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Seasonality is repetitive in the ecological, biological and human systems. Seasonal factors affect both pathogen survival in the environment and host behavior. In this study, we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate. We started by studying the autonomous system by investigating the global stability of steady states. Later, we proved the existence, uniqueness, positivity and boundedness of a periodic orbit in a non-autonomous system. We demonstrate that the global dynamics are determined using the basic reproduction number ${{ℛ}}_0$ which is defined by the spectral radius of a linear integral operator. We showed that if ${{ℛ}}_0<1$, then the disease-free periodic solution is globally asymptotically stable and if ${{ℛ}}_0>1$, then the trajectories converge to a limit cycle reflecting the persistence of the disease. Finally, we present a numerical investigation that support our results.

This paper presents a Respiratory Syncytial Virus (RSV) epidemic model for a continuous human population of super-spreading infected individuals. The model uses a generalized fractal–fractional derivative with the Mittag-Leffler function to analyze transmission, considering different infection stages, and demonstrates the effectiveness of certain individuals as viral spreaders. Using linear growth and Lipschitz requirements, the study determines the existence, uniqueness, positiveness, and boundedness of a proposed fractional-order system in order to assess its viability. The global stability of equilibrium points is verified by the Lyapunov function, and the effects of the model’s parameters are investigated through sensitivity analysis. To comprehend the model’s behavior and create preventative measures for infected people, a numerical simulation of its activity is also carried out. By offering a more realistic depiction of virus behavior at various dimensions and population effects, the fractal–fractional model enhances comprehension and forecasts. The results highlight the importance of fractional order and its fitting function in the model by demonstrating how input parameters affect the dynamic behavior of RSV infection. Such an investigation will aid in understanding the behavior of the RSV and develop prevention measures for infected individuals.

The growing demand in system reliability and survivability under failures has urged ever-increasing research effort on the development of fault diagnosis and accommodation. In this paper, the on-line fault tolerant control problem for dynamic systems under unanticipated failures is investigated from a realistic point of view without any specific assumption on the type of system dynamical structure or failure scenarios. The sufficient conditions for system on-line stability under catastrophic failures have been derived using the discrete-time Lyapunov stability theory. Based upon the existing control theory and the modern computational intelligence techniques, an on-line fault accommodation control strategy is proposed to deal with the desired trajectory-tracking problems for systems suffering from various unknown and unanticipated catastrophic component failures. Theoretical analysis indicates that the control problem of interest can be solved on-line without a complete realization of the unknown failure dynamics provided an on-line estimator satisfies certain conditions. Through the on-line estimator, effective control signals to accommodate the dynamic failures can be computed using only the partially available information of the faults. Several on-line simulation studies have been presented to demonstrate the effectiveness of the proposed strategy. To investigate the feasibility of using the developed technique for unanticipated fault accommodation in hardware under the real-time environment, an on-line fault tolerant control test bed has been constructed to validate the proposed technology. Both on-line simulations and the real-time experiment show encouraging results and promising futures of on-line real-time fault tolerant control based solely upon insufficient information of the system dynamics and the failure dynamics.

The aim of this paper is to present a unifying view of four SVD-neural-computation techniques found in the scientific literature and to present some theoretical results on their behavior. The considered SVD neural algorithms are shown to arise as Riemannian-gradient flows on double Stiefel manifold and their geometric and dynamical properties are investigated with the help of differential geometry.

In this work we present a data-driven modeling of the insulin dynamics in different in silico patients using a recurrent neural network with output feedback. The inputs for the identification is the rate of insulin (μU / dl / min) applied to the patient, and blood glucose concentration. The output is insulin concentration (μU / ml) present in the blood stream. Once completed the off-line modeling, this model could be used for on-line monitoring of the insulin concentration for a better treatment. The learning law of the recurrent neural network is inspired by adaptive observer theory, and proven to be convergent in the parameters and stable in the Lyapunov sense, even with only 13 samples available. Simulation results are shown to validate the presented modeling.

This paper put forward a new discrete-time lattice hydrodynamic model with a steady flow control strategy. The controlled discrete-time lattice hydrodynamic model is theoretically studied by the discrete-time Lyapunov stability theory, and a sufficient stable condition to suppress traffic congestion is given in the form of linear matrix inequality. Also, the traffic flow evolution rules under different conditions of control gain are exhibited through numerical simulation, and the stabilization effect of the steady flow control strategy on traffic flow is verified intuitively.

We focus on the problem of learning equilibria in a particular routing game similar to the Wardrop traffic model. We describe a routing game played by a large number of players and present a distributed learning algorithm that we prove to converge weakly to equilibria for the system. The proof of convergence is based on a differential equation governing the global evolution of the system that is inferred from all the local evolutions of the agents in play. We prove that the differential equation converges with the help of Lyapunov techniques.

