January  2017, 37(1): 591-604. doi: 10.3934/dcds.2017024

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

1. 

Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

2. 

School of Computing and Engineering, Huddersfield University, HD 4, the United Kingdom

* Corresponding author: wangweifrancistju@yahoo.com

Received  July 2015 Revised  September 2016 Published  November 2016

Fund Project: This work is supported by the National Natural Science Foundation of China (Grand No. 11102127,11372210) and Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 12JCYBJC12500)

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

Citation: Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024
References:
[1]

A. Bahrami and A. H. Nayfeh, Nonlinear dynamics of tapping mode atomic force microscopy in the bistable phase, Nonlinear Sci Numer Simulat, 18 (2013), 799-810.  doi: 10.1016/j.cnsns.2012.08.021.  Google Scholar

[2]

M. Belhaq, New analytical technique for predicting homoclinic bifurcations in autonomous dynamical systems, Mech Res Commun, 25 (1998), 49-58.  doi: 10.1016/S0093-6413(98)00006-8.  Google Scholar

[3]

M. Belhaq and A. Fahsi, Homoclinic bifurcations in self-excited oscillators, Mech Res Commun, 23 (1996), 381-386.  doi: 10.1016/0093-6413(96)00035-3.  Google Scholar

[4]

M. BelhaqA. Fashi and F. Lakrad, Predicting homoclinic bifurcations in planar autonomous systems, Nonlinear Dyn, 18 (1999), 303-310.  doi: 10.1023/A:1026428718802.  Google Scholar

[5]

M. BelhaqB. Fiedler and F. Lakrad, Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt-Poincare method, Nonlinear Dyn, 23 (2000), 67-86.  doi: 10.1023/A:1008316010341.  Google Scholar

[6]

Y. Y. CaoK. W. Chung and J. Xu, A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dyn, 64 (2011), 221-236.  doi: 10.1007/s11071-011-9990-9.  Google Scholar

[7]

Y. Y. Chen and S. H. Chen, Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method, Nonlinear Dyn, 58 (2009), 417-429.  doi: 10.1007/s11071-009-9489-9.  Google Scholar

[8]

Y. Y. ChenS. H. Chen and K. Y. Sze, A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators, Acta Mech Sinica, 25 (2009), 721-729.  doi: 10.1007/s10409-009-0276-0.  Google Scholar

[9]

Y. Y. ChenL. W. YanK. Y. Sze and S. H. Chen, Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems, Appl Math Mech Engl Ed, 33 (2012), 1137-1152.  doi: 10.1007/s10483-012-1611-6.  Google Scholar

[10]

A. G. DavodD. D. GanjiR. Azami and H. Babazadeh, Application of improved amplitude frequency formulation to nonlinear differential equation of motion equations, Mod Phys Lett B, 23 (2009), 3427-3440.  doi: 10.1142/S0217984909021466.  Google Scholar

[11]

H. Ding and L. Q. Chen, Galerkin methods for natural frequencies of high-speed axially moving beams, J Sound Vib, 329 (2010), 3484-3494.  doi: 10.1016/j.jsv.2010.03.005.  Google Scholar

[12]

H. DingL. Q. Chen and S. P. Yang, Convergence of galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J Sound Vib, 331 (2012), 2426-2442.  doi: 10.1016/j.jsv.2011.12.036.  Google Scholar

[13]

J. J. FengQ. C. Zhang and W. Wang, Chaos of several typical asymmetric systems, Chaos Solitons Fract, 45 (2012), 950-958.  doi: 10.1016/j.chaos.2012.02.022.  Google Scholar

[14]

J. H. He, Preliminary report on the energy balance for nonlinear oscillations, Mech Res Commun, 29 (2002), 107-111.   Google Scholar

[15]

J. H. He, Homotopy perturbation technique, Comput Methods Appl Mech Eng, 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3.  Google Scholar

[16]

Y. Khan and A. Mirzabeigy, Improved accuracy of He's energy balance method for analysis of conservative nonlinear oscillator, Neural Comput Appl, 25 (2014), 889-895.  doi: 10.1007/s00521-014-1576-2.  Google Scholar

[17]

F. N. Mayoof and M. A. Hawwa, Chaotic behavior of a curved carbon nanotube under harmonic excitation, Chaos Solitons Fract, 42 (2009), 1860-1867.  doi: 10.1016/j.chaos.2009.03.104.  Google Scholar

[18]

Y. V. Mikhlin, Analytical construction of homoclinic orbits of two-and three-dimensional dynamical systems, J Vib Shock, 230 (2000), 971-983.  doi: 10.1006/jsvi.1999.2669.  Google Scholar

