doi: 10.3934/dcdsb.2022037
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Chaotic motion and control of the driven-damped Double Sine-Gordon equation

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

2. 

Department of Mathematics and Computer, Wuyi University, Wuyishan, 354300, China

3. 

Departamento de Matemáticas, Universidad de Chile, Santiago, 7800003, Chile

*Corresponding author: Yonghui Xia. Email: yhxia@zjnu.cn; xiadoc@163.com

Received  December 2021 Early access March 2022

In this paper, the chaotic motion of the driven and damped double Sine-Gordon equation is analyzed. We detect the homoclinic and heteroclinic chaos by Melnikov method. The corresponding Melnikov functions are derived. A numerical method to estimate the Melnikov integral is given and its effectiveness is illustrated through an example. Numerical simulations of homoclinic and heteroclinic chaos are precisely demonstrated through several examples. Further, we employ a state feedback control method to suppress both chaos simultaneously. Finally, numerical simulations are utilized to prove the validity of control methods.

Citation: Hang Zheng, Yonghui Xia, Manuel Pinto. Chaotic motion and control of the driven-damped Double Sine-Gordon equation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022037
References:
[1]

G. L. AlfimovA. S. Malishevskii and E. V. Medvedeva, Discrete set of kink velocities in Josephson structures: The nonlocal double Sine-Gordon model, Physica D, 282 (2014), 16-26.  doi: 10.1016/j.physd.2014.05.005.

[2]

A. R. BishopR. FleschM. G. ForestD. W. McLaughlin and E. A. Overman, Correlations between chaos in a perturbed Sine-Gordon equation and a truncated model system, SIAM J. Math. Anal., 21 (1990), 1511-1536.  doi: 10.1137/0521083.

[3]

B. Bruhn and B. P. Koch, Homoclinic and heteroclinic bifurcations in rf SQUIDs, Z. Naturforsch. A, 43 (1988), 930-938.  doi: 10.1515/zna-1988-1104.

[4]

H. J. CaoX. B. Chi and G. R. Chen, Suppressing or inducing chaos in a model of robot arms and mechanical manipulators, J. Sound Vibration, 271 (2004), 705-724.  doi: 10.1016/S0022-460X(03)00382-1.

[5]

Q. CaoK. DjidjeliW. G. Price and E. H. Twizell, Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg-De Vries and Kadomtsev-Petviashvili equations, Physica D, 125 (1999), 201-221.  doi: 10.1016/S0167-2789(98)00242-5.

[6]

A. CaliniN. M. ErcolaniD. W. McLaughlin and C. M. Schober, Melnikov analysis of numerically induced chaos in the nonlinear Schrödinger equation, Physica D, 89 (1996), 227-260.  doi: 10.1016/0167-2789(95)00223-5.

[7]

Y. C. Charles, Chaos and shadowing around a heteroclinically tubular cycle with an application to Sine-Gordon equation, Stud. Appl. Math., 116 (2006), 145-171.  doi: 10.1111/j.1467-9590.2006.00336.x.

[8]

H. B. Chen and J. H. Xie, Harmonic and subharmonic solutions of the sd oscillator, Nonlinear Dynam., 84 (2016), 2477-2486.  doi: 10.1007/s11071-016-2659-7.

[9]

N. ErcolaniM. G. Forest and D. W. McLaughlin, Modulational stability of two phase sine-Gordon wavetrains, Stud. Appl. Math., 71 (1984), 91-101.  doi: 10.1002/sapm198471291.

[10]

R. Grimshaw and X. Tian, Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-De Vries equation, Proc. R. Soc. Lond. A, 445 (1994), 1-21.  doi: 10.1098/rspa.1994.0045.

[11]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Sprigner-Verlag, New York, 1990.

[12]

Y. X. GuoW. H. Jiang and B. Niu, Bifurcation analysis in the control of chaos by extended delay feedback, J. Franklin Inst., 350 (2013), 155-170.  doi: 10.1016/j.jfranklin.2012.10.009.

[13]

L. Huang and J. Lenells, Nonlinear fourier transforms for the Sine-Gordon equation in the quarter plane, J. Differential Equations, 264 (2018), 3445-3499.  doi: 10.1016/j.jde.2017.11.023.

[14]

J. B. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science, Beijing, 2013.

[15]

J. B. Li and J. P. Shi, Bifurcations and exact solutions of ac-driven complex Ginzburg-Landau equation, Appl. Math. Comput., 221 (2013), 102-110.  doi: 10.1016/j.amc.2013.05.067.

