Limit cycle and bifurcation of nonlinear problems

doi:10.1016/j.chaos.2005.03.007

Chaos, Solitons & Fractals

Volume 26, Issue 3, November 2005, Pages 827-833

https://doi.org/10.1016/j.chaos.2005.03.007 Get rights and content

Abstract

In this paper a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of nonlinear equations. Some examples are given to illustrate its convenience and effectiveness.

Introduction

In this paper we will consider the following nonlinear equation: $\ddot{x} + x + ε f (x, \dot{x}, \ddot{x}) = 0,$ where the parameter ε needs not be small.

This paper aims at finding its limit cycles or bifurcation curves if existed. If we know that the discussed system has limit cycles, then the energy method (or variational method) suggested by He [1] can be powerfully applied to the search for its solution. Homotopy perturbation method [2], [3], [4] was first introduced to find the bifurcation curves [5]. Recently some new analytical methods were appeared to solve periodic solutions of various nonlinear oscillators, e.g., the homotopy perturbation method [2], [3], [4], [6], the variational iteration methods [7], [8], [9], various Lindstedt–Poincare methods [10], [11], [12], [13], variational method [14], [15], [16], [17], [18], [19], extended tan h method [20], [21], Adomian Pade approximation [22], and others.

Section snippets

Limit cycles

In this section, we will propose a new but simple technique, which is quite different from the above mentioned methods, to search for limit cycles. Generally speaking, limit cycles can be determined approximately in the form [1] $x = b + a \cos ω t + \sum_{n = 1}^{m} (C_{n} \cos n ω t + D_{n} \sin n ω t),$ where a, b, C_n and D_n are constants.

We rewrite (1) in the following iteration form ${\ddot{x}}_{n + 1} = - x_{n} - ε f (x_{n}, {\dot{x}}_{n}, {\ddot{x}}_{n}) .$

If the system has limit cycle, then we begin with $x_{0} = b + a \cos ω t + \sum_{i = 1}^{m} (C_{i} \cos i ω t + D_{i} \sin i ω t) .$

Substituting (4) into (3) and integrating

Bifurcation

Recently, a number of methods for finding bifurcation of nonlinear problems were presented, see Refs. [24], [25], [26], [27], [28], [29], in this section, the above iteration method will be applied to finding bifurcation curves.

We consider the following Duffing oscillator in space [5], [30] $\ddot{x} + ε (x - A^{2} x^{3}) = 0, ε ⩾ 0$ with boundary conditions $x (0) = x (π) = 0,$ and normalization condition $x (π / 2) = 1 .$

For any ε ⩾ 0, the above equation has the trivial solution x(t) = 0. The so-called bifurcation occurs when a nontrivial

van der Pol equation

Now we consider the van der Pol equation $\dot{x} = y,$ $\dot{y} = - x + ε (1 - x^{2}) y .$

If we begin with x = A cos ωt, then from (42) we have $y = - A ω \sin ω t .$

Substituting x = A cos ωt and y = −Aω sin ωt into the right side of Eq. (43), we have $\dot{y} = - A \cos ω t - ε A ω (1 - A^{2} \cos^{2} ω t) \sin ω t = - A \cos ω t - ε A ω (1 - \frac{A^{2}}{2} - \frac{A^{2}}{2} \cos 2 ω t) \sin ω t = - A \cos ω t - ε A ω (1 - \frac{A^{2}}{2}) \sin ω t + \frac{ε ω A^{3}}{2} \cos 2 ω t \sin ω t = - A \cos ω t - ε A ω (1 - \frac{A^{2}}{4}) \sin ω t + \frac{ε ω A^{3}}{4} \sin 3 ω t .$

Integrating (45) yields $y = - \frac{A}{ω} \sin ω t + ε A (1 - \frac{A^{2}}{4}) \cos ω t - \frac{ε A^{3}}{12} \cos 3 ω t .$

Comparing (46) with (44) results in $A = 2 and ω = 1 .$

Thus, in the first approximation, we obtain that the

Conclusion

The preceding analysis is of course rather crude but has the virtue of utter simplicity. We conclude from the results 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

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View full text

Limit cycle and bifurcation of nonlinear problems

Abstract

Introduction

Section snippets

Limit cycles

Bifurcation

van der Pol equation

Conclusion

Comput Methods Appl Mech Eng

Int J Non-linear Mech

Appl Math Comput

Int J Nonlinear Mech

Int J Non-Linear Mech

Int J Non-Linear Mech

Chaos, Solitons & Fractals

Chaos, Soliton & Fractals

Chaos, Solitons & Fractals

Determination of limit cycles for strongly nonlinear oscillators

Phys Rev Lett

Homotopy perturbation method for bifurcation of nonlinear problems

Int J Non-linear Sci Numer Simul

Application of He’s homotopy perturbation method to Volterra’s integro-differential equation

Int. J. Non-linear Sci Numer Simul

An approximate solution for 1-D weakly nonlinear oscillations

Int J Non-linear Sci Numer Simul

Polycrystalline solids, nonlinear relaxation phenomena in polycrystalline solids

Int J Non-linear Sci Numer Simul

Modified Lindstedt–Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion

Int J Non-Linear Sci Numer Simul