Nonconstant Periodic Solutions Research Articles

Hopf bifurcation and stability analysis for a delayed equation with φ-Laplacian

A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, implies the non-existence of nonconstant periodic solutions. Our conditions improve previous results known in the literature for the standard case $\varphi(x)=x$.

Complex Dynamics of a Simple Tumor-Immune Model with Tumor Malignancy

One main feature of a malignant tumor is its uncontrolled growth. In this paper, we propose a simple tumor-immune model to study the progressive characteristics of malignant and benign tumors, where the anti-tumor immunity can be described by the Michaelis–Menten function or the mass action law. The model includes only two state variables for the tumor cells and the effector cells representing the immune system. Three quantities with clear biological meanings are given to determine the asymptotic states of the tumor progression. Moreover, differences in asymptotic states between the two anti-tumor immunity descriptions are drawn. Differently from existing simple models, on the one hand, the model exhibits rich dynamical behaviors including super-critical and sub-critical Bogdanov–Takens bifurcations (consisting of Hopf bifurcation, saddle–node bifurcation, and homoclinic bifurcation) and saddle–node bifurcation of nonconstant periodic solutions (leading to the appearance of two periodic orbits) as the parameters vary; on the other hand, the malignant feature, dormancy, and immune escape of the tumor are revealed with numerical simulations. Furthermore, from the perspective of qualitative analysis and numerical simulations, how the obtained results can be applied to the treatment and control of tumors is illustrated.

Non-constant periodic solutions of the Ricker model with periodic parameters

In this paper, we study the Ricker model x n + 1 = x n exp ⁡ [ r n ( 1 − x n ) ] , n = 0 , 1 , 2 … , ( ∗ ) where x 0 ≥ 0 , and { r n } n = 0 ∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that ( ∗ ) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results.

Caputo-fabrizio fractional-order systems: periodic solution and stabilization of non-periodic solution with application to gunn diode oscillator

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

Existence of $ S $-asymptotically $ \omega $-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $ 1 < \alpha < 2 $

<abstract><p>It is known that there is no non-constant periodic solutions on a closed bounded interval for differential equations with fractional order. Therefore, many researchers investigate the existence of asymptotically periodic solution for differential equations with fractional order. In this paper, we demonstrate the existence and uniqueness of the $ S $-asymptotically $ \omega $-periodic mild solution to non-instantaneous impulsive semilinear differential equations of order $ 1 < \alpha < 2 $, and its linear part is an infinitesimal generator of a strongly continuous cosine family of bounded linear operators. In addition, we consider the case of differential inclusion. Examples are given to illustrate the applicability of our results.</p></abstract>

On the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N + 1)-body problem

In this paper, we study the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N+1)-body problem. First, based on the relationship between the potential and the period as a function of the energy, we deduce the properties of the period of the solution of the corresponding autonomous equation (eccentricity e=0) in the prescribed energy range. Then, according to these properties and the stability criteria of symmetric periodic solutions of the time-periodic Newtonian equation, we analytically prove the linear stability/instability of the symmetric (m,p)-periodic solutions which emanated from nonconstant periodic solutions of the corresponding autonomous equation when the eccentricity is small, which indicates that the former stability criteria can be extended to the generalized Sitnikov problem with N≥3.

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

Complex dynamics of a tumor-immune system with antigenicity

With the effect of antigenicity in consideration, we propose and analyze a conceptual model for the tumor-immune interaction. The model is described by a system of two ordinary differential equations. Though simple, the model can have complicated dynamical behaviors. Besides the tumor-free equilibrium, there can be up to three tumor-present equilibria, which can be a saddle or stable node/focus. Sufficient conditions on the nonexistence of nonconstant periodic solutions are provided. Bifurcation analysis including Hopf bifurcation and Bogdanov–Takens bifurcation is carried out. The theoretical results are supported by numerical simulations. Numerical simulations reveal the complexity of the dynamical behaviors of the model, which includes the subcritical/supercritical Hopf bifurcation, homoclinic bifurcation, saddle-node bifurcation at a nonhyperbolic periodic orbit, the appearance of two limit cycles with a singular closed orbit, and so on. Some biological implications of the theoretical results and numerical simulations are also provided.

