Stability Of Periodic Solutions Research Articles

A periodic reaction-diffusion model of hospital infection with crowding effects

In this paper, we propose a periodic reaction-diffusion model of hospital infection with crowding effects. We introduce the basic reproduction number R0 for this model and show that the infection-free periodic solution is globally asymptotically stable if R0≤1, while the system admits a globally asymptotically stable positive periodic solution if R0>1. We also obtain the asymptotic behavior of the basic reproduction number when the diffusion rates go to infinity and zero, respectively. Further, we numerically study the effects of diffusion rates, the outflow rate by patient crowding and other parameters on R0 with and without seasonality, and found that the neglect of seasonality does overestimate the infection risk and the disease may be well controlled by avoiding overcrowding of patients rather than increasing the mobility of individuals or bacteria.

Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy

A nonsmooth Filippov avian-only influenza model with threshold strategies of culling susceptible and/or infected birds under different conditions is proposed to control the spread of avian influenza. The time delay representing the incubation period of avian influenza is incorporated into our model to make it more realistic, which is different from the traditional Filippov model. The stability of various types of equilibria and the existence of Hopf bifurcation are researched. Moreover, the existence of the sliding mode and its dynamics are investigated by Filippov convexity method. The theoretical analysis and numerical simulations indicate that, according to the value of thresholds and time delay, all solutions eventually converge to the regular equilibrium, pseudoequilibrium or stable periodic solution. We also indicate that time delay has a great influence on the sliding mode. Due to the many factors involved, and our main purpose is to study the impact of time delay on system stability, we have only conducted a simple analysis of the impact of time delay on the sliding mode. Furthermore, the boundary bifurcation switching stable regular equilibrium or stable limit cycle to a stable pseudoequilibrium can be exhibited by numerical simulations. Finally, with the increase of time delay, the global bifurcations from grazing bifurcation to buckling bifurcation and then to cross bifurcation are obtained. Our results indicate that although Filippov control strategies can effectively control the number of infected birds in many cases, the existence of time delay may challenge influenza control by the emergence of buckling bifurcation and cross bifurcation.

Double Hopf bifurcation analysis of time delay coupled active control system

In this paper, the stability analysis of double Hopf bifurcation is carried out based on the time delay coupled active control system by using the related theory of bifurcation analysis. We got the stability switching region of the system with respect to time delay. The time delay and coupling strength are selected as the bifurcation parameters, then the normal form of the coupled time delay active control system is obtained by using the multi-scale method, and the dynamic behavior near the bifurcation point is obtained. So the parameter region of the stable periodic solution and the stable quasi-periodic solution are obtained. The analytical solution of the system is then solved by using the multi-frequency homotopy analysis method (MFHAM) to verify the existence of the unstable periodic solution. Finally, numerical simulation is executed to verify the theoretical results.

Deep learning-based state prediction of the Lorenz system with control parameters.

Nonlinear dynamical systems with control parameters may not be well modeled by shallow neural networks. In this paper, the stable fixed-point solutions, periodic and chaotic solutions of the parameter-dependent Lorenz system are learned simultaneously via a very deep neural network. The proposed deep learning model consists of a large number of identical linear layers, which provide excellent nonlinear mapping capability. Residual connections are applied to ease the flow of information and a large training dataset is further utilized. Extensive numerical results show that the chaotic solutions can be accurately forecasted for several Lyapunov times and long-term predictions are achieved for periodic solutions. Additionally, the dynamical characteristics such as bifurcation diagrams and largest Lyapunov exponents can be well recovered from the learned solutions. Finally, the principal factors contributing to the high prediction accuracy are discussed.

Outlet boundary condition and mean temperature gradient effects on the minimum acoustics disturbances energy in triggering nonlinear thermoacoustic instability

In this study, we theoretically investigate the impact of outlet boundary conditions and mean temperature gradients on the maximum transient growth rate of acoustical energy and the critical energy required for triggering. Our analysis encompasses open–open and open–closed thermoacoustic systems. The theoretical models developed focus on horizontal ducts with a mean temperature jump over the heat source, employing the modified King's law. By linearizing the unsteady heat release, the nonlinear thermoacoustic equations transform into linearized-delay ones. This approach enables us to predict optimal initial perturbations for linearized-delay and nonlinear systems, corresponding to maximum transient growth rates of acoustic energy over short and long periods, respectively, thus providing insights into critical energy for triggering. We find that a closed outlet leads to higher transient energy growth and a lower critical energy for triggering compared to an open outlet. The increased mean temperature gradient has a “destructive” impact on triggering in open–open systems but a “constructive” effect in open–closed systems. Raising the mean temperature ratio generally increases the critical energy for triggering in the open–open system, whereas it decreases the critical energy in the open–closed system. The critical energy for nonlinear optimal initial perturbations is notably affected by the minimum energy of critical unstable periodic solutions, while the critical energy for linearized-delay optimal initial perturbations is closely tied to the energy level of stable periodic solutions. Due to the transient energy growth rate, the critical energy for nonlinear optimal initial perturbations is significantly lower than that for linearized-delay optimal initial perturbations.

