American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020252

Melnikov analysis of the nonlocal nanobeam resting on fractional-order softening nonlinear viscoelastic foundations

Guy Joseph Eyebe 1,, , Betchewe Gambo 1, , Alidou Mohamadou 1, and  Timoleon Crepin Kofane 2,

1. 

Department of Physics, Faculty of Science, The University of Maroua, Maroua, 814, Cameroon
2. 

Laboratory of Mechanics, Materials and Structures, Doctoral research unit in Physics and Applications, University of Yaounde I, Yaounde, 812, Cameroon

* Corresponding author: Guy Joseph Eyebe

Received  April 2019 Revised  June 2019 Published  January 2020

In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that the distance between chaotic region and non-chaotic region in this kind of structure depends strongly on the fractional order parameter.

Keywords: Melnikov, chaos, nanobeam, fractional-order, nonlinearity.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Guy Joseph Eyebe, Betchewe Gambo, Alidou Mohamadou, Timoleon Crepin Kofane. Melnikov analysis of the nonlocal nanobeam resting on fractional-order softening nonlinear viscoelastic foundations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020252
References:
[1]

J. D. Achenbach and C. Sun, Moving load on a flexibly supported timochenko beam, Int. J. Solids Struct., 1 (1965), 353-370.   Google Scholar

[2]

L. M. Anague Tabejieu, B. R. Nana Nbendjo and P. Woafo, On the dynamics of rayleigh beams resting on fractional-order viscoelastic pasternak foundations subjected to moving loads, Chaos Solitons Fract., 93 (2016), 39-47.  doi: 10.1016/j.chaos.2016.10.001.  Google Scholar

[3]

A. A. Andronov and A. Witt, Towards mathematical theory of capture, Archiv. fur Electrotechnik, 24 (1930), 99-110.   Google Scholar

[4]

H. Askari, H. Jamchidifar and B. Fidan, High resolution mass identification using nonlinear vibrations of nanoplates, Measurement, 101 (2017), 166-174.  doi: 10.1016/j.measurement.2017.01.012.  Google Scholar

[5]

M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mech. Res. Commun., 43 (2012), 34-40.  doi: 10.1016/j.mechrescom.2012.02.001.  Google Scholar

[6]

M. Aydogdu and M. Arda, Torsional vibration analysis of double walled carbon nanotubes using nonlocal elasticity, Int. J. Mech. Mater. Des., 12 (2016), 71-84.  doi: 10.1007/s10999-014-9292-8.  Google Scholar

[7]

B. E. Demartini, H. E. Butterfield, J. Moehlis and K. L. Turner, Chaos in microelectromechanical oscillator governed by the nonlinear Mathieu equation, J. Microelec. Syst., 16 (2007), 1314-1323.  doi: 10.1109/JMEMS.2007.906757.  Google Scholar

[8]

H. Ding, Q. L. Chen and S. P. Yang, Convergence of garlekin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 331 (2012), 2426-2442.   Google Scholar

[9]

A. C. Eringen, On differential equations of nonlocal elasticity and solution of screw dislocation and surface waves, J. Appl. Phys., 54 (1983), 4703-4710.  doi: 10.1063/1.332803.  Google Scholar

[10]

A. C. Eringen, Nonlocal Continuum Fields Theories, Springer, USA, 2002.  Google Scholar

[11]

E. Ghavanloo, F. Daneshmand and M. Rafiei, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E, 42 (2010), 2218-2224.   Google Scholar

[12]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields, Springer-Verlag, USA, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[13]

Y. Haitao and Y. Yuan, Analytical solution for an infinite euler-bernoulli beam on a visco-elastic foundation subjected to arbitrary dynamic loads, J. Eng. Mech., 140 (2014), 542-551.   Google Scholar

[14]

Z. Hryniewicz, Dynamics of rayleigh beam on nonlinear foundation due to moving load using adomian decomposition and coiflet expansion, Soil Dyn. Earthq. Eng., 31 (2011), 1123-1131.  doi: 10.1016/j.soildyn.2011.03.013.  Google Scholar

[15]

B. Karami, M. Janghorban and L. Li, On guided wave propagation in fully clamped porous functionally graded nanoplates, Acta Astronaut, 143 (2018), 380-390.  doi: 10.1016/j.actaastro.2017.12.011.  Google Scholar