[16]

S. B. LiS. ChaoW. Zhang and Y. X. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application, Nonlinear Dynam., 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[17]

S. B. Li, W. S. Ma, W. Zhang and Y. X. Hao, Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application, Internat. J. Bifur. Chaos, 26 (2016), 1650014, 13pp. doi: 10.1142/S0218127416500140.

[18]

S. B. LiX. X. MaX. L. BianS. K. Lai and W. Zhang, Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator, Nonlinear Dynam., 99 (2020), 1621-1642.  doi: 10.1007/s11071-019-05380-0.

[19]

K. N. Lu and Q. D. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[20]

G. P. Luo and C. R. Zhu, Transversal homoclinic orbits and chaos for partial functional differential equations, Nonlinear Anal., 71 (2009), 6254-6264.  doi: 10.1016/j.na.2009.06.026.

[21]

J. E. Macías-Díaz, Computer simulation of the energy dynamics of a sinusoidally perturbed double Sine-Gordon equation: an application to the transmission of wave signals, Rev. Mex. Fis., 58 (2012), 29-40. 

[22]

M. Marhl and M. Perc, Determining the flexibility of regular and chaotic attractors, Chaos Solitons Fractals, 28 (2006), 822-833.  doi: 10.1016/j.chaos.2005.08.013.

[23]

V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Mosc. Math. Soc., 12 (1963), 3-52. 

[24]

T. MiyajiH. Okamoto and A. D. D. Craik, Three-dimensional forced-damped dynamical systems with rich dynamics: bifurcations, chaos and unbounded solutions, Physica D, 311/312 (2015), 25-36.  doi: 10.1016/j.physd.2015.09.001.

[25]

B. Niu and W. H. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.

[26]

E. OttC. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 2837.  doi: 10.1103/PhysRevLett.64.1196.

[27]

E. A. OvermanD. W. Mclaughlin and A. R. Bishop, Coherence and chaos in the driven damped Sine-Gordon equation: measurement of the soliton spectrum, Physica D, 19 (1986), 1-41.  doi: 10.1016/0167-2789(86)90052-7.

[28]

M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons Fractals, 27 (2006), 395-403.  doi: 10.1016/j.chaos.2005.03.045.

[29]

M. Perc and M. Marhl, Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor, Phys. Rev. E, 70 (2004), 016204.  doi: 10.1103/PhysRevE.70.016204.

[30]

K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.  doi: 10.1016/0375-9601(95)00654-L.

[31]

N. R. QuinteroR. Alvarez-Nodarse and F. G. Mertens, Driven and damped double Sine-Gordon equation: the influence of internal modes on the soliton ratchet mobility, Phys. Rev. E, 80 (2009), 016605.  doi: 10.1103/PhysRevE.80.016605.

[32]

T. S. Raju and K. Porsezian, On solitary wave solutions of ac-driven complex Ginzburg-Landau equation, J. Phys. A: Math. Gen., 39 (2006), 1853-1858.  doi: 10.1088/0305-4470/39/8/005.

[33]

M. Salam and S. Sastry, Dynamics of the forced Josephson junction circuit: The regions of chaos, IEEE Trans. Circuits Syst., 32 (1985), 784-796.  doi: 10.1109/TCS.1985.1085790.

[34]

Y. L. SongH. P. JiangQ. X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive mussel-algae model near turing-hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.

[35]

Q. D. Wang, Periodically forced double homoclinic loops to a dissipative saddle, J. Differential Equations, 260 (2016), 4366-4392.  doi: 10.1016/j.jde.2015.11.011.

[36]

G. W. Wang, K. T. Yang, H. C. Gu, F. Guan and A. H. Kara, A $(2+1)$-dimensional Sine-Gordon and Sinh-Gordon equations with symmetries and kink wave solutions, Nucl. Phys. B, 953 (2020), 114956, 14pp. doi: 10.1016/j.nuclphysb.2020.114956.

[37]

A.-M. Wazwaz, The tanh method and a variable separated ODE method for solving double Sine-Gordon equation, Phys. Lett. A, 350 (2006), 367-370.  doi: 10.1016/j.physleta.2005.10.038.

[38]

Z. C. Wei and W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium, Internat. J. Bifur. Chaos, 24 (2014), 1450127, 14pp. doi: 10.1142/S0218127414501272.

[39]

Z. C. Wei, W. Zhang, Z. Wang and M. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Internat. J. Bifur. Chaos, 25 (2015), 1550028, 11pp. doi: 10.1142/S0218127415500285.

