An analysis of two degenerate double-Hopf bifurcations

Gheorghe Moza; Mihaela Sterpu; Carmen Rocşoreanu; Gheorghe Moza; Mihaela Sterpu; Carmen Rocşoreanu

doi:10.3934/era.2022020

Electronic Research Archive

2022, Volume 30, Issue 1: 382-403. doi: 10.3934/era.2022020

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Research article

An analysis of two degenerate double-Hopf bifurcations

1.
Department of Mathematics, Politehnica University of Timisoara, Romania
2.
Department of Mathematics, University of Craiova, Romania
3.
Department of Statistics and Economic Informatics, University of Craiova, Romania

Academic Editor: Enrique Ponce

Received: 28 October 2021 Revised: 18 December 2021 Accepted: 04 January 2022 Published: 13 January 2022

The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.

Keywords:

Citation: Gheorghe Moza, Mihaela Sterpu, Carmen Rocşoreanu. An analysis of two degenerate double-Hopf bifurcations[J]. Electronic Research Archive, 2022, 30(1): 382-403. doi: 10.3934/era.2022020

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Abstract

1. Introduction

Nonlinear oscillators have been widely used in various engineering and applied sciences, such as mathematics, physics, structural dynamics, mechanical engineering and other related fields of science ^{[1,2,3,4,5,6]}. Nonlinear differential equations (NDEs) can model many phenomena in various scientific aspects to present their effects and behaviors through mathematical principles. Perturbation methods are extremely beneficial when the nonlinear response is small ^[7,8,9,10]. In general, solving strongly nonlinear differential equations is very difficult. In ^[11], Mickens suggested an approximate expression for solving a truly nonlinear Duffing oscillator. Recently, various powerful analytical and numerical approximation techniques have been suggested for dealing with nonlinear oscillator differential equations. These include He's frequency-amplitude formulation ^[12], the harmonic balance method ^[13,14], the straightforward frequency prediction method ^[15], the modified harmonic balance method ^[16,17], the energy balance method ^[18,19,20], the homotopy perturbation method ^{[21,22,23,24,25]}, the Hamiltonian approach ^[26,27], the weighted averaging method ^[28], the global residue harmonic balance method ^[29,30,31], the max-min approach ^[32,33], Newton's harmonic balance method ^[34], the variational iteration method ^[35], the parameter-expansion method ^[36], the Lindstedt-Poincaré method ^[37,38] and the global error minimization method ^{[39,40,41,42]}.

The global error minimization method (GEMM) is one of the most frequently used techniques for dealing with the solutions of nonlinear oscillators, as it provides more accurate results valid for both weakly and strongly nonlinear oscillators than other known methods ^{[39,40,41,42]}. In the previous example, the solution up to the first approximation is calculated.

In this study, we extend and improve the global error minimization method up to a third order approximation to achieve the higher-order analytical solution of strongly nonlinear Duffing-harmonic oscillators. The present method is applied for two different problems, and the analytical results show that the modified global error minimization method (MGEMM) has better agreement with numerical solutions than the other analytical methods. Excellent agreement is observed between the approximate and exact solutions even for large amplitudes of the oscillation. Comparing the exact solutions with the approximate results has proved that the MGEMM is quite an accurate method in strongly nonlinear oscillator systems.

2. Basic idea of the modified global error minimization method (MGEMM)

To describe the proposed modified global error minimization method (MGEMM), we consider general second order nonlinear oscillator differential equations as follows:

$\ddot u + F(\dot u, \, u, \, t) = 0, \, \;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.\;\;\;\;$

(1)

We introducing $E(u)$ as a new function, defined as in ^[40,43], in the following form:

$E(u) = \int\nolimits_0^T \;{\left( {\ddot u + F(\dot u, \, u, \, t)} \right)^2}dt, \, \;\;\;\;\;T = 2\pi {\omega ^{ - 1}}.$

(2)

By assuming that $F(u)$ is an odd function, a general n-th order trial function of Eq (1) can be expressed as a sum of trigonometric functions as follows:

