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This paper proposes a postverification method (PVM) for solving forced Duffing oscillator problems without prescribed periods. Comprising a postverification procedure and small random perturbation, the proposed PVM improves the sensitivity of the convergence of Newton’s iteration. Numerical simulations revealed that the PVM is more accurate and robust than Kubíček’s approach. We applied the PVM to previous research on bifurcation problems.

A Duffing oscillator equation [

There are several applications for nonlinear dynamic systems with ambiguous or unknown force oscillators [

A continuation method involving Newton’s iterative method has frequently been applied to solve the branch of bifurcation points. The approach proposed by Kubíček and Marek [

Forced Duffing oscillators, which lack nonlinear restoring force, were simulated using Kubíček’s approach, the MK approach, and proposed PVM to evaluate the performances of these methods. The numerical results indicated that the proposed approaches can be used to obtain periodic numerical solutions with greater stability. Furthermore, the PVM was applied to autonomous equations to obtain the periodic solutions.

The remainder of this paper is organized as follows. Section

Based on the assumption that the solution is stable, we developed the MK approach [

Denote

An initial value

Subsequently,

Denote

There exist two numbers

The interval [

The aforementioned argument can be summarized as the following algorithm, which is referred to as Kubíček’s approach [

The approach proposed by Kubíček and Marek [

Let

Set an initial estimate

Repeat

Modify

until

When using Kubíček’s approach, a periodic solution is not always obtainable, even when the criterion of convergence is satisfied, potentially because the

The algorithm for the modified Kubíček(MK) approach is given in Algorithm

Let

Set the initial estimate

Repeat

For

Next

until

In simulations, the computational cost of evaluating

Let

Set the initial estimate

Repeat

For

Next

until

The performance of the proposed methods was evaluated by simulating the Duffing oscillator equation with a nonlinear restoring force [

This example illustrates that the results obtained by using Kubíček’s approach, the MK approach, and proposed PVM to solve (

The numerical solution (top panels) and the trajectories in the phase planes (bottom panels) simulated by Kubíček’s approach, MK, and PVM are shown from left to right panels, respectively. These results are identical for these three methods.

The values of the coefficients

These results show the numerical solutions on [

Because the period is unprescribed, the problem (

These results show the numerical solution on [

This example illustrates an autonomous Duffing-type oscillator [

Figure

These results show the numerical solution

Kubíček and Marek [

The eigenvalues

The aforementioned system of

The Duffing equation [

The calculation of bifurcation points depicts the periods of

The remaining of this section is to demonstrate the difficulty on calculation of the unstable periodic solutions. In the left panel of Figure

It illustrates the phase plane with the initial value point

This study proposes a PVM for obtaining the periodic solutions of forced Duffing equations without a prescribed period. Specifically, to solve a system with

The application of the bifurcation problem demonstrated the branching of other periodic solutions that can be traced to the bifurcation points. However, the simulation and autonomous case revealed that the convergent initial values must be near to the initial estimate. Thus, calculating unstable periodic solutions remains challenging.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to anonymous referees for their valuable comments. Chien-Chang Yen was supported in part by the Ministry of Science and Technology, Taiwan, under the Grant NSC 102-2115-M-030-003 and the other FJU Project of A0502004.