[40]

S. Wiggins, Introduction to Applied Non-Linear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[41]

K. Yagasaki, Chaos in a pendulum with feedback control, Nonlinear Dynam., 6 (1994), 125-142.  doi: 10.1007/BF00044981.

[42]

W. ZhangQ. Z. Huo and L. Li, Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator, Appl. Math. Mech., 13 (1992), 217-226.  doi: 10.1007/BF02457367.

[43]

C. Zheng, Numerical solution to the Sine-Gordon equation defined on the whole real axis, SIAM J. Sci. Comput., 229 (2007), 2494-2506.  doi: 10.1137/050640643.

[44]

L. Q. Zhou and F. Q. Chen, Chaotic motions of a damped and driven morse oscillator, Appl. Mech. Mater., 459 (2013), 505-510.  doi: 10.4028/www.scientific.net/AMM.459.505.

[45]

L. Q. Zhou and F. Q. Chen, Subharmonic bifurcations and chaos for the traveling wave solutions of the compound KdV-Burgers equation with external and parametrical excitations, Appl. Math. Comput., 243 (2014), 105-113.  doi: 10.1016/j.amc.2014.05.064.

[46]

C. R. ZhuG. P. Luo and Y. L. Shu, The existences of transverse homoclinic solutions and chaos for parabolic equations, J. Math. Anal. Appl., 335 (2007), 626-641.  doi: 10.1016/j.jmaa.2006.11.057.

[47]

C. R. Zhu and W. N. Zhang, Multiple chaos arising from single-parametric perturbation of a degenerate homoclinic orbit, J. Differential Equations, 268 (2020), 5672-5703.  doi: 10.1016/j.jde.2019.11.024.

show all references

References:
[1]

G. L. AlfimovA. S. Malishevskii and E. V. Medvedeva, Discrete set of kink velocities in Josephson structures: The nonlocal double Sine-Gordon model, Physica D, 282 (2014), 16-26.  doi: 10.1016/j.physd.2014.05.005.

[2]

A. R. BishopR. FleschM. G. ForestD. W. McLaughlin and E. A. Overman, Correlations between chaos in a perturbed Sine-Gordon equation and a truncated model system, SIAM J. Math. Anal., 21 (1990), 1511-1536.  doi: 10.1137/0521083.

[3]

B. Bruhn and B. P. Koch, Homoclinic and heteroclinic bifurcations in rf SQUIDs, Z. Naturforsch. A, 43 (1988), 930-938.  doi: 10.1515/zna-1988-1104.

[4]

H. J. CaoX. B. Chi and G. R. Chen, Suppressing or inducing chaos in a model of robot arms and mechanical manipulators, J. Sound Vibration, 271 (2004), 705-724.  doi: 10.1016/S0022-460X(03)00382-1.

[5]

Q. CaoK. DjidjeliW. G. Price and E. H. Twizell, Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg-De Vries and Kadomtsev-Petviashvili equations, Physica D, 125 (1999), 201-221.  doi: 10.1016/S0167-2789(98)00242-5.

[6]

A. CaliniN. M. ErcolaniD. W. McLaughlin and C. M. Schober, Melnikov analysis of numerically induced chaos in the nonlinear Schrödinger equation, Physica D, 89 (1996), 227-260.  doi: 10.1016/0167-2789(95)00223-5.

[7]

Y. C. Charles, Chaos and shadowing around a heteroclinically tubular cycle with an application to Sine-Gordon equation, Stud. Appl. Math., 116 (2006), 145-171.  doi: 10.1111/j.1467-9590.2006.00336.x.

[8]

H. B. Chen and J. H. Xie, Harmonic and subharmonic solutions of the sd oscillator, Nonlinear Dynam., 84 (2016), 2477-2486.  doi: 10.1007/s11071-016-2659-7.

[9]

N. ErcolaniM. G. Forest and D. W. McLaughlin, Modulational stability of two phase sine-Gordon wavetrains, Stud. Appl. Math., 71 (1984), 91-101.  doi: 10.1002/sapm198471291.

[10]

R. Grimshaw and X. Tian, Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-De Vries equation, Proc. R. Soc. Lond. A, 445 (1994), 1-21.  doi: 10.1098/rspa.1994.0045.

[11]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Sprigner-Verlag, New York, 1990.

[12]

Y. X. GuoW. H. Jiang and B. Niu, Bifurcation analysis in the control of chaos by extended delay feedback, J. Franklin Inst., 350 (2013), 155-170.  doi: 10.1016/j.jfranklin.2012.10.009.