$u(t) = \mathop \sum \limits_{n = 0}^\infty ({a_{2n + 1)}}\cos ((2n + 1)\omega t)),$

(3)

where ${a_{(2n + 1)}}$ are unknown constant values which satisfy the relation

$A = \mathop \sum \limits_{n = 0}^\infty {a_{(2n + 1)}}.$

(4)

The following conditions were used to obtain the unknown parameters (i.e., ${a_{(2n + 1)}}$ and $\omega$ ):

$\frac{{\partial E(u)}}{{\partial \omega }} = 0, \, \;\;\;\;\;\;\;\;\frac{{\partial E(u)}}{{\partial {a_{(2n + 1)}}}} = 0, \, \;\;\;n \geqslant 1.\;\;$

(5)

By solving the n Eq (5) with the aid of Eq (4), the constants ${a_1}, \, \, {a_3}, {a_5}$ and the frequency of vibration $\omega$ are obtained.

3. Examples of two nonlinear Duffing-harmonic oscillators

In this section, two practical examples of nonlinear Duffing-harmonic oscillators are illustrated to show the effectiveness, accuracy and applicability of the proposed approach.

3.1. Example 1

In this application, we consider the following nonlinear Duffing-harmonic oscillator:

$\ddot u + {k_1}u + \frac{{{k_3}u}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0, \,$

(6)

where dots denote differentiation with respect to $t$ . Now, we shall study some different relevant cases considering Eq (6).

3.1.1. Case 1

First, we consider ${k_1} = 1$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Then, we have a nonlinear oscillator system having an irrational elastic item ^[12,16].

$\ddot u + u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0.$

(7)

According to the basic idea of the global error minimization method, the minimization problem of Eq (7) is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{u}{{1 + {u^2}}}} \right)^2}dt, \, \;\;\;T = 2\pi /\omega .$

(8)

The first-order approximate solution for Eq (7) can be represented as a trial function in the form

${u_1}(t) = {a_1}\cos (\omega t).$

(9)

Substituting Eq (9) into Eq (8) and choosing ${a_1} = A,$ it follows that

$E({u_1}) = \frac{{4{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{8\omega }} - 4{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{4}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(10)

Applying $\partial E({u_1})/\partial \omega = 0$ , the frequency of the nonlinear oscillator is obtained as follows:

$\omega = {\omega _1} = \sqrt {\frac{{16 + 18{A^2} + 5{A^4} + 2\sqrt {256 + 576{A^2} + 487{A^4} + 180{A^6} + 25{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(11)

In order to illustrate the capacity of the global error minimization method, the second-order approximation is applied to the Duffing-harmonic oscillator by using the following new trial solution:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t), \,$

(12)

where $A = {a_1} + {a_3}.$ By substituting Eq (12) into Eq (8), we have

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \tfrac{\pi }{{8\omega }}\left( {5{{({\omega ^2} - 1)}^2}a_1^6 + 5(11{\omega ^4} - 14{\omega ^2} + 3)a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4})} \right.} \\ { + \left. {3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + 3a_1^4\left( {4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2} \right)} \\ { + a_3^2\left( {8{{(2 - 9{\omega ^2})}^2} + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4} \right)} \\ { + \left. {a_1^2\left( {8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2 + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4} \right)} \right) = 0.} \end{array}} \right)\,$

(13)