Analysis of a Delayed Multiscale AIDS/HIV-1 Model Coupling Between-Host and Within-Host Dynamics

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h<1, the disease-free equilibrium is globally asymptotically stable; if R˜0h>1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.

Delayed nonmonotonic immune response in HIV infection system

In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune‐free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle‐node‐Hopf bifurcation. Numerical simulations are carried out to verify our results.

On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales

Abstract This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.

Steady-state bifurcations and patterns formation in a diffusive toxic-phytoplankton–zooplankton model

In this paper, we study a diffusive toxic-phytoplankton–zooplankton model with prey-taxis under Neumann boundary condition. By analyzing the characteristic equation, we discuss the local stability of the positive constant solutions and show the repulsive prey-taxis is the key factor that destabilizes the solutions. By means of the abstract bifurcation theorem, we investigate the existence of non-constant positive steady-state solutions bifurcating from the constant coexistence equilibrium. Furthermore, we obtain the criterion for the stability of the branching solutions near the bifurcation point. Numerical simulations support our theoretical results, together with some interesting phenomena, stable heterogeneous periodic solutions emerge when prey-tactic sensitivity coefficient is well below the critical value, and zooplankton populations present extinction and continued transitions as habitat size increases.

Numerical Analysis of a Drop-Shaped Aquatic Robot

Finite-dimensional equations constructed earlier to describe the motion of an aquatic drop-shaped robot due to given rotor oscillations are studied. To study the equations of motion, we use the Poincaré map method, estimates of the Lyapunov exponents, and the parameter continuation method to explore the evolution of asymptotically stable solutions. It is shown that, in addition to the so-called main periodic solution of the equations of motion for which the robot moves in a circle in a natural way, an additional asymptotically stable periodic solution can arise under the influence of highly asymmetric impulsive control. This solution corresponds to the robot’s sideways motion near the circle. It is shown that this additional periodic solution can lose stability according to the Neimark–Sacker scenario, and an attracting torus appears in its vicinity. Thus, a quasiperiodic mode of motion can exist in the phase space of the system. It is shown that quasiperiodic solutions of the equations of motion also correspond to the quasiperiodic motion of the robot in a bounded region along a trajectory of a rather complex shape. Also, strange attractors were found that correspond to the drifting motion of the robot. These modes of motion were found for the first time in the dynamics of the drop-shaped robot.

Gompertz models with periodical treatment and applications to prostate cancer.

In this paper, Gompertz type models are proposed to understand the temporal tumor volume behavior of prostate cancer when a periodical treatment is provided. Existence, uniqueness, and stability of periodic solutions are established. The models are used to fit the data and to forecast the tumor growth behavior based on prostate cancer treatments using capsaicin and docetaxel anticancer drugs. Numerical simulations show that the combination of capsaicin and docetaxel is the most efficient treatment of prostate cancer.

Periodic solution analysis of a population dynamics system model for pulsating organisms

Abstract Population dynamics has a wide range of applications in ecological theory, especially in the fields of plant and animal conservation and the management and development of ecological environments. Periodic solution analysis of a population dynamics model for pulsating organisms. The influence of impulsive dynamics on the periodic solution of the system is investigated in this paper, which considers several types of population dynamics systems with impulsive effects. First, the impulsive differential modeling of the model of a constantator in a polluted environment considering time-lagged growth response and impulsive inputs proves that only ̄t needs to be sufficiently large to have x(t) > m x, such that, the constantator seeks a unique periodic solution for microbial extinction and persistent survivability. Next, a model of integrated pest control is modeled to find, a periodic solution for pest extinction and the existence of (0, I* (t)) is globally stable. Then, a Lur’e system with impulsive biodynamics is explored, modeled with uncertain parameters, and simulated with Chua’s circuit system to determine that the state trajectory lines all eventually converge to 0 and have stable periodic solutions. Finally, the Beddington-DeAngelis predator-prey model with impulsive effects is used to argue, using correlation priming, for the existence of z i * ( t ) = exp { x i * ( t ) } z_i^*\left( t \right) = \exp \left\{ {x_i^*\left( t \right)} \right\} , i = 1, 2, such that z * ( t ) = ( z 1 * ( t ) , z 2 * ( t ) ) T {z^*}\left( t \right) = {\left( {z_1^*\left( t \right),\,z_2^*\left( t \right)} \right)^T} there is a positive ω − periodic solution for this system.

