Nonconstant Periodic Solutions Research Articles

A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system

In this paper, we consider the multiplicity result of minimal periodic solutions for the first order Hamiltonian system x˙(t)=J∇H(x(t)). We point out a mistake in the literatures [M. Girardi, M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Anal. T.M.A. 10 (4) (1986) 371–382] and [Tianqing An, On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems, J. Math. Anal. Appl. 329 (2007) 1273–1284]. When the potential is even, under some stronger restricted conditions, we give the correct proof and obtain more distinct nonconstant periodic solutions with minimal period T.

The Characterization of the Variational Minimizers for Spatial RestrictedN+1-Body Problems

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restrictedN+1-body problems with a zero mass on the vertical axis of the plane forNequal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any2≤N≤472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.

Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.

Non-constant periodic solutions for second order Hamiltonian system with a p-Laplacian

Abstract In this paper, some existence theorems are obtained for nonconstant periodic solutions of second order Hamiltonian system with a p-Laplacian by using the Linking Theorem.

Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay

In this paper, a modified Holling–Tanner predator–prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator–prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator–prey models with Beddington–DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.

Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems

In this paper we present a general framework for applications of the twisted equivariant degree (with one free parameter) to an autonomous $\Gamma$-symmetric system of functional differential equations in order to detect and classify (according to their symmetric properties) its periodic solutions. As an example we establish the existence of multiple non-constant periodic solutions of delay Lotka-Volterra equations with $\Gamma$-symmetries. We also include some computational examples for several finite groups $\Gamma$.

Periodic and homoclinic travelling waves in infinite lattices

Consider an infinite chain of particles subjected to a potential f , where nearest neighbours are connected by nonlinear oscillators. The nonlinear coupling between particles is given by a potential V . The dynamics of the system is described by the infinite system of second order differential equations q ̈ j + f ′ ( q j ) = V ′ ( q j + 1 − q j ) − V ′ ( q j − q j − 1 ) , j ∈ Z . We investigate the existence of travelling wave solutions. Two kinds of such solutions are studied: periodic and homoclinic ones. On one hand, we prove under some growth conditions on f and V , the existence of non-constant periodic solutions of any given period T > 0 , and speed c > c 0 , where the constant c 0 depends on f ″ ( 0 ) and V ″ ( 0 ) . On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed c > c 0 , emanating from the origin. Moreover, we prove that these homoclinics decay exponentially at infinity. Each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity.

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

Global Continua of Rapidly Oscillating Periodic Solutions of State-Dependent Delay Differential Equations

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions.

Sharp bounds for periodic solutions of Lipschitzian differential equations

A general system of Lipschitzian differential equations, containing simultaneously terms without delay and with arbitrary constant and time-varying delays, is considered. For the autonomous case, a lower bound for the period of nonconstant periodic solutions, expressed in the respective supremum Lipschitz constants, is found. For nonautonomous periodic equations, explicit upper bounds for the amplitudes and maximum derivatives of periodic solutions are obtained. For all n ⩾ 2, the bounds do not depend on n and, in general, are different from that for n = 1. All the bounds are sharp; they are attained in linear differential equations with piece-wise constant deviating arguments. A relation between the obtained bounds and the sharp bounds in other metrics is established.

Hopf bifurcations in a predator–prey system with multiple delays

This paper is concerned with a two species Lotka–Volterra predator–prey system with three discrete delays. By regarding the gestation period of two species as the bifurcation parameter, the stability of positive equilibrium and Hopf bifurcations of nonconstant periodic solutions are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs). In addition, the global existence of bifurcated periodic solutions are also established by employing the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, to verify our theoretical predictions, some numerical simulations are also included.

Geometric method for a global repellor in competitive Lotka–Volterra systems

The geometric method of involving the relative positions of the nullcline planes is used to analyze the global asymptotic behavior of solutions of autonomous Lotka–Volterra competitive systems. Some observations about limit sets are made and, based on these observations, new criteria are established for the system to have a single point global repellor. Clearly, these criteria can be used to preclude the existence of nonconstant periodic solutions.

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R 2 → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found.

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho [L.A.V. Carvalho, On a method to investigate bifurcation of periodic solution in retarded differential equations, J. Difference Equ. Appl. 4 (1998) 17–27] to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka-Valterra type, followed by a system of 2 equations in 2 unknowns that could model the interactions of 2 identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cook and Ladeira [K.L. Cook, L.A.C. Laderia, Applying Carvalho’s method to find periodic solutions of difference equations, J. Difference Equ. Appl. 2 (1996) 105–115. [2]].

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho (A method to investigate bifurcation of periodic solution in retarded differential equations, J. Differ. Equ. Appl. 4 (1998), pp. 17–27) to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka–Valterra type; then a system of two equations in two unknowns that could model the interactions of two identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cooke and Ladeira (Applying Carvalho's method to find periodic solutions of difference equations, J. Differ. Equ. Appl. 2 (1996), pp. 105–115). We first state Carvalho's result.

Periodicity and knots in delay models of population growth

Recently, we investigated the effect of delay on the asymptotic behavior of the model x ̇ + x = f ( x ( ⋅ − τ ) ) of population growth, when the nonlinearity f is a unimodal function. Now we prove that for large delay, there are several nonconstant (positive) periodic solutions. We also use knots theory to study periodic solutions with period 3 τ . Some of our results do not rely on the continuity of f and thus are applicable to wider range of biological problems in which the growth functions are piecewise continuous.

On a result of Leizarowitz and Mizel

Leizarowitz and Mizel (1989) studied a class of one-dimensional infinite horizon variational problems arising in continuum mechanics and established that these problems possess periodic solutions. They considered a one-parameter family of integrands and show the existence of a constant $c$ such that if a parameter is larger than or equal to $c$, then the corresponding variational problem has a solution which is a constant function, while if a parameter is less than $c$, then the corresponding variational problem possesses only non-constant periodic solutions. In this paper we generalize this result for a large class of families of integrands.

Stability, bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models

In this paper a system of three delay differential equations representing a Hopfield type general model for three neurons with two-way (bidirectional) time delayed connections between the neurons and time delayed self-connection from each neuron to itself is studied. Delay independent and delay dependent sufficient conditions for linear stability, instability and the occurrence of a Hopf bifurcation about the trivial equilibrium are addressed. The partition of the resulting parametric space into regions of stability, instability, and Hopf bifurcation in the absence of self-connection is realized. To extend the local Hopf branches for large delay values a particular bidirectional delayed tri-neuron model without self-connection is investigated. Sufficient conditions for global existence of multiple non-constant periodic solutions are obtained for such a model using the global Hopf-bifurcation theorem for functional differential equations due to J. Wu and the Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney, and following the approach developed by Wei and Li.