Setting $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ leads to

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(20\omega ( - 1 + {\omega ^2})a_1^6 + 5( - 28\omega + 44{\omega ^3})a_1^5{a_3} + 2a_1^3{a_3}(8( - 18\omega + 20{\omega ^3}) + 3( - 100\omega + 436{\omega ^3})a_3^2} \\ { + 3a_1^4(4( - 6\omega + 4{\omega ^3}) + ( - 220\omega + 636{\omega ^3})a_3^2) + a_3^2( - 288\omega (2 - 9{\omega ^2}) + 12( - 54\omega + 324{\omega ^3})a_3^2} \\ { - 180\omega (1 - 9{\omega ^2})a_3^4) + a_1^2(32\omega ( - 2 + {\omega ^2}) + 16( - 90\omega + 236{\omega ^3})a_3^2 + 3( - 380\omega + 2236{\omega ^3})a_3^4))} \\ { - \tfrac{1}{{8{\omega ^2}}}\pi (5{{( - 1 + {\omega ^2})}^2}a_1^6 + 5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3\left( {5 - 50{\omega ^2}} \right.} \\ {\left. { + 109{\omega ^4}} \right)\, a_3^2) + 3a_1^4(4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2) + a_3^2(8{{(2 - 9{\omega ^2})}^2}} \\ { + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4) + a_1^2(8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2} \\ { + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4)) = 0, } \end{array}} \right)\,$

(14)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5 + 6(15 - 110{\omega ^2} + 159{\omega ^4})a_1^4{a_3} + 12(5 - 50{\omega ^2} + 109{\omega ^4})a_1^3a_3^2} \\ { + 2a_1^3\left( {8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + a_3^2\left( {24(2 - 27{\omega ^2}81{\omega ^4}){a_3} \\+ 20{{(1 - 9{\omega ^2})}^2}a_3^3} \right)} \\ { + a_1^2\left( {32(6 - 45{\omega ^2} + 59{\omega ^4}){a_3} + 12(15 - 190{\omega ^2} + 559{\omega ^4})a_3^3} \right) + 2{a_3}\left( {8{{(2 - 9{\omega ^2})}^2}} \right.} \\ { + \left. {12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^2)} \right) = 0.} \end{array}} \right)\,$

(15)

For a known amplitude, the parameters of ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained by using the condition $A = {a_1} + {a_3}$ and solving Eqs (14) and (15). The computations were performed using the Mathematica software program, version 9.

To illustrate the capacity of this method, the third order approximation is applied by using the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t), \,$

(16)

where $A = {a_1} + {a_3} + {a_5}$ . Bringing Eq (16) into Eq (8) results in

$\left. \begin{gathered} \begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + a_3^2{{\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8\left( {2 - 9{\omega ^2}} \right){\omega ^2}} \right)}^2}} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \right.} \\ {\left. { + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right) + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \\ {4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(17)

Applying $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ yields

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {20a_1^6\omega \left( {{\omega ^2} - 1} \right) + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2a_1^5\omega \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 2a_1^3{a_3}\left( {3a_3^2\left( {436{\omega ^3} - 100\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 460\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right.} \\ {\left. { + 10a_5^2\left( {1004{\omega ^3} - 124\omega } \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 860\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {11576{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 52\omega } \right) + 8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 3a_5^2\left( {13596{\omega ^3} - 940\omega } \right)} \right) + a_5^2\left( {12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 800\omega \left( {2 - 25{\omega ^2}} \right)} \right.} \\ {\left. { - 500a_5^4\omega \left( {1 - 25{\omega ^3}} \right)} \right) + a_3^2\left( {3a_5^4\left( {22524{\omega ^3} - 1180\omega } \right) + 16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) \\- 288\omega \left( {2 - 9{\omega ^2}} \right)} \right)} \\ \begin{gathered} + a_1^4\left( {a_3^2\left( {1908{\omega ^3} - 660\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 380\omega } \right) + 3\left( {3a_5^2\left( {1108{\omega ^3} - 180\omega } \right) \\+ 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) \hfill \\ \begin{array}{*{20}{l}} {a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 380\omega } \right) + 18{a_5}a_3^3\left( {1364{\omega ^3} - 180\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) \\+ 32\omega \left( {{\omega ^2} - 2} \right)} \right.} \\ { + 9a_5^4\left( {5268{\omega ^3} - 340\omega } \right) + 4a_3^2\left( {3a_5^2\left( {6972{\omega ^3} - 700\omega } \right) + 4\left( {236{\omega ^3} - 90\omega } \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {10876{\omega ^3} - 860\omega } \right) + 8\left( {196{\omega ^3} - 54\omega } \right)} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) \\+ 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_3^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \end{array}} \right)\,$

