Uniqueness Of Periodic Solutions Research Articles

Stability and periodicity in dynamic delay equations

Let T be an arbitrary time scale that is unbounded above. By means of a variation of Lyapunov’s method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations x Δ ( t ) = − a ( t ) x ( δ ( t ) ) δ Δ ( t ) . Moreover, if T is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation.

Partial characterization of the global dynamic of a predator-prey model with non constant mortality rate

In this paper we characterize partially the global dynamic of a predator prey model with non constant mortality rate. Concretely, we give necessary and sufficient conditions in order the system be dissipative and permanent. We study the global stability of the nontrivial equilibrium, when it is unique. We show that it is possible the existence of a unique periodic solution which arises from a supercritical Hopf bifurcation and end up through a subcritical Hopf bifurcation; suggesting that the model exhibits new dynamical features which are not present in the classical model; i.e., in the model with constant mortality rate.

Limit cycles in piecewise-affine gene network models with multiple interaction loops

In this article, we consider piecewise affine differential equations modelling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds, etc.). Our main result is an alternative theorem showing that if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases.

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.

Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays

In the current paper, BAM neural networks with time-varying coefficients and distributed time delays are studied. Sufficient conditions guaranteeing the exponential componentwise convergence and existence of one unique periodic solution are obtained by the comparison principle, continuation theorem of topological degree and inequality techniques. The boundedness and differentiability of activation functions are removed. The obtained sufficient criteria are easy to verify and are hence very useful in applications.

Existence and Uniqueness of Periodic Solutions of Mixed Monotone Functional Differential Equations

This paper deals with the existence and uniqueness of periodic solutions for the first-order functional differential equation with periodic coefficients and delays. We choose the mixed monotone operator theory to approach our problem because such methods, besides providing the usual existence results, may also sometimes provide uniqueness as well as additional numerical schemes for the computation of solutions.

Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations

This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.

On periodic solutions of neural networks via differential inclusions

Discontinuous dynamical systems, especially neural networks with discontinuous activation functions, arise in a number of applications and have received considerable research attention in recent years. However, there still remain some fundamental issues to be investigated, for instance, how to define the solutions of such discontinuous systems and what conditions can guarantee the existence and stability of the solutions. In this paper, based on the concept of Filippov solution, the dynamics of a general class of neural networks with discontinuous activation functions is investigated. Sufficient conditions are obtained to ensure the existence and stability of the unique periodic solution for the neural networks by using the differential inclusions theory, the Lyapunov-Krasovskii functional method and linear matrix inequality (LMI) technique. Two numerical examples are given to illustrate the theoretical results.

Periodic solutions for a class of neutral functional differential equations with infinite delay

In this paper, we study the existence and uniqueness of periodic solutions of the nonlinear neutral functional differential equation with infinite delay of the form d d t ( x ( t ) − ∫ − ∞ 0 g ( s , x ( t + s ) ) d s ) = A ( t , x ( t ) ) x ( t ) + f ( t , x t ) . In the process we use the fundamental matrix solution of x ′ ( t ) = A ( t , u ( t ) ) x ( t ) and construct appropriate mappings, where u ∈ C ( R , R n ) is an ω -periodic function. Then we employ matrix measure and the Leray–Schauder fixed point theorem to show the existence of periodic solutions of this neutral differential equation. In the special case where g ( s , u ) ≡ 0 and A ( t , x ) = A ( t ) , some sufficient conditions which ensure the uniform stability and global attractivity of a unique periodic solution are derived.

On the existence of a unique periodic solution to a Liénard type p-Laplacian non-autonomous equation

In this work, we investigate the following Liénard type p -Laplacian non-autonomous equation: ( φ p ( x ′ ( t ) ) ) ′ + f ( x ( t ) ) x ′ ( t ) + g ( t , x ( t ) ) = e ( t ) . Some criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using the Manásevich–Mawhin continuation theorem and some new techniques. Our results are new and extend some known results from the literature. In the last section, two illustrative examples will be provided to demonstrate the applications of our results.

BAM-type impulsive neural networks with time-varying delays

By using the continuation theorem of coincidence degree theory and constructing a suitable Lyapunov functional, we derive some sufficient conditions for the existence and global exponential stability of a unique periodic solution of BAM neural networks, which assumes neither the monotony nor the boundedness of the activation functions. It is believed that these results are significant and useful for the design and applications of BAM neural networks.

Existence and uniqueness of periodic solutions for forced Liénard-type equations

By using topological degree theory and some analysis skill, some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Lienard-type equations are obtained.

New results of periodic solutions for a Rayleigh equation with a deviating argument

In this work, we study the following Rayleigh equation with a deviating argument: $$x^{\prime\prime}(t) + f(t, x^\prime(t))+ g(t, x(t-\tau(t))) = e(t)$$ . Some criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation is given by using Mawhin’s continuation theorem and some new techniques. Comparing with the previous literature, our results are new and complement some known results.