[16]

M. H. Kargonovin, D. Younesian, D. J. Thompson and C. J. C. Jones, Response of beams on the nonlinear viscoelastic foundations to harmonic moving loads, Comput. Struct., 83 (2005), 1865-1877.   Google Scholar

[17]

M. A. Kazemi-Lari, S. A. Fazelzadeh and E. Ghavanloo, Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation, Physica E, 44 (2012), 1623-1630.  doi: 10.1016/j.physe.2012.04.007.  Google Scholar

[18]

K. Kiani, meshless approach for free transverse vibration of embedded single walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect, Int. J. Mech. Sci., 52 (2010), 1343-1356.  doi: 10.1016/j.ijmecsci.2010.06.010.  Google Scholar

[19]

K. Kiani, Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subject to axial load using nonlocal shear deformable beam theories, Int. J. Mech. Sci., 68 (2013), 16-34.   Google Scholar

[20]

K. Kiani, Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles, Nonlinear Dyn., 76 (2014), 1885-1903.  doi: 10.1007/s11071-014-1255-y.  Google Scholar

[21]

H. L. Lee and W. J. Chang, Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium, Physica E, 41 (2009), 529-532.  doi: 10.1016/j.physe.2008.10.002.  Google Scholar

[22]

P. M. Mathews, Vibrations of a beam on elastic foundation, J. Appl. Math. Mech., 38 (1958), 105-115.  doi: 10.1002/zamm.19580380305.  Google Scholar

[23]

I. Mehdipour, A. Barari, A. Kimiaeifar and G. Domairry, Vibrational analysis of curved single-walled carbon nanotube on a pasternak elastic foundation, Adv. Eng. Softw., 48 (2012), 1-5.  doi: 10.1016/j.advengsoft.2012.01.004.  Google Scholar

[24]

V. K. Melnikov, On the stability of the center of some periodic pertubation, Trans. Moscow Math. Soc., 12 (1963), 1-57.   Google Scholar

[25]

G. Mikhasev, On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium., Z Angew Math. Mech., 94 (2014), 130-141.  doi: 10.1002/zamm.201200140.  Google Scholar

[26]

M. Mir and M. Tahani, Chaotic behavior of nonlocal nanobeam resting on a nonlinear viscoelastic foundation subjected to harmonic excitation, Modares Mech. Eng., 18 (2018), 264-272.   Google Scholar

[27]

T. Murmu and S. C. Pradhan, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E, 41 (2009), 1232-1239.  doi: 10.1016/j.physe.2009.02.004.  Google Scholar

[28]

K. B. Mustapha and Z. W. Zhong, Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two parameter elastic medium, Comput. Mater. Sci., 50 (2010), 742-751.  doi: 10.1016/j.commatsci.2010.10.005.  Google Scholar

[29]

A. H. Nayfeh, Introduction to Pertubation Techniques,, John Wiley, New York, 1981.  Google Scholar

[30]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, 1979.  Google Scholar

[31]

B. R. Nana Nbendjo and P. Woafo, Active control with delay of horseshoes chaos using piezoelectric absorber buckled beam under parametric excitation, Chaos Solitons Fract, 32 (2007), 73-79.  doi: 10.1016/j.chaos.2005.10.070.  Google Scholar

[32]

B. R. Nana Nbendjo and P. Woafo, Modelling of the dynamics of Euler's beam by $ \phi^5 $ potential, Mech. Res. Commun., 38 (2011), 542-545.   Google Scholar

[33]

H. Niknam and M. M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation, Compos. Struct., 119 (2015), 452-462.  doi: 10.1016/j.compstruct.2014.09.023.  Google Scholar

[34]

I. Petras, Fractional nonlinear systems: Modeling, analysis and simulation, Higher Education Press , Beijing, 2011. Google Scholar

[35]

S. C. Pradhan and G. K. Reddy, Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM, Comput. Mater. Sci., 50 (2011), 1052-1056.  doi: 10.1016/j.commatsci.2010.11.001.  Google Scholar

[36]

M. Rafiei, S. R. Mohebpour and F. Daneshmand, Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium, Physica E, 44 (2012), 1372-1379.  doi: 10.1016/j.physe.2012.02.021.  Google Scholar