(18)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {5a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {11{\omega ^2} - 3} \right) + 30a_3^5{{\left( {1 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {2{a_3}\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right)} \right) + a_1^4\left( {2{a_3}\left( {477{\omega ^4} - 330{\omega ^2} + 45} \right) + 4{a_5}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {6{a_3}\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right) + 2a_1^3\left( {8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right)} \\ { + 12a_3^3\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + \\ 4{a_1}{a_3}{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + 2{a_3}\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + \\ 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(19)

$\begin{gathered} \left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {3a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {9{\omega ^2} - 1} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 6{a_5}\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + \\ 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ { + \left. {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {1 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 36a_5^3\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 12{a_3}a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 24a_3^2{a_5} \\\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \\ {\left. {\left. { + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right) \hfill \\ \, \hfill \\ \end{gathered}$

(20)

Now, by solving Eqs (18)–(20) and applying the condition $A = {a_1} + {a_3} + {a_5}$ , the parameters ${a_1}, \, {a_3}, \, {a_5}$ and the angular frequency $\omega$ can be obtained for the known amplitude $A,$ using the Mathematica software program, version 9. To examine the accuracy of the MGEMM solutions, the obtained results are compared with the frequency-amplitude formulation (FAF) ^[12], the energy balance method (EBM) ^[19], the modified harmonic balance method (MHBM) ^[16] and the exact solutions, as presented in Table 1 and Figure 1. We conclude that the third order approximation provides an excellent accuracy with respect to the exact numerical solutions.

Table 1. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{FAF}}$	${\omega _{EBM}}$	${\omega _{MHBM}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[12]	^[19]	^[16]	present	Exact
0.01	1.41419	1.41419	1.41419	1.41419	1.41419
0.1	1.41158	1.41158	1.41158	1.41158	1.41158
0.2	1.40388	1.40389	1.40390	1.40390	1.40390
0.4	1.37581	1.37595	1.37616	1.37616	1.37616
0.6	1.33694	1.33743	1.33827	1.33827	1.33827
0.8	1.29448	1.29550	1.29744	1.29743	1.29743
1	1.25375	1.25514	1.25845	1.25840	1.25842
5	1.02500	1.02588	1.03148	1.02945	1.03139
10	1.00656	1.00681	1.00895	1.00790	1.00893
100	1.00007	1.0000	1.00010	1.00010	1.00010
1000	1.00000	1.00000	1.00000	1.00000	1.00000

| Show Table

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Figure 1. Comparison of the approximate solution (red line) with the numerical solution (blue line).

DownLoad: Full-Size Img PowerPoint

3.1.2. Case 2

Now, if we put ${k_1} = 0$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6), we obtain the following nonlinear oscillator, in which the restoring force has a rational expression ^[21,26].

$\ddot u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(21)

Using the previously mentioned procedure, the solution up to a third-order approximation is calculated. Depending on the analytical approximation, first, second or third, the approximate solution is assumed in the forms of (9), (12) and (16), respectively.

Finally, as in Case 1, the third order approximate solutions are compared with the homotopy perturbation method (HPM) ^[21], the Hamiltonian approach (HA) ^[26] and the exact solutions, as displayed in Table 2.

Table 2. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{HPM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[21]	^[26]	present	Exact
0.01	0.999963	0.9999625	0.999963	0.99999
0.1	0.996271	0.99627403	0.996273	0.9991208
1	0.755929	0.765366864	0.761539	0.76157808
10	0.114708	0.13420106	0.11948	0.123322
100	0.0115462	0.01407125	0.0120357	0.0125265

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DownLoad: CSV

3.2. Example 2

In the second application, we will consider the following nonlinear Duffing-harmonic oscillator ^[18]:

$\ddot u + {k_1}u + \frac{{{k_3}{u^3}}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0, \,$

(22)

where dots denote differentiation with respect to $t$ . Now, we consider some following cases to compare the present solutions with published solutions using different approximate analytical methods.