Global Dynamics of a Lotka–Volterra Competition Diffusion System with Nonlocal Effects

This paper is concerned with the global dynamics of a Lotka–Volterra competition diffusion system having nonlocal intraspecies terms. Based on the reconstructed comparison principle and monotone iteration, the existence and uniqueness of the solution for the corresponding Cauchy problem are established. In addition, the spreading speed of the system with compactly supported initial data is considered, which admits uniform upper and lower bounds. Finally, some sufficient conditions for guaranteeing the existence and nonexistence of Turing bifurcation are given, which depend on the intensity of nonlocality. Comparing with the classical Lotka–Volterra competition diffusion system, our results indicate that a nonconstant periodic solution may exist if the nonlocality is strong enough, which are also illustrated numerically.

Puu System of Fractional Order and Its Chaos Suppression

In this paper, the fractional-order variant of Puu’s system is introduced, and, comparatively with its integer-order counterpart, some of its characteristics are presented. Next, an impulsive chaos control algorithm is applied to suppress the chaos. Because fractional-order continuous-time or discrete-time systems have not had non-constant periodic solutions, chaos suppression is considered under some numerical assumptions.

On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem

Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd (2,1)-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even (2,1)-periodic solutions are elliptic and linearized stable.

Spatio-temporal coexistence in the cross-diffusion competition system

<p style='text-indent:20px;'>We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

Existence of nonconstant periodic solutions for p(t)-Laplacian Hamiltonian system

The purpose of this paper is to consider the existence of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some results are obtained by using the least action principle and the minimax methods.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

Hopf bifurcation analysis in a predator\u2013prey model with time delay and food subsidies

We analyze the stability of positive equilibrium in a predator-prey model with time delay τ and subsidies. The sufficient conditions of the local Hopf bifurcations at the positive equilibrium are obtained. By center manifold theorem and normal form theory, we analyze the direction of Hopf bifurcations and stability of the bifurcating periodic solution. Using the global Hopf bifurcation theorem, we find that each connected component is unbounded. High-dimensional Bendixson theorem is used to prove that the system has no nonconstant periodic solutions of τ-period, then we obtain the global existence of periodic solutions. Finally, a numerical example is performed to support the theoretical results, and the effect of the food subsidy is discussed. We find that the food subsidy will make the stable interval [0,tau _{0}) of positive equilibrium larger with tau _{0} the first Hopf bifurcation value.

Plastic Dynamical Model for Bulk Metallic Glasses

Based on previous experimental results of the plastic dynamic analysis of metallic glasses upon compressive loading, a dynamical model is proposed. This model includes the sliding speed of shear bands in the plastically strained metallic glasses, the shear resistance of shear bands, the internal friction resulting from plastic deformation, and the influences from the testing machine. This model analysis quantitatively predicts that the loading rate can influence the transition of the plastic dynamics in metallic glasses from chaotic (low loading rate range) to stable behavior (high loading rate range), which is consistent with the previous experimental results on the compression tests of a Cu50Zr45Ti5 metallic glass. Moreover, we investigate the existence of a nonconstant periodic solution for plastic dynamical model of bulk metallic glasses by using Manásevich–Mawhin continuation theorem.

Asymptotically Periodic Solutions for Caputo Type Fractional Evolution Equations

Abstract In this paper, we prove that Caputo type linear fractional evolution equations do not have nonconstant periodic solutions. Then, we study asymptotically periodic solutions of semilinear fractional evolution equations and establish existence and uniqueness results by using theory of semigroup and fixed point theorems. Finally, two examples are given to illustrate the theoretical results.

Periodic orbits in virtually contact structures

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

Infinitely many non-constant periodic solutions with negative fixed energy for Hamiltonian systems

ABSTRACTIn this paper, we obtain the existence of infinitely many non-constant periodic solutions with negative fixed energy for a class of second-order Hamiltonian systems, which greatly improves the existing results such as Zhang [Periodic solutions for some second order Hamiltonian systems. Nonlinearity. 2009;22(9):2141–2150, Theorem 1.5]. Moreover, we exhibit two simple and instructive examples to make our result more clear, which have not been solved by known results.

Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems

<p style='text-indent:20px;'>In the past years, there were very few works on the existence of nonconstant periodic solutions with fixed energy of singular second-order Hamiltonian systems, and now we attempt to ingeniously use Ekeland's variational principle to prove the existence of nonconstant periodic solutions with any fixed energy for singular second-order Hamiltonian systems, and our results greatly generalize some well known results such as [<xref ref-type="bibr" rid="A90">1</xref>,Theorem 3.6]. Moreover, we exhibit two simple and instructive singular potential examples to make our result more clear, which have not been solved by known results.