[37]

G. Romano and R. Barreta, Nonlocal elasticity in nanobeams: The stress-driven integral model, Int. J. Eng. Sci., 115 (2017), 14-27.  doi: 10.1016/j.ijengsci.2017.03.002.  Google Scholar

[38]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems,, Springer Science and Business, New York, 2007.  Google Scholar

[39]

E. J. Sapountzakis and A. Kampitsis, Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads, J. Sound Vib., 330 (2011), 5410-5426.  doi: 10.1016/j.jsv.2011.06.009.  Google Scholar

[40]

Y. Shen, H. Y. Sing and H. Ma, Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives, Int. J. Non-linear Mech., 47 (2012), 975-983.   Google Scholar

[41]

M. S. Siewe and U. H. Hegazy, Homoclinic bifurcation and chaos control in MEMS resonators, Appl. Math. Model., 35 (2011), 5533–5552. doi: 10.1016/j.apm.2011.05.021.  Google Scholar

[42]

N. Togun and S. M. Bagdatli, Nonlinear vibration of a nanobeam on pasternak elastic foundation based on nonlocal euler-bernoulli beam theory, Math. Comput. Appl., 21 (2016), 1-19.  doi: 10.3390/mca21010003.  Google Scholar

[43]

B. L. Wang and K. F. Wang, Vibration analysis of embedded nanotubes using nonlocal continuum theory, Composites Part B Eng., 47 (2013), 96-101.  doi: 10.1016/j.compositesb.2012.10.043.  Google Scholar

[44]

M. H. Yas and N. Samadi, Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation, Int. J. Press Vessels Piping, 98 (2012), 119-128.  doi: 10.1016/j.ijpvp.2012.07.012.  Google Scholar

[45]

X. Zhang and L. Zhou, Melnikovs method for chaos of the nanoplate postulating nonlinear foundation, Appl. Math. Model., 61 (2018), 744-749.  doi: 10.1016/j.apm.2018.05.003.  Google Scholar

show all references

References:
[1]

J. D. Achenbach and C. Sun, Moving load on a flexibly supported timochenko beam, Int. J. Solids Struct., 1 (1965), 353-370.   Google Scholar

[2]

L. M. Anague Tabejieu, B. R. Nana Nbendjo and P. Woafo, On the dynamics of rayleigh beams resting on fractional-order viscoelastic pasternak foundations subjected to moving loads, Chaos Solitons Fract., 93 (2016), 39-47.  doi: 10.1016/j.chaos.2016.10.001.  Google Scholar

[3]

A. A. Andronov and A. Witt, Towards mathematical theory of capture, Archiv. fur Electrotechnik, 24 (1930), 99-110.   Google Scholar

[4]

H. Askari, H. Jamchidifar and B. Fidan, High resolution mass identification using nonlinear vibrations of nanoplates, Measurement, 101 (2017), 166-174.  doi: 10.1016/j.measurement.2017.01.012.  Google Scholar

[5]

M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mech. Res. Commun., 43 (2012), 34-40.  doi: 10.1016/j.mechrescom.2012.02.001.  Google Scholar

[6]

M. Aydogdu and M. Arda, Torsional vibration analysis of double walled carbon nanotubes using nonlocal elasticity, Int. J. Mech. Mater. Des., 12 (2016), 71-84.  doi: 10.1007/s10999-014-9292-8.  Google Scholar

[7]

B. E. Demartini, H. E. Butterfield, J. Moehlis and K. L. Turner, Chaos in microelectromechanical oscillator governed by the nonlinear Mathieu equation, J. Microelec. Syst., 16 (2007), 1314-1323.  doi: 10.1109/JMEMS.2007.906757.  Google Scholar

[8]

H. Ding, Q. L. Chen and S. P. Yang, Convergence of garlekin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 331 (2012), 2426-2442.   Google Scholar

[9]

A. C. Eringen, On differential equations of nonlocal elasticity and solution of screw dislocation and surface waves, J. Appl. Phys., 54 (1983), 4703-4710.  doi: 10.1063/1.332803.  Google Scholar

[10]

A. C. Eringen, Nonlocal Continuum Fields Theories, Springer, USA, 2002.  Google Scholar

[11]