3.2.1. Case 1

First, we consider ${k_1} = 1,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (22). Then, we have the following nonlinear Duffing-harmonic equation ^[18,28]:

$\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \, \, \dot u(0) = 0.$

(23)

The minimization problem is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}}} \right)^2}dt, \, \;\;T = \frac{{2\pi }}{\omega }.$

(24)

For the first-order approximation, assume that the trial function is given by

${u_1}(t) = {a_1}\cos (\omega t), \,$

(25)

where $A = {a_1}$ . By inserting Eq (25) into Eq (24), we obtain

$E({u_1}) = \frac{{{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{2\omega }} - 2{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{2}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(26)

The frequency can be found through the condition $\partial E({u_1})/\partial \omega = 0$ , as follows:

$\omega = {\omega _1} = \sqrt {\frac{{8 + 18{A^2} + 10{A^4} + 2\sqrt {64 + 288{A^2} + 487{A^4} + 360{A^6} + 100{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(27)

To improve the analytical approximation, we add additional terms to the trial function:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t).$

(28)

The constraint of this minimization is $A = {a_1} + {a_3}$ . Substituting the above new trial function into Eq (24), we obtain

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 2} \right)}^2} + 5a_1^5{a_3}\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \\ { + a_3^2\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. { + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(29)

By using $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ , it follows that

$\left. {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5{a_3}a_1^5} \right.\, \left( {44{\omega ^3} - 56\omega } \right) + 3a_1^4\left( {a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right) + 20a_1^6\omega \left( {{\omega ^2} - 2} \right)} \end{array}} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right) + a_3^2\left( {12a_3^2\left( {324{\omega ^3} - 54\omega } \right) - 288\omega \left( {1 - 9{\omega ^3}} \right)} \right.} \\ {\left. {\left. { - 180a_3^4\omega \left( {2 - 9{\omega ^3}} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 16a_3^2\left( {236{\omega ^3} - 90\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + 5{a_3}a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right)} \right.} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ {\left. {\left. { + 5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0, } \end{array}} \right)\,$

(30)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 6{a_3}a_1^4\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4a_3^2a_1^3\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \\ { + 2a_1^3\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {24{a_3}\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 20a_3^3{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 32{a_3}\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. { + 2{a_3}\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(31)

The minimization problem's conditions can be easily achieved by replacing ${a_1} = A - {a_3}$ , and the parameters ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained for a known amplitude $A$ .

To show the accuracy of the MGEM method in higher order approximations, we apply the third order approximation and consider the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t).$

(32)

Using Eq (32) as the trial function in Eq (24), where $A = {a_1} + {a_3} + {a_5}$ , leads to

$\left. {\begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{{\pi \omega }}{8}\left( {5a_1^6} \right.\left( {{\omega ^2} - 4} \right) + a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 28} \right) + 3{a_5}\left( {9{\omega ^2} - 20} \right)} \right) + 45a_3^6\left( {9{\omega ^2} - 4} \right)} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^2} - 300} \right) + 10a_5^2\left( {251{\omega ^2} - 124} \right) + 3{a_3}{a_5}\left( {663{\omega ^2} - 460} \right) + 40{\omega ^2}} \right)} \\ { + a_1^4\left( {a_3^2\left( {477{\omega ^2} - 660} \right) + 9a_5^2\left( {277{\omega ^2} - 180} \right) + 4{a_3}{a_5}\left( {367{\omega ^2} - 380} \right) + 12{\omega ^2}} \right)} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^2} - 860} \right) + 324{\omega ^2}} \right) + 2{a_1}a_3^2{a_5}\left( {2a_3^2\left( {1447{\omega ^2} - 620} \right) + 2296{\omega ^2}} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^2} - 940} \right) + 15{a_3}{a_5}\left( {153{\omega ^2} - 52} \right)} \right) + 125a_5^2\left( {60a_5^2{\omega ^2} + 40{\omega ^2}} \right.} \\ {\left. { + a_5^4\left( {25{\omega ^2} - 4} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^2} - 380} \right) + 944a_3^2{\omega ^2} + 8{\omega ^2}} \right.} \\ { + 6{a_3}{a_5}\left( {3a_3^2\left( {341{\omega ^2} - 180} \right) + 392{\omega ^2}} \right) + 12a_5^2\left( {7a_3^2\left( {249{\omega ^2} - 100} \right) + 484{\omega ^2}} \right)} \\ {\left. { + 9a_5^4\left( {1317{\omega ^2} - 340} \right) + 6{a_3}a_5^3\left( {2719{\omega ^2} - 860} \right)} \right) + a_3^2\left( {12848a_5^2{\omega ^2} + 648{\omega ^2}} \right.} \\ {\left. {\left. { + 3a_5^4\left( {5631{\omega ^2} - 1180} \right)} \right)} \right) = 0.} \end{array}} \right)\,$

