Existence and stability of periodic solutions in a mosquito population suppression model based on the release of Wolbachia-infected males

In this chapter we study the existence, stability and isolation of periodic solutions belonging to n-dimensional systems of periodic nonlinear differential equations of the form ẋ = f (t, x) where f is periodic in t with some period T > 0: f (t + T, x) = f (t,x). One may believe that an autonomous system of the form ẋ = f (x) is a special case since, obviously, here the right-hand side is periodic in t with arbitrary positive period. Though this is true, autonomous systems cannot be treated similarly to periodic non-autonomous ones. This is so because in the case of an autonomous system we do not know a priori what may be the period of a periodic solution if there exists any, and also because the integral curve belonging to a non-constant periodic solution of an autonomous system can never be “isolated”. More will be said about these problems at the appropriate places. We have mentioned these problems here in order to explain why autonomous systems will be treated in the next chapter. The methods developed with the aim of establishing the existence and stability of periodic solutions can be classified in two groups. The first is the group of topological methods based on degree theory and fixed point theorems. These methods will be presented in the first Section of this chapter. The background material can be found in Appendix 2. The second group consists of (small) perturbation methods. These are more effective but have the disadvantage that they work under the assumption that the given differential equation is a “perturbation” of another one whose periodic solution is known. Both methods have their origin in the works of H. Poincaré [1899]. We shall treat the perturbation methods separately in Chapter 6. In the second section of this chapter we study the stability and isolation problems of periodic solutions. In Sections 3, 4 and 5, applications will be presented.KeywordsPeriodic SolutionAutonomous SystemPeriodic SystemIntegral CurveGlobal Asymptotic StabilityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the stability of periodic solutions with defined sign in MEMS via lower and upper solutions

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

Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays

In this paper, we initiate the study of a class of neural networks with impulses. A sufficient condition for the existence and global exponential stability of a unique periodic solution of the networks is established. Our condition does not assume the differentiability or monotonicity of the activation functions.

The existence of periodic solutions for periodic reaction-diffusion systems with time delay by the periodic upper-lower solution method is investigated. Some methods for proving the uniqueness and the stability of the periodic solution are also given. Two examples are used to show how to use our methods.

In this paper, we consider the sufficient conditions for the stability of periodic solutions of static recurrent neural networks with impulsive delay. In this paper, we study the time - delay static recurrent neural network affected by pulse. The results show that the neural network is stable when the pulse function is linear and relatively small, and a condition for the periodic solution with exponential stability is obtained. This paper introduces the research status of artificial neural network, summarizes the research background and development of static recurrent neural network dynamic system, and introduces the main work of this paper. Using the fixed point theory, M. The existence of periodic solutions and the global robust exponential stability of the static recursive neural network with variable delays and the existence of almost periodic solutions of the static recursive neural network of the partitioned time are studied by combining the properties of the matrix and the Lyapunov function combined with the inequality technique. Global exponential stability, the stability conditions of the corresponding problem are obtained respectively, and the results of the related research are generalized. Using Lyapunov. The stability of the quasi - static neural recursive neural network and the stability of the periodic solution are studied. The condition of the stationary static recursive neural network is obtained and the correctness of the condition is illustrated. Considering the influence of stochastic perturbation on the dynamic behavior of static recurrent neural network, the static recursive neural network with time delay and the static recursive neural network with distributed time delay are studied by using the infinitesimal operator, Ito formula and the convergence theorem of martingales. Global critical exponential stability of quasi - static neural network with stochastic perturbation. The static recursive neural network with Markovian modulation and the time-delay static recurrent neural network model considering both random perturbation and Markovian switching are studied. The linear matrix inequality, the finite state space Markov chain property and the Lyapunov-krasovskii function, The judgment condition of the global exponential stability of the system is obtained. Firstly, the global exponential stability problem of quasi - static neural neural network with time - delay and recursive neural network is studied by using the generalized Halanay inequality. Then the stability of the Markovian response sporadic static recurrent neural network is studied by combining the properties of Markov chain.