[13]

L. Huang and J. Lenells, Nonlinear fourier transforms for the Sine-Gordon equation in the quarter plane, J. Differential Equations, 264 (2018), 3445-3499.  doi: 10.1016/j.jde.2017.11.023.

[14]

J. B. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science, Beijing, 2013.

[15]

J. B. Li and J. P. Shi, Bifurcations and exact solutions of ac-driven complex Ginzburg-Landau equation, Appl. Math. Comput., 221 (2013), 102-110.  doi: 10.1016/j.amc.2013.05.067.

[16]

S. B. LiS. ChaoW. Zhang and Y. X. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application, Nonlinear Dynam., 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[17]

S. B. Li, W. S. Ma, W. Zhang and Y. X. Hao, Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application, Internat. J. Bifur. Chaos, 26 (2016), 1650014, 13pp. doi: 10.1142/S0218127416500140.

[18]

S. B. LiX. X. MaX. L. BianS. K. Lai and W. Zhang, Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator, Nonlinear Dynam., 99 (2020), 1621-1642.  doi: 10.1007/s11071-019-05380-0.

[19]

K. N. Lu and Q. D. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differential Equations, 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.

[20]

G. P. Luo and C. R. Zhu, Transversal homoclinic orbits and chaos for partial functional differential equations, Nonlinear Anal., 71 (2009), 6254-6264.  doi: 10.1016/j.na.2009.06.026.

[21]

J. E. Macías-Díaz, Computer simulation of the energy dynamics of a sinusoidally perturbed double Sine-Gordon equation: an application to the transmission of wave signals, Rev. Mex. Fis., 58 (2012), 29-40. 

[22]

M. Marhl and M. Perc, Determining the flexibility of regular and chaotic attractors, Chaos Solitons Fractals, 28 (2006), 822-833.  doi: 10.1016/j.chaos.2005.08.013.

[23]

V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Mosc. Math. Soc., 12 (1963), 3-52. 

[24]

T. MiyajiH. Okamoto and A. D. D. Craik, Three-dimensional forced-damped dynamical systems with rich dynamics: bifurcations, chaos and unbounded solutions, Physica D, 311/312 (2015), 25-36.  doi: 10.1016/j.physd.2015.09.001.

[25]

B. Niu and W. H. Jiang, Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations, J. Math. Anal. Appl., 398 (2013), 362-371.  doi: 10.1016/j.jmaa.2012.08.051.

[26]

E. OttC. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 2837.  doi: 10.1103/PhysRevLett.64.1196.

[27]

E. A. OvermanD. W. Mclaughlin and A. R. Bishop, Coherence and chaos in the driven damped Sine-Gordon equation: measurement of the soliton spectrum, Physica D, 19 (1986), 1-41.  doi: 10.1016/0167-2789(86)90052-7.

[28]

M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons Fractals, 27 (2006), 395-403.  doi: 10.1016/j.chaos.2005.03.045.

[29]

M. Perc and M. Marhl, Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor, Phys. Rev. E, 70 (2004), 016204.  doi: 10.1103/PhysRevE.70.016204.

[30]

K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.  doi: 10.1016/0375-9601(95)00654-L.

[31]

N. R. QuinteroR. Alvarez-Nodarse and F. G. Mertens, Driven and damped double Sine-Gordon equation: the influence of internal modes on the soliton ratchet mobility, Phys. Rev. E, 80 (2009), 016605.  doi: 10.1103/PhysRevE.80.016605.

[32]

T. S. Raju and K. Porsezian, On solitary wave solutions of ac-driven complex Ginzburg-Landau equation, J. Phys. A: Math. Gen., 39 (2006), 1853-1858.  doi: 10.1088/0305-4470/39/8/005.

[33]

M. Salam and S. Sastry, Dynamics of the forced Josephson junction circuit: The regions of chaos, IEEE Trans. Circuits Syst., 32 (1985), 784-796.  doi: 10.1109/TCS.1985.1085790.

[34]

Y. L. SongH. P. JiangQ. X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive mussel-algae model near turing-hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.

[35]

Q. D. Wang, Periodically forced double homoclinic loops to a dissipative saddle, J. Differential Equations, 260 (2016), 4366-4392.  doi: 10.1016/j.jde.2015.11.011.

[36]

G. W. Wang, K. T. Yang, H. C. Gu, F. Guan and A. H. Kara, A $(2+1)$-dimensional Sine-Gordon and Sinh-Gordon equations with symmetries and kink wave solutions, Nucl. Phys. B, 953 (2020), 114956, 14pp. doi: 10.1016/j.nuclphysb.2020.114956.