A novel fractional-order adaptive non-singular terminal sliding mode control (FONTSMC) method is investigated for the synchronization of two nonlinear fractional-order chaotic systems in the presence of external disturbance. The proposed controller consists of a fractional-order non-singular terminal sliding mode surface and an adaptive gain adjusted with sliding surface. Based on Lyapunov stability theory and stability theorem for fractional-order dynamic systems, the controlled system’s stable synchronization is guaranteed. A dual-channel secure communication system is presented to transmit useful signals based on the proposed synchronization controller. Finally, numerical simulations and comparison with fractional-order PID controller, fractional-order PD sliding mode controller and adaptive terminal sliding mode controller are given to demonstrate the effectiveness and the robustness of the proposed FONTSMC control. The application of the proposed synchronization method is studied in the dual-channel secure communication.

This paper is inspired on the structure of the field-oriented control by presenting an analytical and practical study of an adaptive nested controller for trajectory tracking control of a permanent magnet synchronous motor (PMSM) driving a single-link arm. The originality of the new approach relies in the use of adaptive control to compensate the electrical and mechanical dynamics and in the presentation of a rigorous closed-loop system stability analysis based on Lyapunov theory. It is worthwhile to notice that the new controller resembles the pure field-oriented control except for the adaptive terms. The new scheme is compared with other methodologies as with the classical nonadaptive field-oriented control algorithm and with an adaptive controller proposed in the literature. A better tracking accuracy is obtained with the proposed adaptive scheme.

We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient:

Then for k = 2,3,… we extend our stability result to the k-order equation,

In this paper, a symbolic dynamical model of the average queue size of the random early detection (RED) algorithm is proposed. The conditions for both the system parameters and the initial conditions that the average queue size of the RED algorithm would converge to a fixed point are derived. These results are useful for network engineers to design both the system parameters and the initial conditions so that internet networks can achieve a good performance.

In this work, a zero static error proportional controller for a two-cell DC-DC buck converter is synthesized and analyzed. Under a traditional proportional control scheme, the system presents a constant error of the current supplying the output load. As the proportional feedback gain is increased, the average static error decreases. However, subharmonic oscillations and chaotic behavior emerge beyond successive bifurcations. To achieve zero current and zero voltage static error, we suggest a modification of the traditional proportional controller. By optimizing the feedback gain, the settling time is also decreased. Then, using nonlinear analysis and Lyapunov stability theory, we prove that zero static error is achieved even in the presence of duty cycle saturation. Numerical simulations are presented to confirm our theoretical results.

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

In this paper, the ultimate bound and globally exponentially attractive sets of a modified Lorenz–Stenflo system are studied based on the Lyapunov stability theory and optimization theory. Comparing with the best results in the current literature, our new results include the existing results as special cases. Furthermore, the new results offer a theoretical support to studying the Hausdorff dimension of attractor of this modified Lorenz–Stenflo system. These theoretical results are also important and useful for chaos control and chaos synchronization.

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses $m_1$, $m_2$ and $m_3$ (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass $m_1=0.01$ and $0.99$ of the primary, fixed on the horizontal axis. Moreover, the $2$:$-1$ four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces.

In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

This note follows up on the study in the paper [Zepeda Ramírez et al., 2021], where we investigate the nonlinear stability of equilibrium points for the planar restricted equilateral four-body problem for the specific values of masses $m_1=0.01$ and $0.99$ of the primary, fixed on the horizontal axis. We give an improvement of the previous results.

Admissible perturbations refer to the perturbations that do not change the Mironenko reflecting function of the system, which are obtained for an autonomous three-dimensional quadratic generalized Langford system with five parameters. The obtained nonautonomous perturbed systems retain many of the qualitative properties of the original system solutions. In particular, the instability (in the sense of Lyapunov) of the equilibrium point, the presence of a periodic solution and its asymptotic stability (instability) are proved for the perturbed systems. The presence of similar chaotic attractors in the original and perturbed systems is shown by numerical simulation.

We consider a chemostat model of phytoplankton competing for nitrogen taking into account effects of both intra- and interspecific crowding and the light limitation. We consider crowding as an additive density-dependent mortality rate. Crowding effects may be classified into intra- and interspecific crowding depending on whether the additional mortality is caused by the same or alternate species.

We analyze the existence and local and global stability of single species and coexistence equilibria using the linearization and stability method of Lyapunov. We present a numerical example illustrating the fact that the crowding effects may lead to the bistable coexistence of two phytoplankton species. We demonstrate that the crowding effects and the light limitation affect the outcome of exploitative competition for a single resource and promote coexistence. We also show that while the crowding has a stabilizing effect on phytoplankton community, the light limitation may destabilize the system and produce sustained oscillations.

In an emergency, fear can spread among crowds through one-on-one encounters, with negative societal consequences. The purpose of this research is to create a novel theoretical model of fear (panic) spread in the context of epidemiology during an emergency using the fractal fractional operator. For quantitative analysis, the system’s boundedness and positivity are checked. According to the Arzela Ascoli theorem, the model is completely continuous. As a result of the discovery of Schauder’s fixed point, it has at least one solution. The existence and uniqueness of the concerned solution have been examined using the fixed point theory technique. Numerical simulations are used to demonstrate the accuracy of the proposed techniques using a generalized form of Mittag-Leffler kernel with a fractal fractional operator. Finally, simulations are utilized to represent the spread of group emotional contagion (spontaneous spread of emotions and related behaviors) dynamically.