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Volumn 65, Issue 3, 2011, Pages 247-254

Complex order van der Pol oscillator

Author keywords

Complex order derivative; Dynamical behavior; Van der Pol oscillator

Indexed keywords

COMPLEX DERIVATIVE; COMPLEX-ORDER DERIVATIVES; DYNAMICAL BEHAVIORS; HIGH-DIMENSIONAL; PARAMETER SPACES; PERIODIC SOLUTION; TWO OSCILLATORS; VAN DER POL OSCILLATOR;

EID: 80051788613     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-010-9886-0     Document Type: Article
Times cited : (63)

References (46)
  • 1
    • 34247477202 scopus 로고    scopus 로고
    • On fractional order differential equations model for nonlocal epidemics
    • DOI 10.1016/j.physa.2007.01.010, PII S0378437107000623
    • E. Ahmed A.S. Elgazzar 2007 On fractional order differential equations model for nonlocal epidemics Phys. A: Stat. Mech. Appl. 379 2 607 614 10.1016/j.physa.2007.01.010 (Pubitemid 46654665)
    • (2007) Physica A: Statistical Mechanics and its Applications , vol.379 , Issue.2 , pp. 607-614
    • Ahmed, E.1    Elgazzar, A.S.2
  • 2
    • 2542604518 scopus 로고    scopus 로고
    • Acoustic and seismic signals of heavy military vehicles for co-operative verification
    • 10.1016/j.jsv.2003.05.002
    • J. Altmann 2004 Acoustic and seismic signals of heavy military vehicles for co-operative verification J. Sound Vib. 273 4-5 713 740 10.1016/j.jsv.2003. 05.002
    • (2004) J. Sound Vib. , vol.273 , Issue.45 , pp. 713-740
    • Altmann, J.1
  • 3
    • 37349026876 scopus 로고    scopus 로고
    • Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis
    • DOI 10.1016/j.physb.2007.08.034, PII S0921452607005753
    • B. Appelbe D. Rachinskii A. Zhezherun 2008 Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis Phys. B: Condens. Matter 403 2-3 301 304 10.1016/j.physb.2007.08.034 (Pubitemid 350297746)
    • (2008) Physica B: Condensed Matter , vol.403 , Issue.2-3 , pp. 301-304
    • Appelbe, B.1    Rachinskii, D.2    Zhezherun, A.3
  • 4
    • 77954348949 scopus 로고    scopus 로고
    • Analysis of a fractional order Van der Pol-like oscilattor via describing function method
    • 10.1007/s11071-009-9647-0 1204.70018
    • M. Attari M. Haeri M.S. Tavazoei 2010 Analysis of a fractional order Van der Pol-like oscilattor via describing function method Nonlinear Dyn. 10.1007/s11071-009-9647-0 1204.70018
    • (2010) Nonlinear Dyn.
    • Attari, M.1    Haeri, M.2    Tavazoei, M.S.3
  • 5
    • 58749115373 scopus 로고    scopus 로고
    • About fractional quantization and fractional variational principles
    • 10.1016/j.cnsns.2008.10.002
    • D. Baleanu 2009 About fractional quantization and fractional variational principles Commun. Nonlinear Sci. Numer. Simul. 14 6 2520 2523 10.1016/j.cnsns.2008.10.002
    • (2009) Commun. Nonlinear Sci. Numer. Simul. , vol.14 , Issue.6 , pp. 2520-2523
    • Baleanu, D.1
  • 6
    • 52349119862 scopus 로고    scopus 로고
    • Discretization of complex-order algorithms for control applications
    • 2463068 10.1177/1077546307087445
    • R.S. Barbosa J.A. Tenreiro Machado M.F. Silva 2008 Discretization of complex-order algorithms for control applications J. Vib. Control 14 9-10 1349 1361 2463068 10.1177/1077546307087445
