nine terms including only one control parameter, two nonlinear terms. Some dynamical behaviors of this system were analyzed theoretically and numerical simulations such as equilibrium points, Lyapunov exponents, Lyapunov dimension, phase portraits, multistability, and hybrid projective synchronization. [ABSTRACT FROM AUTHOR]

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.

This paper concentrates on the stability and stabilization of short memory fractional differential equations with delayed impulses. The sufficient conditions for asymptotic stability of short memory fractional differential equations with two kinds of delayed impulses are derived, respectively. The results show that the delayed impulses in short memory fractional differential equations exhibit double effects on system performance. For an unstable system, one can stabilize the system by inputting delays in impulses; for a stable system, the stability would be destroyed if the delays were too long. Further, a class of fractional chaotic systems is presented to test the validity of the established theoretical results, some criteria for impulsive synchronization of fractional chaotic systems are derived, and the corresponding impulsive controllers are designed. Finally, a fractional Chua chaotic oscillator is presented to illustrate the practicability of the established impulsive controllers. [ABSTRACT FROM AUTHOR]

This paper studies exact synchronization and asymptotic synchronization problems for a controlled linear system of ordinary differential equations. In this paper, we build up necessary and sufficient conditions for exact synchronization and asymptotic synchronization problems. When a system is not controllable but exactly synchronizable, it can be asymptotically synchronized in any given rate and the state of exact synchronization is given. However, when a system is not controllable and can be asymptotically synchronized in any given rate, it may not be exactly synchronizable. [ABSTRACT FROM AUTHOR]

Analyzing the validity and success of a data assimilation algorithm when somestate variable observations are not available is an important problem inmeteorology and engineering. We present an improved data assimilation algorithmfor recovering the exact full reference solution (i.e. the velocity andtemperature) of the 3D Planetary Geostrophic model, at an exponential rate intime, by employing coarse spatial mesh observations of the temperature alone.This provides, in the case of this paradigm, a rigorous justification to anearlier conjecture of Charney which states that temperature history of theatmosphere, for certain simple atmospheric models, determines all other statevariables.

We study the synchronization properties of the random double rotations on tori. We give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher.

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtain improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present a few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the convergence analysis of the data assimilation algorithm.