Stability Of Periodic Solutions Research Articles

Rich dynamics of a Filippov plant disease model with time delay

We propose a nonsmooth Filippov plant disease model with threshold strategy of roguing of the infected plants with a saturated roguing rate. Different from traditional Filippov models, here we incorporate time delay which represents the incubation period of the plant disease to make the model more realistic. Analytical results show that the two delayed subsystems might admit Hopf bifurcations when the delay passes through some critical values. Then we investigate some key elements to our model, including the regular/virtual equilibrium, the sliding segment, sliding mode dynamics and pseudoequilibrium. The results indicate that all solutions finally converge to either the regular equilibrium, the pseudoequilibrium or a stable standard periodic solution depending on the values of the threshold level and the time delay. Furthermore, we show that the time delay plays a significant role in discontinuity-induced bifurcations. That is, as the time delay is varied, the model exhibits complex local sliding bifurcation (boundary bifurcation) and global sliding bifurcations from a standard periodic solution to a sliding bifurcation then to a crossing bifurcation. In conclusion, a Filippov system with time delay can provide some new insights for plant disease control.

Bifurcations and Chaos in Periodically Forced Coupled Ramp Neurons

Bifurcations and chaos in periodically forced coupled ramp neurons (rectified linear units) are examined as one of minimal chaotic neural networks. The first ramp neuron connects to the second ramp neuron excitatorily while the second ramp neuron connects to the first neuron inhibitorily. When the first ramp neuron is forced by a rectangular wave, a stable periodic solution is generated. The system is piecewise linear and its periodic solutions are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. Their Poincaré maps and Jacobian matrices are also derived rigorously, by which the bifurcations of the periodic solutions are calculated. The periodic solution undergoes period-doubling bifurcations, which leads to the generation of a chaotic attractor. A period-three solution is also generated through saddle-node bifurcations, which is characteristic of a chaotic system. A one-dimensional phase return map is derived and it is shown that the bifurcations of its fixed points agree with the bifurcations of the multiple periodic solutions of the original forced system. Further, qualitatively the same bifurcations and chaos occur when the first ramp neuron is replaced by a step neuron, in which a chaotic attractor is also generated through a border collision bifurcation. The linear response and the dead zone of the ramp neurons are essential for the emergence of chaos.

Modeling the impact of awareness programs on the transmission dynamics of dengue and optimal control

In this work, we first propose a mathematical model to study the impact of awareness programs on dengue transmission. The basic reproduction number [Formula: see text] is derived. The existence and stability of equilibria are investigated. The uniform persistence is established when [Formula: see text] is larger than one. Our results suggest that awareness programs have significant impacts on dengue transmission dynamics although they cannot affect [Formula: see text]. When [Formula: see text] is less than one, awareness programs can shorten the prevailing time effectively. When [Formula: see text] is larger than one, awareness programs may destabilize the unique interior equilibrium and a stable periodic solution appears due to Hopf bifurcation. In particular, we find that the occurrence of Hopf bifurcation depends not only on the intensity of awareness programs but also on the level of [Formula: see text]. Besides, large fluctuations in the number of infected individuals caused by the stable periodic solution may bring pressure on limited medical resources. Therefore, different from intuitive ideas, blindly increasing the intensity of awareness programs is not necessarily conducive to control the transmission of dengue. The decision-making department should decide to adopt different publicity strategies according to the current level of [Formula: see text]. Finally, we consider the optimal control problem of the model and find the optimal control strategy corresponding to awareness programs by Pontryagin’s Maximum Principle. The results manifest that the optimal control strategy can effectively mitigate the transmission of dengue.

Oscillatory wave patterns and spiral breakup in the Brusselator model using numerical bifurcation analysis

This article addresses the core breakup analysis of the spiral dynamics of an oscillatory system of chemical reactions, the so-called Brusselator model. We first obtain stable oscillatory periodic solutions corresponding to the stable limit cycle near a Hopf bifurcation point when the diffusion terms are neglected. Then, we investigate the occurrence of periodic traveling wave solutions of the model and perform the stability analysis of these solutions on the parameter plane. A stability boundary is also identified on the parameter plane. In the two-dimensional spatial domain, we illustrate rotating spiral waves and their instability that leads to a spiral breakup from the center of rotation.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Bifurcation analysis of a competitive system with general toxic production and delayed toxic effects

Population interaction may release poisonous chemicals to inhibit other species’ growth in the ecosystem, especially for the competitive populations. The negative effect of toxic chemical substances may not display immediately and appear with time lag during the species’ growth. In this work, we investigate a competitive system with the delayed toxic effects of the chemicals from species interaction. Theoretical results obtained in this work help us reveal the delayed toxic factors on species’ growth. We first consider the existence and the stability of the equilibria. The influence of delay terms on the positive steady state is validated. The delayed toxic effects here will contribute to the oscillation for the concentration of species when the value of time delay passes through a critical point. Besides, the stability of periodic solutions from the Hopf bifurcation and the direction of the Hopf bifurcation are also determined. Finally, several numerical examples are provided to validate the theoretical conclusions.