(33)

By setting $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ , we obtain

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {20a_1^6\omega } \right.\left( {{\omega ^2} - 2} \right) - 180a_3^6\omega \left( {2 - 9{\omega ^2}} \right) + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2{\omega _3}a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 6} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 1720\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 2a_3^2\left( {5788{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 104\omega } \right) + 3a_5^2\left( {13596{\omega ^3} - 1880\omega } \right)} \right) + a_5^2\left( { - 800\omega \left( {1 - 25{\omega ^2}} \right)} \right.} \\ {\left. { + 12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 500a_5^4\omega \left( {2 - 25{\omega ^2}} \right)} \right) + a_3^2\left( {16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) - 288\omega \left( {1 - 9{\omega ^2}} \right)} \right.} \\ {\left. { + 3a_5^4\left( {22524{\omega ^3} - 2360\omega } \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 920\omega } \right)} \right.} \\ {\left. { + 2\left( {5a_5^2\left( {1004{\omega ^3} - 248\omega } \right) + 4\left( {20{\omega ^3} - 18\omega } \right)} \right)} \right) + a_1^4\left( {3a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 760\omega } \right)} \right.} \\ {\left. { + 3\left( {a_5^2\left( {3324{\omega ^3} - 1080\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ {\left. { - \tfrac{{1 - \pi }}{{8{\omega ^2}}}} \right)\, 5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 6} \right) + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 5a_3^6{{\left( {2 - 9{\omega ^2}} \right)}^2}} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_3^2\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right. + } \\ {\left. {3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^4\left( {3a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right. + 3\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {a_5^2 + \left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right) + 6{a_3}{a_5}\left( {8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {\left. { + a_5^2 + \left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(34)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\, \left( {3a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {9{\omega ^2} - 2} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right) + 6{a_5}\left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 6{a_5}\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right) + 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {2 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 24{a_5}a_3^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 12a_5^2{a_3}\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \\ {\left. {\left. { + 36a_5^3\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(35)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {\, 5a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {11{\omega ^2} - 6} \right) + 30a_3^5{{\left( {2 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {4{a_3}\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + a_1^4\left( {6{a_3}\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {2{a_3}\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 3{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right) + 12a_3^3\left( {4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right)} \right) + 4{a_1}{a_3}{a_5}\left( {2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right) + 15{a_3}{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + 2{a_3}\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3\left( {a_3^2\left( {327\omega _3^4 - 300{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(36)

Putting ${a_1} = A - {a_3} - {a_5}$ in the minimization problem changes the constraint minimization problem to an unconstrained minimization problem, which is easier to find the solution of Eq (33) by using the conditions $\partial E({u_3})/\partial \omega = 0,$ $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ .

We plot the analytical solutions obtained from MGEMM (red line) Eq (32) and compare them with numerical solutions of Eq (23) obtained using the fourth order Runge-Kutta method (blue line). It is observed that for all different values of the amplitude A, the approximate solutions match extremely well with the numerical solutions (see Figure 2).

Figure 2. Comparison of the approximate solution (red line) with the numerical solution (blue line).