    • (2008) J. Vib. Control , vol.14 , Issue.910 , pp. 1349-1361
    • Barbosa, R.S.1    Tenreiro MacHado, J.A.2    Silva, M.F.3
  • 7
    • 34748856163 scopus 로고    scopus 로고
    • Analysis of the Van der Pol oscillator containing derivatives of fractional order
    • DOI 10.1177/1077546307077463
    • R.S. Barbosa J.A. Tenreiro Machado B.M. Vinagre A.J. Calderón 2007 Analysis of the Van der Pol oscillator containing derivatives of fractional order J. Vib. Control 13 9-10 1291 1301 1158.70009 10.1177/1077546307077463 (Pubitemid 47482814)
    • (2007) JVC/Journal of Vibration and Control , vol.13 , Issue.9-10 , pp. 1291-1301
    • Barbosa, R.S.1    MacHado, J.A.T.2    Vinagre, B.M.3    Caldern, A.J.4
  • 9
    • 34548530905 scopus 로고    scopus 로고
    • Chaotic dynamics of the fractionally damped van der Pol equation
    • DOI 10.1016/j.chaos.2006.05.010, PII S0960077906004607
    • J.-H. Chen W.-C. Chen 2008 Chaotic dynamics of the fractionally damped van der Pol equation Chaos Solitons Fractals 35 188 198 10.1016/j.chaos.2006.05. 010 (Pubitemid 47385209)
    • (2008) Chaos, Solitons and Fractals , vol.35 , Issue.1 , pp. 188-198
    • Chen, J.-H.1    Chen, W.-C.2
  • 11
    • 24744435151 scopus 로고    scopus 로고
    • On quadratic integral equation of fractional orders
    • DOI 10.1016/j.jmaa.2005.02.012, PII S0022247X05001071
    • M.A. Darwish 2005 On quadratic integral equation of fractional orders J. Math. Anal. Appl. 311 1 112 119 2165466 1080.45004 10.1016/j.jmaa.2005.02.012 (Pubitemid 41288764)
    • (2005) Journal of Mathematical Analysis and Applications , vol.311 , Issue.1 , pp. 112-119
    • Darwish, M.A.1
  • 12
    • 38649105169 scopus 로고    scopus 로고
    • Synchronization of van der Pol oscillator and Chen chaotic dynamical system
    • DOI 10.1016/j.chaos.2006.08.039, PII S0960077906008484
    • E.M. Elabbasy M.M. El-Dessoky 2008 Synchronization of van der Pol oscillator and Chen chaotic dynamical system Chaos Solitons Fractals 36 5 1425 1435 2388980 1148.37023 10.1016/j.chaos.2006.08.039 (Pubitemid 351172796)
    • (2008) Chaos, Solitons and Fractals , vol.36 , Issue.5 , pp. 1425-1435
    • Elabbasy, E.M.1    El-Dessoky, M.M.2
  • 13
    • 33846839489 scopus 로고    scopus 로고
    • Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor
    • DOI 10.1016/j.chaos.2005.12.040, PII S0960077906000336
    • Z.-M. Ge W.-R. Jhuang 2007 Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor Chaos Solitons Fractals 33 1 270 289 1152.34355 10.1016/j.chaos.2005.12.040 (Pubitemid 46209796)
    • (2007) Chaos, Solitons and Fractals , vol.33 , Issue.1 , pp. 270-289
    • Ge, Z.-M.1    Jhuang, W.-R.2
  • 14
    • 33947178355 scopus 로고    scopus 로고
    • Chaos in a generalized van der Pol system and in its fractional order system
    • DOI 10.1016/j.chaos.2006.03.028, PII S0960077906002323
    • Z.-M. Ge M.-Y. Hsu 2007 Chaos in a generalized van der Pol system and in its fractional order system Chaos Solitons Fractals 33 1711 1745 1131.65100 10.1016/j.chaos.2006.03.028 (Pubitemid 46413055)
    • (2007) Chaos, Solitons and Fractals , vol.33 , Issue.5 , pp. 1711-1745
    • Ge, Z.-M.1    Hsu, M.-Y.2
  • 15