Periodic oscillation for BAM neural networks with impulses

In this paper, a class of neural networks called bidirectional associative memory (BAM) networks with impulses is discussed. Some sufficient conditions are established for the existence and global exponential stability of a unique periodic solution and the exponentially convergent rate is estimated. The sufficient conditions are easy to be verified and when the impulsive jumps are absent, the results reduce to those of the non-impulsive systems. The approaches are based on employing inequality analysis, matrix theory and its spectral theory.

Stability analysis for periodic solution of neural networks with discontinuous neuron activations

In this paper, we present a general class of neural networks with discontinuous neuron activations and varying coefficients, where the neuron activation function is a discontinuous monotone increasing and bounded function. By using the fixed point theorem in differential inclusion theory and constructing suitable Lyapunov functions, a condition is derived which ensures the existence and global exponential stability of a unique periodic solution for the neural network. Furthermore, under certain conditions global convergence in finite time of the state is investigated. The obtained results show that Forti’s conjecture for neural networks without delays is true. Finally, two numerical examples are given to demonstrate the effectiveness of the results obtained in this paper.

On the existence of periodic solutions to some nonlinear thermal control problems

1 We consider the heat equation with a boundary condition involving a control function. The control function satisfies an ordinary differential equation with a right-hand side containing a nonlinear functional that provides the hysteresis phenomenon. The dependence of the functional on the “mean” temperature over the domain causes nonlocal effects. The problem under consideration occurs in the modeling of thermal control processes in chemical reactors and climate control systems. The solvability of the problem and the periodicity of its solutions are considered. 1. In chemical reactors and climate control systems, there arises the problem of temperature control inside a volume by means of some thermal elements on the boundary of the volume. We consider a mathematical model for such a thermal control process. In our model, the temperature distribution inside the domain obeys the heat equation, while the boundary condition involves a control function. The control function satisfies an ordinary differential equation whose right-hand side is given by a nonlinear functional depending on the “mean” temperature over the domain, which provides the so-called hysteresis phenomenon. The existence and uniqueness of solutions to twophase Stefan problems involving a boundary hysteresis control were studied in [1‐3]. In the present paper, we establish an existence and uniqueness result for the heat equation and investigate the periodicity of its solutions. We suggest the concepts of a strong periodic solution 1 The article was translated by the authors. and a mean-periodic solution and show that the existence of a mean-periodic solution implies the existence of a strong periodic solution with the same period. We also give an example in which a unique mean-periodic solution (hence, a unique strong periodic solution) exists.

Existence and Global Asymptotical Stability of Periodic Solution for theT-Periodic Logistic System with Time-Varying Generating Operators andT0-Periodic Impulsive Perturbations on Banach Spaces

This paper studies the existence and global asymptotical stability of periodic PC-mild solution for theT-periodic Logistic system with time-varying generating operators andT0-periodic impulsive perturbations on Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator corresponding to homogenous well-posedT-periodic system with time-varying generating operators andT0-periodic impulsive perturbations are given. It is shown that the system have a unique periodic PC-mild solution which is globally asymptotically stable whenTandT0are rational dependent and its period must benT0for somen∈N. At last, an example is given for demonstration.

New results on the existence and uniqueness of periodic solutions for Liénard type -Laplacian equation

Abstract By means of topological theory, a class of Lienard type p-Laplacian equation with a deviating argument of the form ( ϕ p ( x ′ ( t ) ) ) ′ + f ( x ( t ) ) x ′ ( t ) + β ( t ) g ( x ( t - τ ( t ) ) ) = e ( t ) is studied. The authors establish some criteria to guarantee the existence and uniqueness of periodic solutions for the above equation. It is meaningful that, compared with the corresponding ones in the known literature, the approaches used to estimate a prior bounds of periodic solutions are essentially new.

The development of slugging in two-layer hydraulic flows

In this paper, we develop a hydraulic theory to describe the occurrence and structure of slugging in a confined two-layer gas-liquid flow generated by prescribed, constant, upstream volumetric flow rates in each layer. A linearized theory for the uniform flow is established, after which we use bifurcation theory to study fully non-linear periodic travelling wave structures. We find that a two-parameter family of such travelling wave solutions exists. Under given conditions, the volumetric flow rate constraint provides a relation between these two parameters. To select a unique periodic travelling wave solution, we require a further relation. We first investigate the conjecture that the periodic travelling wave solution selected in the initial value problem has the same wavelength as the linearly most temporally unstable mode. To do this, we solve the initial value problem numerically on a periodic domain. We find that the separation of the liquid slugs that form is much longer than the wavelength of the most unstable temporal mode. We then develop a different conjecture based on the convective instability of the long ‘tails’ of the available periodic travelling wave solutions, which leads to a better understanding of the wavelength selection process.