[19]

V. K. Melnikov, On the stability of the center for some periodic perturbations, Trans Moscow Math Soc, 12 (1963), 3-52.   Google Scholar

[20]

A. H. NayfehH. M. OuakadF. NajarS. Choura and E. M. Abdel-Rahman, Nonlinear Dynamics of a Resonant Gas Sensor, Nonlinear Dyn, 59 (2010), 607-618.  doi: 10.1007/s11071-009-9567-z.  Google Scholar

[21]

A. F. Vakakis and M. F. A. Azeez, Analytic approximation of the homoclinic orbits of the Lorenz system at σ = 10, b = 8/3 and ρ = 13.926, Nonlinear Dyn, 15 (1998), 245-257.  doi: 10.1023/A:1008202529152.  Google Scholar

[22]

Q. C. ZhangW. Wang and X. J. He, The application of the undetermined fundamental frequency for analyzing the critical value of chaos, Acta Phys Sin, 58 (2009), 5162-5168.   Google Scholar

[23]

Q. C. ZhangW. Wang and W. Y. Li, Heteroclinic bifurcation of strongly nonlinear oscillator, Chin Phys Lett, 25 (2008), 1905-1907.   Google Scholar

[24]

D. YounesianH. AskariZ. Saadatnia and Y. M. Kalami, Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method, Comput Math Appl, 59 (2010), 3222-3228.  doi: 10.1016/j.camwa.2010.03.013.  Google Scholar

show all references

References:
[1]

A. Bahrami and A. H. Nayfeh, Nonlinear dynamics of tapping mode atomic force microscopy in the bistable phase, Nonlinear Sci Numer Simulat, 18 (2013), 799-810.  doi: 10.1016/j.cnsns.2012.08.021.  Google Scholar

[2]

M. Belhaq, New analytical technique for predicting homoclinic bifurcations in autonomous dynamical systems, Mech Res Commun, 25 (1998), 49-58.  doi: 10.1016/S0093-6413(98)00006-8.  Google Scholar

[3]

M. Belhaq and A. Fahsi, Homoclinic bifurcations in self-excited oscillators, Mech Res Commun, 23 (1996), 381-386.  doi: 10.1016/0093-6413(96)00035-3.  Google Scholar

[4]

M. BelhaqA. Fashi and F. Lakrad, Predicting homoclinic bifurcations in planar autonomous systems, Nonlinear Dyn, 18 (1999), 303-310.  doi: 10.1023/A:1026428718802.  Google Scholar

[5]

M. BelhaqB. Fiedler and F. Lakrad, Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt-Poincare method, Nonlinear Dyn, 23 (2000), 67-86.  doi: 10.1023/A:1008316010341.  Google Scholar

[6]

Y. Y. CaoK. W. Chung and J. Xu, A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dyn, 64 (2011), 221-236.  doi: 10.1007/s11071-011-9990-9.  Google Scholar

[7]

Y. Y. Chen and S. H. Chen, Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method, Nonlinear Dyn, 58 (2009), 417-429.  doi: 10.1007/s11071-009-9489-9.  Google Scholar

[8]

Y. Y. ChenS. H. Chen and K. Y. Sze, A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators, Acta Mech Sinica, 25 (2009), 721-729.  doi: 10.1007/s10409-009-0276-0.  Google Scholar

[9]

Y. Y. ChenL. W. YanK. Y. Sze and S. H. Chen, Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems, Appl Math Mech Engl Ed, 33 (2012), 1137-1152.  doi: 10.1007/s10483-012-1611-6.  Google Scholar

[10]

A. G. DavodD. D. GanjiR. Azami and H. Babazadeh, Application of improved amplitude frequency formulation to nonlinear differential equation of motion equations, Mod Phys Lett B, 23 (2009), 3427-3440.  doi: 10.1142/S0217984909021466.  Google Scholar

[11]

H. Ding and L. Q. Chen, Galerkin methods for natural frequencies of high-speed axially moving beams, J Sound Vib, 329 (2010), 3484-3494.  doi: 10.1016/j.jsv.2010.03.005.  Google Scholar

[12]

H. DingL. Q. Chen and S. P. Yang, Convergence of galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J Sound Vib, 331 (2012), 2426-2442.  doi: 10.1016/j.jsv.2011.12.036.  Google Scholar

[13]

J. J. FengQ. C. Zhang and W. Wang, Chaos of several typical asymmetric systems, Chaos Solitons Fract, 45 (2012), 950-958.  doi: 10.1016/j.chaos.2012.02.022.  Google Scholar