    • 37349088581 scopus 로고    scopus 로고
    • Chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems
    • DOI 10.1016/j.chaos.2006.06.093, PII S0960077906006813
    • Z.-M. Ge M.-Y. Hsu 2008 Chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems Chaos Solitons Fractals 36 592 604 1142.37027 10.1016/j.chaos.2006.06.093 (Pubitemid 350297922)
    • (2008) Chaos, Solitons and Fractals , vol.36 , Issue.3 , pp. 592-604
    • Ge, Z.-M.1    Hsu, M.-Y.2
  • 16
    • 33846377845 scopus 로고    scopus 로고
    • Chaos in a modified van der Pol system and in its fractional order systems
    • DOI 10.1016/j.chaos.2005.12.024, PII S0960077905012191
    • Z.-M. Ge A.-R. Zhang 2007 Chaos in a modified van der Pol system and in its fractional order systems Chaos Solitons Fractals 32 1791 1822 10.1016/j.chaos.2005.12.024 (Pubitemid 46136969)
    • (2007) Chaos, Solitons and Fractals , vol.32 , Issue.5 , pp. 1791-1822
    • Ge, Z.-M.1    Zhang, A.-R.2
  • 20
    • 53349102813 scopus 로고
    • Impulses and physiological states in theoretical models of nerve membrane
    • 10.1016/S0006-3495(61)86902-6
    • R. FitzHugh 1961 Impulses and physiological states in theoretical models of nerve membrane Biophys. J. 1 445 466 10.1016/S0006-3495(61)86902-6
    • (1961) Biophys. J. , vol.1 , pp. 445-466
    • Fitzhugh, R.1
  • 21
    • 51249192747 scopus 로고
    • Mathematical models of threshold phenomena in the nerve membrane
    • 10.1007/BF02477753
    • R. FitzHugh 1955 Mathematical models of threshold phenomena in the nerve membrane Bull. Math. Biophys. 17 257 278 10.1007/BF02477753
    • (1955) Bull. Math. Biophys. , vol.17 , pp. 257-278
    • Fitzhugh, R.1
  • 22
    • 0014359129 scopus 로고
    • Motion picture of nerve impulse propagation using computer animation
    • R. FitzHugh 1968 Motion picture of nerve impulse propagation using computer animation J. Appl. Physiol. 25 628 630
    • (1968) J. Appl. Physiol. , vol.25 , pp. 628-630
    • Fitzhugh, R.1
  • 23
    • 0002847893 scopus 로고    scopus 로고
    • Fractional calculus, integral and differential equations of fractional order
    • A. Carpenteri F. Mainardi (eds). Springer Berlin
    • Gorenflo, R., Mainardi, F.: Fractional calculus, integral and differential equations of fractional order. In: Carpenteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin (1997)
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics
    • Gorenflo, R.1    Mainardi, F.2
  • 24
    • 0036650850 scopus 로고    scopus 로고
    • Time fractional diffusion: A discrete random walk approach
    • DOI 10.1023/A:1016547232119, Fractional Order Calculus and Its Applications
    • R. Gorenflo F. Mainardi D. Moretti P. Pardisi 2002 Time-fractional diffusion: a discrete random walk approach Nonlinear Dyn. 29 1-4 129 143 1009.82016 10.1023/A:1016547232119 (Pubitemid 34945395)
    • (2002) Nonlinear Dynamics , vol.29 , Issue.1-4 , pp. 129-143
    • Gorenflo, R.1    Mainardi, F.2    Moretti, D.3    Paradisi, P.4
  • 25
    • 77956684069 scopus 로고    scopus 로고
    • Theory and Applications of Fractional Differential Equations
    • Elsevier Amsterdam 1092.45003 10.1016/S0304-0208(06)80001-0
    • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