A Cluster Based Algorithm Coupled With Shooting Method for Estimation of Parametric Clusters Yielding Optimal Stable Periodic Solutions in Nonlinear Vibrating Systems

Abstract This work presents a novel cluster based optimization procedure for estimating parameter values that yield stable, periodic responses with desired amplitude in nonlinear vibrating systems. The parameter values obtained by conventional nonlinear optimization schemes, with minimization of amplitude as the objective, may not furnish periodic and stable responses. Moreover, global optimization strategies may converge to isolated optima that are sensitive to parametric perturbations. In practical engineering systems, unstable or isolated optimal orbits are not practically realizable. To overcome these limitations, the proposed method tries to converge to a cluster in the r-dimensional parameter space in which the design specifications including the specified optimality, periodicity, stability and robustness are satisfied. Thus, it eliminates the need for computationally expensive bifurcation studies to locate stable, periodic parameter regimes before optimization. The present method is based on a hybrid scheme which involves the algebraic form of the governing equations in screening phase and its differential form in the selection phase. In the screening phase, force and energy balance conditions are used to rephrase the nonlinear governing equations in terms of the design parameter vector. These rephrased equations are reduced to algebraic form using a harmonic balance procedure which also specifies the desired amplitude and frequency of the response. An error norm based on this algebraic form is defined and is used to contract the search bounds in the parameter space leading to convergence to a cluster. The selection phase of the algorithm uses shooting method coupled with evaluation of Floquet multipliers to retain only those vectors in the arrived cluster yielding stable periodic solutions. The method is validated with Den Hartog's vibration absorbers and is then applied to vibration absorbers with material nonlinearity and vibration isolators with geometric nonlinearity. In both the cases, the converged cluster is shown to yield stable, periodic responses satisfying the amplitude condition. Parametric perturbation studies are conducted on the cluster to illustrate its robustness. The use of algebraic form of governing equations in the screening phase drastically reduces the computational time needed to converge to the cluster. The fact that the present method converges to a cluster in the parameter space rather than to a single parameter value offers the designer more freedom to choose the design vector from inside the cluster. It also ensures that the design is robust to small changes in parameter values.

Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method

In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass.

Complex dynamics in a Filippov pest control model with group defense

In this paper, an integrated pest management Filippov model with group defense behavior is established, which takes the population density of pests as the control index of integrated pest management. First, under the condition that both subsystems have a globally asymptotically stable equilibrium, the dynamics of the established model are systematically analyzed, including the sliding mode dynamics, the existence and global stability of the real, virtual and pseudo equilibrium, as well as boundary-node and boundary-focus bifurcation. Next, we study the complex dynamics in the Filippov model when an unstable node (focus) or a stable limit cycle occurs in the subsystem by using numerical simulations. The results show that although there are no closed orbits in subsystem, a stable periodic solution may exist for Filippov system after switching perturbation. Finally, we conclude that the group defense behavior of pest makes it harder to control.

Global Dynamics of a Predator–Prey Model with Fear Effect and Impulsive State Feedback Control

In this paper, a predator–prey model with fear effect and impulsive state control is proposed and analyzed. By constructing an appropriate Poincaré map, the dynamic properties of the system, including the existence, nonexistence, and stability of periodic solutions are studied. More specifically, based on the biological meaning, the pulse and the phase set are firstly defined in different regions as well as the corresponding Poincaré map. Subsequently, the properties of the Poincaré map are analyzed, and the existence of a periodic solution for the system is investigated according to the properties of the Poincaré map. We found that the existence of the periodic solution for the system completely depends on the property of the Poincaré map. Finally, several examples containing numerical simulations verify the obtained theoretical result.

Existence and stability of periodic solutions of a shifted comb-drive finger actuator

Existence and stability of periodic solutions of a shifted comb-drive finger actuator

Dynamics of a new impulsive predator–prey model with predator population seasonally large-scale migration

In this paper, we present a new impulsive predator–prey model with predator population seasonally large-scale migration. By theories of impulsive differential equations, we obtain the sufficient conditions for the permanence and existence of globally asymptotically stable prey-extinct boundary periodic solution. Our results provide theoretical depicts for the management in population ecology.