In this paper, the existence and global exponential stability of periodic solution is investigated for a class of impulsive bidirectional associative memories neural networks that possesses a Cohen-Grossberg dynamics incorporating variable delays and time-variant coefficients. By using compressive mapping and Lyapunov functional, sufficient conditions are obtained to guarantee the existence and uniqueness of the periodic solution and its global exponential stability. We can see that impulses contribute to the existence and stability of periodic solution for this system. Some comparisons and examples are given to demonstrate the effectiveness of the obtained results. The model studied in this paper is a generalization of some existing models in literature, including Hopfield neural networks, BAM neural networks with impulse and time delays, Cohen-Grossberg neural networks, and thus, the main results of this paper generalize some results in literature.

Consideration is being given here to existence and stability of nonlinear periodic solutions in autonomous hysteresis of ordinary differential equations.

\n Existence and stability of periodic solutions are studied for a system of delay\n differential equations with two delays, with periodic coefficients. It models the\n evolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenous\n leukemia under a periodic treatment that acts only on mature cells. Existence of a guiding\n function leads to the proof of the existence of a strictly positive periodic solution by a\n theorem of Krasnoselskii. The stability of this solution is analysed.\n

In this paper we study the existence, uniqueness and stability of periodic solutions for a two-neuron network system with or without external inputs. The system consists of two identical neurons, each possessing nonlinear feedback and connected to the other neuron via a nonlinear sigmoidal activation function. In the absence of external inputs but with appropriate conditions on the feedback and connection strengths, we prove the existence, uniqueness and stability of periodic solutions by using the Poincaré–Bendixson theorem together with Dulac's criterion. On the other hand, for the system with periodic external inputs, combining the techniques of the Liapunov function with the contraction mapping theorem, we propose some sufficient conditions for establishing the existence, uniqueness and exponential stability of the periodic solutions. Some numerical results are also provided to demonstrate the theoretical analysis.

Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delay

Some new results on periodic solution of Cohen–Grossberg neural network with impulses

Integrated pest management (IPM) is a long-term management strategy and has been proved to be more effective in pest control. To well-understand the mechanism and effect of the action of IPM, the geometric theory of the involved semi-continuous dynamic systems is becoming more and more important. In this work, a geometric approach is applied to analyze the stability of the positive order-one periodic solution in semi-continuous dynamic systems. A stability criterion to test the stability of the order-one periodic solution is established. As an application, a stage-structure model involved chemical control is presented to show the efficiency of the proposed method. The sufficient conditions to insure the existence of the periodic solution are provided. In addition, the number and the stability of the periodic solutions are discussed accordingly. The simulations are carried out to verify the results.

Данная работа посвящена проблеме устойчивости малого периодического решения нормальной автономной системы обыкновенных дифференциальных уравнений. При исследовании устойчивости периодического решения автономной системы естественно анализировать локальную динамику пересечений возмущенных траекторий с ортогональными сечениями соответствующего цикла. Путем введения специальной системы координат, в которой одна из осей направлена по касательной к траектории периодического решения, задача об орбитальной устойчивости периодического решения сводится к задаче об устойчивости по Ляпунову нулевого решения вспомогательной системы с периодической по t правой частью. Для вспомогательной системы, размерность которой на единицу меньше размерности исходной системы, в линейном приближении вопрос об устойчивости нулевого решения сводится к оценке мультипликаторов матрицы монодромии. Таким образом, по теореме Андронова — Витта реализуется классический подход к исследованию орбитальной устойчивости периодического решения. При этом имеет место некритический случай орбитальной устойчивости. Такой подход традиционно используется и в условиях бифуркации типа Хопфа для систем с параметром. В данной работе для автономной системы с параметром получены условия бифуркации малого решения, период которого близок к периоду решений соответствующей линейной однородной системы. Сформулировано определение свойства орбитальной устойчивости по параметру, согласно которому возмущенные правые полутраектории сколь угодно близки к исследуемому циклу не только за счет малости возмущений начальных значений, но и за счет малости параметра. При этом использована идея ослабления требований определения устойчивости ляпуновского типа, предложенная М.М. Хапаевым. Свойство орбитальной устойчивости по параметру может иметь место и при наличии орбитальной неустойчивости исследуемого цикла в классическом смысле. Для исследования орбитальной устойчивости малого периодического решения по параметру использовано нелинейное приближение упомянутой выше вспомогательной системы возмущенных движений.