[37]

A.-M. Wazwaz, The tanh method and a variable separated ODE method for solving double Sine-Gordon equation, Phys. Lett. A, 350 (2006), 367-370.  doi: 10.1016/j.physleta.2005.10.038.

[38]

Z. C. Wei and W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium, Internat. J. Bifur. Chaos, 24 (2014), 1450127, 14pp. doi: 10.1142/S0218127414501272.

[39]

Z. C. Wei, W. Zhang, Z. Wang and M. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Internat. J. Bifur. Chaos, 25 (2015), 1550028, 11pp. doi: 10.1142/S0218127415500285.

[40]

S. Wiggins, Introduction to Applied Non-Linear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[41]

K. Yagasaki, Chaos in a pendulum with feedback control, Nonlinear Dynam., 6 (1994), 125-142.  doi: 10.1007/BF00044981.

[42]

W. ZhangQ. Z. Huo and L. Li, Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator, Appl. Math. Mech., 13 (1992), 217-226.  doi: 10.1007/BF02457367.

[43]

C. Zheng, Numerical solution to the Sine-Gordon equation defined on the whole real axis, SIAM J. Sci. Comput., 229 (2007), 2494-2506.  doi: 10.1137/050640643.

[44]

L. Q. Zhou and F. Q. Chen, Chaotic motions of a damped and driven morse oscillator, Appl. Mech. Mater., 459 (2013), 505-510.  doi: 10.4028/www.scientific.net/AMM.459.505.

[45]

L. Q. Zhou and F. Q. Chen, Subharmonic bifurcations and chaos for the traveling wave solutions of the compound KdV-Burgers equation with external and parametrical excitations, Appl. Math. Comput., 243 (2014), 105-113.  doi: 10.1016/j.amc.2014.05.064.

[46]

C. R. ZhuG. P. Luo and Y. L. Shu, The existences of transverse homoclinic solutions and chaos for parabolic equations, J. Math. Anal. Appl., 335 (2007), 626-641.  doi: 10.1016/j.jmaa.2006.11.057.

[47]

C. R. Zhu and W. N. Zhang, Multiple chaos arising from single-parametric perturbation of a degenerate homoclinic orbit, J. Differential Equations, 268 (2020), 5672-5703.  doi: 10.1016/j.jde.2019.11.024.

Figure 1.  Bifurcations and the phase portraits of system (10)
Figure 2.  Homoclinic orbits of system (10) when $h = h_{a_n}$, $\nu = 2$ and $\lambda = -2$
Figure 3.  Heteroclinic orbits of system (10) when $h = h_{b_n}$, $\nu = 2$ and $\lambda = 0.8$
Figure 4.  Heteroclinic bifurcation curves computed by two algorithms for $\alpha = 1$
Figure 5.  The phase portraits of system (9) and time history curves of $u$, $y$ for $\nu = 2$, $\lambda = -2$, $\varepsilon = 0.01$, $\theta_1 = 0.4$, $\theta_2 = 0.3$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 1$, $f_2 = 40$
Figure 6.  The Poincaré section of system (9) with $\nu = 2$, $\lambda = -2$, $\varepsilon = 0.01$, $\theta_1 = 0.4$, $\theta_2 = 0.3$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 1$, $f_2 = 40$
Figure 7.  The phase portraits of system (9) and time history curves of $u$, $y$ for $\nu = 2$, $\lambda = 0.8$, $\varepsilon = 0.01$, $\theta_1 = 1$, $\theta_2 = 2$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 10$, $f_2 = 10$
Figure 8.  The Poincaré section of system (9) with $\nu = 2$, $\lambda = 0.8$, $\varepsilon = 0.01$, $\theta_1 = 1$, $\theta_2 = 2$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 10$, $f_2 = 10$
Figure 9.  The phase portraits of system (9) and time history curves of $u$, $y$ for $\nu = 2$, $\lambda = -2$, $\varepsilon = 0.01$, $\theta_1 = 0.4$, $\theta_2 = 0.3$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 1$, $f_2 = 40$, $P = 20$
Figure 10.  The phase portraits of system (9) and time history curves of $u$, $y$ for $\nu = 2$, $\lambda = 0.8$, $\varepsilon = 0.01$, $\theta_1 = 1$, $\theta_2 = 2$, $\omega = 0.65$, $\alpha = 2$, $f_1 = 10$, $f_2 = 10$, $P = 30$
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