DownLoad: Full-Size Img PowerPoint

3.2.2. Case 2

Second, we consider ${k_1} = 0,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Hence, we have the one-dimensional nonlinear oscillator governed by ^{[21,22,23,24,25,26]}

$\ddot u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(37)

Finally, we can obtain the first and second-order approximations to Eq (37) given by Eqs (25) and (28), respectively. We remark that the third-order approximation in Eq (32) can be given by MGEMM in a similar manner. Generally, after three steps of MGEMM, one can obtain the approximated solutions to Eq (37) with sufficient accuracy. The analytical results of Eq (37) are compared with the iterative homotopy harmonic balance method (IHHBM) ^[34], the energy balance method (EBM) ^[20], the max-min approach (MMA) ^[32], the global residue harmonic balance method (GRHBM) ^[29], the Hamiltonian approach (HA) ^[26] and the exact solutions, as shown in Table 3.

Table 3. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{EBM}}$	${\omega _{IHHBM}}$	${\omega _{MMA}}$	${\omega _{GRHBM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[20]	^[34]	^[32]	^[29]	^[26]	present	Exact
0.01	0.00866	0.008478	0.00866	0.008472	0.00865	0.00847	0.00847
0.1	0.08627	0.084418	0.08627	0.084394	0.08624	0.08439	0.08439
1	0.65164	0.63136	0.65465	0.636795	0.64359	0.636783	0.63678
5	0.97343	0.96667	0.97435	0.968107	0.96731	0.969202	0.96698
10	0.99314	0.99090	0.99340	0.991591	0.99095	0.992005	0.99092
50	0.99973	0.99961	0.99973	0.999657	0.999608	0.999676	0.99961
100	0.99999	0.999901	0.99993	0.999914	0.99990	0.999919	0.99990

| Show Table

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4. Results and discussion

In this paper, we test the analytical solutions of strongly nonlinear Duffing-harmonic oscillators to show the effectiveness of MGEMM. Comparisons of the analytical solutions and the exact numerical solutions of the Duffing-harmonic oscillators for small and large values of the amplitude have been illustrated in Figures 1 and 2 and Tables 1–3. In these Figures, the analytical solution is indicated by a red line, while the numerical solution is represented by a blue line. There is good compatibility between the analytical and numerical solutions, which confirms the accuracy of our results, and excellent matching is observed in these calculations. Moreover, as shown in Tables 1–3, the results produced using the MGEMM are in better agreement with those obtained using exact solutions than other existing ones in the literature. The calculations of the above applications were done using Mathematica software.

5. Conclusions

In this paper, the modified global error minimization method has been presented successfully to obtain higher order approximate periodic solutions of strongly nonlinear Duffing-harmonic oscillators. The present method, which is proved to be a powerful mathematical tool to study nonlinear oscillators, can be easily extended to any nonlinear equation because of its efficiency and convenient applicability. We demonstrated the accuracy and efficiency of the proposed method by solving some examples. We showed that the obtained solutions are valid for the whole domain. Comparisons of the obtained solutions, whether numerical or analytical, revealed a clear match, highlighting the precision of the modified GEMM.