[14]

J. H. He, Preliminary report on the energy balance for nonlinear oscillations, Mech Res Commun, 29 (2002), 107-111.   Google Scholar

[15]

J. H. He, Homotopy perturbation technique, Comput Methods Appl Mech Eng, 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3.  Google Scholar

[16]

Y. Khan and A. Mirzabeigy, Improved accuracy of He's energy balance method for analysis of conservative nonlinear oscillator, Neural Comput Appl, 25 (2014), 889-895.  doi: 10.1007/s00521-014-1576-2.  Google Scholar

[17]

F. N. Mayoof and M. A. Hawwa, Chaotic behavior of a curved carbon nanotube under harmonic excitation, Chaos Solitons Fract, 42 (2009), 1860-1867.  doi: 10.1016/j.chaos.2009.03.104.  Google Scholar

[18]

Y. V. Mikhlin, Analytical construction of homoclinic orbits of two-and three-dimensional dynamical systems, J Vib Shock, 230 (2000), 971-983.  doi: 10.1006/jsvi.1999.2669.  Google Scholar

[19]

V. K. Melnikov, On the stability of the center for some periodic perturbations, Trans Moscow Math Soc, 12 (1963), 3-52.   Google Scholar

[20]

A. H. NayfehH. M. OuakadF. NajarS. Choura and E. M. Abdel-Rahman, Nonlinear Dynamics of a Resonant Gas Sensor, Nonlinear Dyn, 59 (2010), 607-618.  doi: 10.1007/s11071-009-9567-z.  Google Scholar

[21]

A. F. Vakakis and M. F. A. Azeez, Analytic approximation of the homoclinic orbits of the Lorenz system at σ = 10, b = 8/3 and ρ = 13.926, Nonlinear Dyn, 15 (1998), 245-257.  doi: 10.1023/A:1008202529152.  Google Scholar

[22]

Q. C. ZhangW. Wang and X. J. He, The application of the undetermined fundamental frequency for analyzing the critical value of chaos, Acta Phys Sin, 58 (2009), 5162-5168.   Google Scholar

[23]

Q. C. ZhangW. Wang and W. Y. Li, Heteroclinic bifurcation of strongly nonlinear oscillator, Chin Phys Lett, 25 (2008), 1905-1907.   Google Scholar

[24]

D. YounesianH. AskariZ. Saadatnia and Y. M. Kalami, Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method, Comput Math Appl, 59 (2010), 3222-3228.  doi: 10.1016/j.camwa.2010.03.013.  Google Scholar

Figure 1.  (a) The phase portrait of homoclinic orbit; (b) The phase portrait of heteroclinic orbit
Figure 2.  (a) The phase portrait of heteroclinic orbit; (b) The curve of $u$; (c) The curve of $\dot u$; 1-without perturbation, 2-with perturbation
Figure 3.  The flow chart of the new approach
Figure 4.  Heteroclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method
Figure 5.  Heteroclinic orbit of the oscillator; 1-Second-order direct method, 2-Numerical simulation method
Figure 6.  Critical bifurcation values; 1-Second-order direct method, 2-Numerical simulation method, 3-Melnikov integration method
Figure 7.  Homoclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method
Table 1.  Comparison of the variables obtained by different methods
GroupParameter values$\omega_{10} $$\theta $ $\mu $
$G_{i}$$\omega_{0} $$\varepsilon $$\alpha $ $\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
11.52211.154-0.1920.2200.2280.225
221.5531.596-0.2300.4810.4880.480
321541.556-0.2050.6300.6480.640
422531.745-0.2980.5060.4930.480
Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method
GroupParameter values$\omega_{10} $$\theta $ $\mu $
$G_{i}$$\omega_{0} $$\varepsilon $$\alpha $ $\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
11.52211.154-0.1920.2200.2280.225
221.5531.596-0.2300.4810.4880.480
321541.556-0.2050.6300.6480.640
422531.745-0.2980.5060.4930.480
Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method
Table 2.  Comparison of the variables obtained by different methods
GroupParameter values$\omega_{10} $$\theta $$\mu $
$G_{i}$ $\omega_{0} $ $\varepsilon $$\alpha $$\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
321541.559-0.2110.6300.6480.645
422531.737-0.2910.5060.4930.494
Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method
GroupParameter values$\omega_{10} $$\theta $$\mu $
$G_{i}$ $\omega_{0} $ $\varepsilon $$\alpha $$\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
321541.559-0.2110.6300.6480.645
422531.737-0.2910.5060.4930.494
Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method
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