E. Ghavanloo, F. Daneshmand and M. Rafiei, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E, 42 (2010), 2218-2224.   Google Scholar

[12]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields, Springer-Verlag, USA, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[13]

Y. Haitao and Y. Yuan, Analytical solution for an infinite euler-bernoulli beam on a visco-elastic foundation subjected to arbitrary dynamic loads, J. Eng. Mech., 140 (2014), 542-551.   Google Scholar

[14]

Z. Hryniewicz, Dynamics of rayleigh beam on nonlinear foundation due to moving load using adomian decomposition and coiflet expansion, Soil Dyn. Earthq. Eng., 31 (2011), 1123-1131.  doi: 10.1016/j.soildyn.2011.03.013.  Google Scholar

[15]

B. Karami, M. Janghorban and L. Li, On guided wave propagation in fully clamped porous functionally graded nanoplates, Acta Astronaut, 143 (2018), 380-390.  doi: 10.1016/j.actaastro.2017.12.011.  Google Scholar

[16]

M. H. Kargonovin, D. Younesian, D. J. Thompson and C. J. C. Jones, Response of beams on the nonlinear viscoelastic foundations to harmonic moving loads, Comput. Struct., 83 (2005), 1865-1877.   Google Scholar

[17]

M. A. Kazemi-Lari, S. A. Fazelzadeh and E. Ghavanloo, Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation, Physica E, 44 (2012), 1623-1630.  doi: 10.1016/j.physe.2012.04.007.  Google Scholar

[18]

K. Kiani, meshless approach for free transverse vibration of embedded single walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect, Int. J. Mech. Sci., 52 (2010), 1343-1356.  doi: 10.1016/j.ijmecsci.2010.06.010.  Google Scholar

[19]

K. Kiani, Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subject to axial load using nonlocal shear deformable beam theories, Int. J. Mech. Sci., 68 (2013), 16-34.   Google Scholar

[20]

K. Kiani, Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles, Nonlinear Dyn., 76 (2014), 1885-1903.  doi: 10.1007/s11071-014-1255-y.  Google Scholar

[21]

H. L. Lee and W. J. Chang, Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium, Physica E, 41 (2009), 529-532.  doi: 10.1016/j.physe.2008.10.002.  Google Scholar

[22]

P. M. Mathews, Vibrations of a beam on elastic foundation, J. Appl. Math. Mech., 38 (1958), 105-115.  doi: 10.1002/zamm.19580380305.  Google Scholar

[23]

I. Mehdipour, A. Barari, A. Kimiaeifar and G. Domairry, Vibrational analysis of curved single-walled carbon nanotube on a pasternak elastic foundation, Adv. Eng. Softw., 48 (2012), 1-5.  doi: 10.1016/j.advengsoft.2012.01.004.  Google Scholar

[24]

V. K. Melnikov, On the stability of the center of some periodic pertubation, Trans. Moscow Math. Soc., 12 (1963), 1-57.   Google Scholar

[25]

G. Mikhasev, On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium., Z Angew Math. Mech., 94 (2014), 130-141.  doi: 10.1002/zamm.201200140.  Google Scholar

[26]

M. Mir and M. Tahani, Chaotic behavior of nonlocal nanobeam resting on a nonlinear viscoelastic foundation subjected to harmonic excitation, Modares Mech. Eng., 18 (2018), 264-272.   Google Scholar

[27]

T. Murmu and S. C. Pradhan, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E, 41 (2009), 1232-1239.  doi: 10.1016/j.physe.2009.02.004.  Google Scholar

[28]

K. B. Mustapha and Z. W. Zhong, Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two parameter elastic medium, Comput. Mater. Sci., 50 (2010), 742-751.  doi: 10.1016/j.commatsci.2010.10.005.  Google Scholar

[29]

A. H. Nayfeh, Introduction to Pertubation Techniques,, John Wiley, New York, 1981.  Google Scholar

[30]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, 1979.  Google Scholar

[31]

B. R. Nana Nbendjo and P. Woafo, Active control with delay of horseshoes chaos using piezoelectric absorber buckled beam under parametric excitation, Chaos Solitons Fract, 32 (2007), 73-79.  doi: 10.1016/j.chaos.2005.10.070.  Google Scholar

[32]

