Periodic solutions and bifurcations of delay-differential equations

doi:10.1016/j.physleta.2005.08.014

Physics Letters A

Volume 347, Issues 4–6, 5 December 2005, Pages 228-230

https://doi.org/10.1016/j.physleta.2005.08.014 Get rights and content

Abstract

In this Letter a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of delay-differential equations. An example is given to illustrate its convenience and effectiveness.

Introduction

This Letter considers a non-linear delay-differential system in the form $\dot{x} = - μ \frac{x (t - 1)}{1 + {[x (t - 1)]}^{4}} .$

This system was studied by Peng and Ucar [1] by numerical approach. The numerical result shows that the system, Eq. (1), has multiple bifurcations, stable limit cycles (periodic or quasiperiodic solutions), and chaotic behavior.

Recently many methods were suggested to deal with non-linear equations, for example, homotopy perturbation method [2], [3], [4], variational iteration method [5], [6], various Lindstedt–Poincaré methods [7], [8], [9], variational method [10], [11], [12], [13], [14], [15], extended tanh method [16], [17], Adomian Pade approximation [18]. There also exist many approaches to bifurcation of various non-linear problems [19], [20], [21], [22], [23], [24].

This Letter suggests a simple but effective iteration approach to Eq. (1) to find its periodic solutions and bifurcations.

Section snippets

An iteration method

Delay systems are widely found in engineering, see Ref. [25] and references cited thereby. In this Letter we suggest an iteration method for the discussed problem.

We rewrite Eq. (1) in the form $\dot{x} = - μ x (t - 1) - \dot{x} {[x (t - 1)]}^{4},$ and construct an iteration formulation for (2) as follows ${\dot{x}}_{n + 1} = - μ x_{n} (t - 1) - {\dot{x}}_{n} {[x_{n} (t - 1)]}^{4} .$ We feel interest in its periodic solutions (or limit cycles) and its bifurcations. If we know that the discussed system has limit cycles, then the energy method (or variational method)

Conclusion

The preceding analysis has the virtue of utter simplicity. We conclude from the result obtained that the method developed here is extremely simple in its principle, quite easy to use, and gives a very good accuracy in the whole solution domain, even with the simplest trial functions. Theoretically any accuracy can be achieved by a suitable choice of the trial functions. With the help of some mathematical software, such as Mathematica, Matlab, the method provides a powerful mathematical tool to

Acknowledgement

The author thanks an unknown reviewer for his careful reading and helpful comments. This work is supported by Program for New Century Excellent Talents in University.

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Periodic solutions and bifurcations of delay-differential equations

Abstract

Introduction

Section snippets

An iteration method

Conclusion

Acknowledgement

Chaos Solitons Fractals

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

Chaos Solitons Fractals

Chaos Soliton Fractals

Int. J. Non-Linear Sci. Numer. Simul.

Int. J. Non-Linear Sci. Numer. Simul.

Int. J. Non-Linear Sci. Numer. Simul.

Int. J. Non-Linear Sci. Numer. Simul.

Int. J. Non-Linear Sci. Numer. Simul.

Int. J. Non-Linear Sci. Numer. Simul.