Integrated pest management for Jatropha Carcus plant: An impulsive control approach

In this research, an integrated pest management model using impulsive differential equations has been investigated for Jatropha curcas plantation to control its natural pests through relying on the release of infective pest individuals and spraying of chemical pesticides. First, identify susceptible pest eradication solutions and assess their feasibility. We then show that all system variables are bounded. Using Floquet's theory and the small amplitude perturbation method, it is obtained that there exists an asymptotically stable susceptible pest eradication periodic solution when the release amount of infected pest is larger than the critical maximum value (or strength of chemical pesticide spraying is larger than some critical maximum value). Also, we have established the permanence of the system. After comparison, it is explored that integrated pest management is more effective than biological control or chemical control. Finally, verify the analytical results through numerical simulation.

A discrete tick population dynamics model with continuous and seasonal birth breeding

In this paper, we formulate a discrete two-stage model with two types of birth mechanisms, continuous and seasonal. We divide tick population into two subgroups, i.e. immature and mature. We also consider diapause as an important adaptive process for ticks in respond to climate change in both stages. We derive a formula for the inherent net reproductive numbers and explore their stability analysis. When breeding is continuous, there exists a unique globally asymptotically stable positive fixed point provided that the inherent net reproductive number is larger than one; the extinction fixed point is globally asymptotically stable if the inherent net reproductive number is less than one. When breeding is seasonal with 2-periodic birth function, there exists a unique globally asymptotically stable periodic solution provided that the inherent net reproductive number is larger than one, while the population goes to extinction if this value is less than one. For the cases with multi-periodic birth function, we use numerical simulation to compare their complex behaviors.

Existence, Stability and Regularity of Periodic Solutions for Nonlinear Fokker–Planck Equations

We consider a class of nonlinear Fokker–Planck equations describing the dynamics of an infinite population of units with mean-field interaction. Relying on a slow–fast viewpoint and on the theory of approximately invariant manifolds we obtain the existence of a stable periodic solution for the PDE, consisting of probability measures. Moreover we establish the existence of a smooth isochron map in the neighborhood of this periodic solution.

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, this paper presents the most probable transition paths to characterize the regime shifts. The most probable transition paths are obtained by minimizing the Freidlin–Wentzell (FW) action functional and Onsager–Machlup (OM) action functional, respectively. The most probable path shows the movement trend of the lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.

Bifurcation Analysis of a Diffusive Virus Infection and Immune Response Model with Two Delays

A reaction–diffusion system with two delays, which describes viral infection spreading in the lymphoid tissue, is investigated. The delays promote very complex dynamics of the model. We get the Hopf bifurcation curves and the stability region for the coexisting steady state on a two-parameter plane by finding the stability switching curves and the subsets of stable region. When two delays cross the boundary of the stable region, the system will undergo stability switches. It is shown that two types of bistability are possible: the coexistence of the stable virus-free steady state and the stable coexisting steady state; the coexistence of the stable virus-free steady state and a stable periodic solution. Numerical simulations suggest that delays can also induce tristability including a steady state and two stable periodic solutions.

Stability and Bifurcation Analysis for a Nitrogen-Fixing Evolutionary Game with Environmental Feedback and Discrete Delays

In this paper, a nitrogen fixation game system with two time delays under nitrogen limitation is investigated. Firstly, we discuss the existence and local stability of the equilibrium points for the nondelay system. Then the nitrogen fixation delay and strategy-dependent delay are used as bifurcation parameters to analyze the local stability and the Hopf bifurcation. In addition, we obtain the direction of Hopf bifurcation, the change of the period for periodic solution and the stability of periodic solution via center manifold theory and normal form method. Finally, numerical simulation is employed to visualize the theoretical analysis results and we find that the nitrogen fixation strategy is the dominant strategy when the values of the two time delays are large enough. This study promotes the investigation of the effects of two time delays on the dynamics of evolutionary games with environmental feedback, especially on stability and bifurcation.

Existence and stability of periodic solutions in a mosquito population suppression model with time delay

By including the maturation period τ of wild mosquitoes, we develop a delay differential equation model to study the suppression of wild mosquito population by releasing Wolbachia-infected male mosquitoes with a release period T=τ/m, where m is a positive integer. We obtain sufficient conditions for non-existence of periodic solutions and existence of a unique or exactly two periodic solutions by taking the initial function as a solution to the delay-free model. We prove the global asymptotic stability of the unique periodic solution, and that one periodic solution is stable and the other is unstable when there are two periodic solutions.

Hopf Bifurcation Analysis of a Diffusive Nutrient–Phytoplankton Model with Time Delay

In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.