Nonlinear Analysis: Modelling and Control
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Turing pattern dynamics in a fractional-diffusion oregonator model under PD control
Full text linkIn this paper, fractional-order diffusion and proportional-derivative (PD) control are introduced in oregonator model, and the Turing pattern dynamics is investigated for the first time. We take the cross-diffusion coefficient as the bifurcation parameter and give some necessary conditions for Turing instability of the fractional-diffusion oregonator model under PD control. At the same time, we construct the amplitude equations near the bifurcation threshold and determine the pattern formation of the fractional-diffusion oregonator model under PD controller. It is observed by numerical simulations that in the absence of control, the pattern formation changes with the variation of the cross-diffusion coefficient in two-dimensional space. Meanwhile, it is verified that the PD control has a significant impact on Turing instability, and the pattern structure can be changed by manipulating the control gain parameters for the fractional-diffusion oregonator model
Symmetry analysis, soliton solutions and conservation laws of the Q(L, m, n) equation
Full text linkIn the present paper, symmetry and soliton solutions of the Q(L, m, n) equation are investigated. The infinitesimal operator of this equation is obtained by virtue of Lie group analysis. Taking different values of the parameters for the coefficients, the corresponding vector fields are obtained. Subsequently, soliton solutions of this equation are obtained for different parameters relying on the solitary wave ansatz method. According to different parameters, new soliton solutions are obtained. Also, conservation laws are also derived. Reciprocal Bäcklund transformations of conservation laws presented from the known conservation laws
Finite-time synchronization and quasi-synchronization of fractional-order fuzzy BAM neural networks with time delays
Full text linkThis paper concentrates on the finite-time synchronization (FTS) and the quasi-synchronization (QS) problems for a kind of fractional-order fuzzy BAM neural networks with time delays (FOFBAMNNs). In order to reach the goals of synchronization, two novel controllers are designed. Then, based on finite-time stability theorem, Lyapunov function theory, and several inequality techniques, through the application of two different designed controllers, several criteria for both FTS and QS are established. Moreover, more precise error level and settling times are given. The effectiveness of the derived criteria is ultimately validated through two simulations
On the h-manifolds for impulsive conformable neural networks with reaction–diffusion terms: Practical stability analysis
Full text linkIn this paper, we consider a new class of conformable impulsive reaction–diffusion neural networks. The stable behavior of h-manifolds with respect to the model is investigated, and sufficient conditions are proposed by constructing suitable Lyapunov-like functions. Our results are new and contribute to the development of the theory of impulsive conformable models. Examples are also presented to illustrate the proposed criteria
Controllability of stochastic impulsive integro-differential systems involving nonlocal conditions and conformable derivatives
Full text linkIn this paper, the controllability of a stochastic impulsive integro-differential system involving nonlocal conditions and conformable derivatives is analyzed. The solution of the system is derived by Duhamel’s formula using Laplace and inverse Laplace transforms. The controllability result for the linear system is proved by using controllability Grammian matrix, and for the nonlinear integro-differential system, fixed point techniques are used. The applicability of the system is verified by means of an example
Analysis of exponential stability and L1-gain performance of positive switched impulsive systems with all unstable subsystems
Full text linkThis article investigates the exponential stability and L1-gain performance of time-varying positive switched impulsive systems even when all modes are unstable. By employing the discretized switched copositive Lyapunov function approach and the analytical method developed for positive systems, we derive a sufficient condition to ensure the exponential stability for such systems. An algorithm for computing the stability region of admissible dwell time is also introduced. Furthermore, building on this stability result, the L1-gain performance of positive switched impulsive systems is further studied. Finally, numerical examples are presented to demonstrate the validity of our results
Distributed constraint optimization for discrete-time multiagent systems with event-triggered communication
Full text linkThis paper investigates the distributed optimization problem (DOP) with equality constraint in discrete-time multiagent systems (MASs) in which the global optimization objective is constituted by the summation of local objective functions. Firstly, by employing the Lagrange multiplier method, we convert the convex optimization problem with equality constraint into a consensus problem of MASs. Secondly, to reduce the communication burden, a type of event-triggered control protocol is proposed to enable all agents achieving consensus. Thirdly, by employing the Lyapunov function method and a set of inequality techniques, we establish some sufficient conditions to ensure that all agents converge to consensus and successfully solve the original DOP. Finally, a numerical simulation example is presented to validate the effectiveness of the theoretical analysis
Arnold tongues of divergence in the Caputo fractional standard map of nilpotent matrices
Full text linkArnold tongues of divergence in the Caputo fractional standard map of nilpotent matrices are explored in this paper. The scalar iterative variables in the Caputo fractional standard map are replaced by iterative matrix variables. The divergence effects induced by the nilpotent matrices result in specific patterns of Arnold tongues. Automatic machine classification techniques help to identify different types of Arnold tongues according to the dynamics of the transient processes of the system. Computational experiments are used to validate theoretical insights and to reveal the patterns of Arnold tongues of divergence
Message spreading modeling from the perspective of social psychology, differential equation dynamics, and deep learning technique
Full text linkTo gain a deeper understanding of the characteristics of message spreading, it is crucial to explore various methods of modeling this phenomenon. Given that message spreading is significantly influenced by social media, we propose a modified spreading model informed by social psychology analysis. This approach also incorporates differential equation dynamics and deep learning technique. The proposed model accounts for a cross-transmission mechanism between individuals and social media platforms, as well as a nonlinear spreading rate, to effectively characterize the saturation effect of messages. Utilizing Lyapunov functionals, we carry out a dynamical analysis of the message spreading model. Furthermore, we develop physics-informed neural networks based on deep learning technique that merges the efficiency inherent in data-driven modeling with the precision offered by mathematical modeling. Numerical simulations demonstrate that our prediction method can accurately capture real-time changes in data while correcting deviations observed in data-driven predictions, which highlights the robust potential for multidisciplinary integration among social psychology, differential equation dynamics, and deep learning technique
Mitigating atmospheric carbon dioxide through deployment of renewable energy: A mathematical model
Full text linkIn recent decades, the widespread reliance on fossil fuels has grown substantially, leading to a rise in atmospheric carbon dioxide (CO2), which poses a major global concern. In this study, we develop and analyze a novel mathematical model to examine the interactions between atmospheric CO2, human population, and energy demand. The model assumes that human activities and energy production from traditional sources (oil, coal, and gas) contribute to increasing CO2 level, while a shift in energy dependence from traditional to renewable sources (hydro, solar, etc.) occurs as a result of environmental awareness. We derive sufficient conditions for both local and global stability of the system’s interior equilibrium. Numerical simulations demonstrate that when reliance on renewable energy sources is low, the system can exhibit oscillatory dynamics and various bifurcations. However, beyond a critical threshold of renewable energy dependency, the system stabilizes around the interior equilibrium, leading to a reduction in atmospheric CO2. Additionally, an optimal control problem is formulated to reduce atmospheric CO2 level while minimizing the associated implementation costs