Acknowledgments

M. Zayed and G. M. Ismail extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through the research groups program under grant R.G.P.2/207/43.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third Edition, Springer–Verlag, New York, 2004. https://doi.org/10.1007/978-1-4757-3978-7
[2]	S. N. Chow, C. Li, D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge and New York, 1994. https://doi.org/10.1017/CBO9780511665639
[3]	J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[4]	G. Revel, D. M. Alonso, J. L. Moiola, A degenerate 2:3 resonant Hopf–Hopf bifurcation as organizing center of the dynamics: numerical semiglobal results, SIAM J. Appl. Dyn. Syst., 14 (2015), 1130–1164. https://doi.org/10.1137/140968197 doi: 10.1137/140968197
[5]	N. K. Gavrilov, Bifurcations of an equilibrium with one zero and a pair of pure imaginary roots, Methods Qual. Theory Differ. Equations, 1987.
[6]	N. K. Gavrilov, Bifurcations of an equilibrium with two pairs of pure imaginary roots, Methods Qual. Theory Differ. Equations, (1980), 17–30,
[7]	J. Keener, Infinite period bifurcation and global bifurcation branches, SIAM J. Appl. Math., 41 (1981), 127–144. https://doi.org/10.1137/0141010 doi: 10.1137/0141010
[8]	W. Langford, Periodic and steady mode interactions lead to tori, SIAM J. Appl. Math., 37 (1979), 22–48. https://doi.org/10.1137/0137003 doi: 10.1137/0137003
[9]	T. G. Molnar, Z. Dombovari, T. Insperger, G. Stepan, On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction, Proc. Roy. Society A, 473 (2017), 1–20. https://doi.org/10.1098/rspa.2017.0502 doi: 10.1098/rspa.2017.0502
[10]	S. Ruan, Nonlinear dynamics in tumor-immune system interaction models with delays, Discrete Cont. Dyn. Syst. B, 26 (2021), 541–602. https://doi.org/10.3934/dcdsb.2020282 doi: 10.3934/dcdsb.2020282
[11]	H. Xu, G. Wen, Q. Qin, H. Zhou, New explicit critical criterion of Hopf–Hopf bifurcation in a general discrete time system, Commun. Nonlinear Sci. Num. Simul., 18 (2013), 2120–2128. https://doi.org/10.1016/j.cnsns.2012.12.019 doi: 10.1016/j.cnsns.2012.12.019
[12]	Y. Zhou, Double Hopf bifurcation of a simply supported rectangular thin plate with parametrically and externally excitations, J. Space Explor., 6 (2017), 1–16.
[13]	L. Perko, Differential Equations and Dynamical Systems, third edition, Springer Verlag, New York, 2001.
[14]	J. Sotomayor, Generic bifurcations of dynamic systems, in Dynamical Systems (ed. M. M. Peixoto), Academic Press, 1973. https://doi.org/10.1016/B978-0-12-550350-1.50047-3
[15]	H. W. Broer, G. Vegter, Subordinate Silnikov bifurcations near some singularities of vector fields having low codimension, Ergod. Theor. Dyn. Syst., 4 (1984), 509–525. https://doi.org/10.1017/S0143385700002613 doi: 10.1017/S0143385700002613
[16]	H. W. Broer, S. J. Holtman, G. Vegter, R. Vitolo, Geometry and dynamics of mildly degenerate Hopf–Neimarck–Sacker families near resonance, Nonlinearity, 22 (2009). https://doi.org/10.1088/0951-7715/22/9/007 doi: 10.1088/0951-7715/22/9/007
[17]	V. Kirk, Breaking of symmetry in the saddle-node Hopf bifurcation, Physics Letters A, 154 (1991), 243–248. https://doi.org/10.1016/0375-9601(91)90814-O doi: 10.1016/0375-9601(91)90814-O
[18]	V. Kirk, Merging of resonance tongues, Physica D, 66 (1993), 267–28l. https://doi.org/10.1016/0167-2789(93)90069-D doi: 10.1016/0167-2789(93)90069-D

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4.	Nazmul Sharif, Helal Uddin Molla, M.S. Alam, Analytical solution of couple-mass-spring systems by novel homotopy perturbation method, 2024, 167, 00207462, 104923, 10.1016/j.ijnonlinmec.2024.104923
5.	Gamal M. Ismail, Nadia M. Farea, Mahmoud Bayat, Ji Wang, Investigation of the highly complex nonlinear problems via modified energy balance method, 2024, 23071877, 10.1016/j.jer.2024.07.006
6.	Gamal M. Ismail, Alwaleed Kamel, Abdulaziz Alsarrani, Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method, 2024, 22, 2391-5471, 10.1515/phys-2024-0011
7.	Zhenbo Li, Jin Cai, Linxia Hou, A modified generalized harmonic function perturbation method and its application in analyzing generalized Duffing–Harmonic–Rayleigh–Liénard oscillator, 2024, 166, 00207462, 104832, 10.1016/j.ijnonlinmec.2024.104832

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