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Periodic solutions and bifurcations of a tristable flutter-based energy harvester

This paper designs a tristable flutter-based energy harvester that can harvest large amplitude interwell wind-induced vibration energy. To explore its periodic responses and bifurcation characteristics, the harmonic balance method and the incremental harmonic balance method are employed to solve its approximate solutions by combining the two-point tracing algorithm. By analyzing the incremental harmonic balance solutions, three different response regions and five vibration types are found. The wind tunnel experiment verifies the existence of these vibration types and the correctness of incremental harmonic balance solutions. Five bifurcation points are analyzed using the Floquet theory, and two evolution paths of its periodic solutions are studied. Finally, the influences of the potential well depths on the stable periodic solutions are studied. The shallow potential energy well is beneficial to improving the response stability. This study aims to fill the gap in the semi-analytical and analytical solution of nonlinear flutter-based energy harvesters, contribute to the theoretical analysis of wind-induced vibration energy harvesting, and provide more possibilities for the design of high-performance wind-induced vibration energy harvesters.

New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

Spatiotemporal Dynamics of a Diffusive Immunosuppressive Infection Model with Nonlocal Competition and Crowley–Martin Functional Response

In this paper, we introduce the Crowley–Martin functional response and nonlocal competition into a reaction–diffusion immunosuppressive infection model. First, we analyze the existence and stability of the positive constant steady states of the systems with nonlocal competition and local competition, respectively. Second, we deduce the conditions for the occurrence of Turing, Hopf, and Turing–Hopf bifurcations of the system with nonlocal competition, as well as the conditions for the occurrence of Hopf bifurcations of the system with local competition. Furthermore, we employ the multiple time scales method to derive the normal forms of the Hopf bifurcations reduced on the center manifold for both systems. Finally, we conduct numerical simulations for both systems under the same parameter settings, compare the impact of nonlocal competition, and find that the nonlocal term can induce spatially inhomogeneous stable periodic solutions. We also provide corresponding biological explanations for the simulation results.

Effect of damping and rotor moment of inertia on stability of self-synchronization for two unbalanced rotors

The self-synchronization of rotors in a nonlinear system has both academic and practical importance because of its applicability as a source of excitation for vibratory machines. Although the characteristics of such self-synchronization can be clarified by analyzing the periodic solutions of the equations of motion, only self-synchronization corresponding to the stable periodic solution is realized in actual systems. The stability of the solution is thus one of the most important characteristics of self-synchronization. In this paper, to stabilize periodic solutions suitable for application to vibratory machines, the effects of the damping of the rigid body and the moment of inertia of the rotors on stability are examined using a system that consists of two unbalanced rotors installed in a rigid body. High-precision analysis conducted using the shooting method shows that it is possible to stabilize the instabilities caused by period-doubling and Hopf bifurcations while keeping other characteristics, such as the synchronous frequency and the rigid body amplitude, mostly unchanged. The findings of this study can serve as a guide to extending the range of stable periodic solutions and are thus expected to be useful in the design of systems that utilize the self-synchronization of rotors.

Evolution of trust in a hierarchical population with different investors based on investment behavioral theory

The study of trust behavior between investors and trustees through trust games has been a hot topic recently. In previous research, the standard N−player trust game model has been used to reveal some aspects of trust evolution, but does not consider the risk perception of investors. According to behavioral investment theory, investors have different sensitivities to investment risk due to their individual environments and characteristics. Here, we construct an N-player trust game model where we divide investors into two categories, namely, venture investors and conservative investors. Conservative investors are willing to pay an information cost upfront to avoid investment risk, while venture investors are not. By analyzing replicator equations, we find that the system has an asymptotically stable internal equilibrium point when the group size is large, while when the group size is small, the system becomes a Hamiltonian system with a stable periodic solution. Our result shows that a lower information cost leads to a higher proportion of trustworthy trustees and thus reduces the portion of untrustworthy trustees.

Analysis and simulations of the HANDY model with social mobility, renewables and nonrenewables

We expand the HANDY (Human And Nature DYnamics) model for the socioeconomic dynamics of a large stratified society. The basic model was introduced in Motesharrei (Dissertation 2014) and Motesharrei et al. (2016). It is a nonlinear system of ODEs for a `simple society' of Elites, Workers, Wealth, and Natural Resources. Following Ali Kadhim (Dissertation 2021), we add social mobility between the classes and split natural resources into renewables and nonrenewables. We establish the existence, boundedness and positivity of the solutions, and investigates the stability of the steady states. The model admits stable steady states, and there is numerical evidence of stable periodic solutions and limit cycles. Simulations depict the different qualitative types of model behavior: convergence to steady states, periodic oscillations, collapse.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/59/abstr.html

Zero-Hopf bifurcation and ultimate boundness of an asymmetrical hyperchaotic Lorenz system

The complex dynamics of a newly proposed 4D hyperchaotic Lorenz-type system are studied in this paper. The sufficient conditions for the emergence and stability of periodic solutions at bifurcation points are derived using averaging theory. The ultimate boundedness of the proposed system is also established.