B. R. Nana Nbendjo and P. Woafo, Modelling of the dynamics of Euler's beam by $ \phi^5 $ potential, Mech. Res. Commun., 38 (2011), 542-545.   Google Scholar

[33]

H. Niknam and M. M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation, Compos. Struct., 119 (2015), 452-462.  doi: 10.1016/j.compstruct.2014.09.023.  Google Scholar

[34]

I. Petras, Fractional nonlinear systems: Modeling, analysis and simulation, Higher Education Press , Beijing, 2011. Google Scholar

[35]

S. C. Pradhan and G. K. Reddy, Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM, Comput. Mater. Sci., 50 (2011), 1052-1056.  doi: 10.1016/j.commatsci.2010.11.001.  Google Scholar

[36]

M. Rafiei, S. R. Mohebpour and F. Daneshmand, Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium, Physica E, 44 (2012), 1372-1379.  doi: 10.1016/j.physe.2012.02.021.  Google Scholar

[37]

G. Romano and R. Barreta, Nonlocal elasticity in nanobeams: The stress-driven integral model, Int. J. Eng. Sci., 115 (2017), 14-27.  doi: 10.1016/j.ijengsci.2017.03.002.  Google Scholar

[38]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems,, Springer Science and Business, New York, 2007.  Google Scholar

[39]

E. J. Sapountzakis and A. Kampitsis, Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads, J. Sound Vib., 330 (2011), 5410-5426.  doi: 10.1016/j.jsv.2011.06.009.  Google Scholar

[40]

Y. Shen, H. Y. Sing and H. Ma, Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives, Int. J. Non-linear Mech., 47 (2012), 975-983.   Google Scholar

[41]

M. S. Siewe and U. H. Hegazy, Homoclinic bifurcation and chaos control in MEMS resonators, Appl. Math. Model., 35 (2011), 5533–5552. doi: 10.1016/j.apm.2011.05.021.  Google Scholar

[42]

N. Togun and S. M. Bagdatli, Nonlinear vibration of a nanobeam on pasternak elastic foundation based on nonlocal euler-bernoulli beam theory, Math. Comput. Appl., 21 (2016), 1-19.  doi: 10.3390/mca21010003.  Google Scholar

[43]

B. L. Wang and K. F. Wang, Vibration analysis of embedded nanotubes using nonlocal continuum theory, Composites Part B Eng., 47 (2013), 96-101.  doi: 10.1016/j.compositesb.2012.10.043.  Google Scholar

[44]

M. H. Yas and N. Samadi, Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation, Int. J. Press Vessels Piping, 98 (2012), 119-128.  doi: 10.1016/j.ijpvp.2012.07.012.  Google Scholar

[45]

X. Zhang and L. Zhou, Melnikovs method for chaos of the nanoplate postulating nonlinear foundation, Appl. Math. Model., 61 (2018), 744-749.  doi: 10.1016/j.apm.2018.05.003.  Google Scholar

Figure 1.  Boundary conditions for different beam supports. (a) Simple-simple case and (b) clamped-clamped case.
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Figure 2.  First three vibration mode shapes for the simple-simple case boundary condition.
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Figure 3.  First three vibration modes shapes for the clamped-clamped case boundary condition.
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Figure 4.  Criticals curves separting the chaotic and non-chaotic regions.
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Figure 5.  Criticals surfaces separating the chaotic and non-chaotic regions.
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Figure 6.  Critical curve separating the chaotic and non-chaotic regions.
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Figure 7.  Regular motion when $ f = 80 $: (a) the phase portrait (b) the Poincare section for the initial conditions $ (q_0,p_0) = (0.1,0.1) $.
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Figure 8.  Regular motion when $ f = 80 $: (c) the waveform $ (t, q(t)) $ (d)the waveform $ (t, p(t)) $ for the initial conditions $ (q_0,p_0) = (0.1,0.1) $.
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Figure 9.  Chaotic motion when $ f = 400 $: (a) the phase portrait (b) the Poincare section for the initial conditions $ (q_0,p_0) = (0.1,0.1) $.
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Figure 10.  Chaotic motion when $ f = 400 $: (c) the waveform $ (t, q(t)) $ (d)the waveform $ (t, p(t)) $ for the initial conditions $ (q_0,p_0) = (0.1,0.1) $.
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