    • (2006) North Holland Mathematics Studies , vol.204
    • Kilbas, A.A.1    Srivastava, H.M.2    Trujillo, J.J.3
  • 26
    • 0030651602 scopus 로고    scopus 로고
    • Analysis and design of fractional-order digital control systems
    • 0875.93154
    • J.T. Machado 1997 Analysis and design of fractional-order digital control systems J. Syst. Anal. Model. Simul. 27 107 122 0875.93154
    • (1997) J. Syst. Anal. Model. Simul. , vol.27 , pp. 107-122
    • MacHado, J.T.1
  • 27
    • 0030464353 scopus 로고    scopus 로고
    • Fractional relaxation-oscillation and fractional diffusion-wave phenomena
    • DOI 10.1016/0960-0779(95)00125-5
    • F. Mainardi 1996 Fractional relaxation-oscillation and fractional diffusion-wave phenomena Chaos Solitons Fractals 7 1461 1477 1409912 1080.26505 10.1016/0960-0779(95)00125-5 (Pubitemid 126377337)
    • (1996) Chaos, Solitons and Fractals , vol.7 , Issue.9 , pp. 1461-1477
    • Mainardi, F.1
  • 29
    • 33646878106 scopus 로고    scopus 로고
    • Analytical approach to linear fractional partial differential equations arising in fluid mechanics
    • DOI 10.1016/j.physleta.2006.02.048, PII S0375960106003148
    • S. Momani Z. Odibat 2006 Analytical approach to linear fractional partial differential equations arising in fluid mechanics Phys. Lett. A 355 4-5 271 279 05675858 10.1016/j.physleta.2006.02.048 (Pubitemid 43783194)
    • (2006) Physics Letters, Section A: General, Atomic and Solid State Physics , vol.355 , Issue.4-5 , pp. 271-279
    • Momani, S.1    Odibat, Z.2
  • 30
    • 6344238035 scopus 로고
    • An active pulse transmission line simulating nerve axon
    • 10.1109/JRPROC.1962.288235
    • J. Nagumo S. Arimoto S. Yoshizawa 1962 An active pulse transmission line simulating nerve axon Proc. IRE 50 2061 2070 10.1109/JRPROC.1962.288235
    • (1962) Proc. IRE , vol.50 , pp. 2061-2070
    • Nagumo, J.1    Arimoto, S.2    Yoshizawa, S.3
  • 31
    • 33745841179 scopus 로고    scopus 로고
    • The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness?
    • DOI 10.1016/j.sigpro.2006.02.003, PII S0165168406000405, Fractional Calculus Applications in Signals and Systems
    • R. Nigmatullin 2006 The statistics of the fractional moments: is there any chance to "read quantitatively" any randomness? Signal Process. 86 10 2529 2547 1172.62303 10.1016/j.sigpro.2006.02.003 (Pubitemid 44026628)
    • (2006) Signal Processing , vol.86 , Issue.10 , pp. 2529-2547
    • Nigmatullin, R.R.1
  • 32
    • 77951983849 scopus 로고    scopus 로고
    • Is it possible to derive newtonian equations of motion with memory?
    • 2608551 1190.83017 10.1007/s10773-010-0249-x
    • R. Nigmatullin D. Baleanu 2010 Is it possible to derive newtonian equations of motion with memory? Int. J. Theor. Phys. 49 4 701 708 2608551 1190.83017 10.1007/s10773-010-0249-x
    • (2010) Int. J. Theor. Phys. , vol.49 , Issue.4 , pp. 701-708
    • Nigmatullin, R.1    Baleanu, D.2
  • 33
    • 70350567183 scopus 로고    scopus 로고
    • Fractional differential equations in electrochemistry
    • 1177.78041 10.1016/j.advengsoft.2008.12.012
    • K.B. Oldham 2010 Fractional differential equations in electrochemistry Adv. Eng. Softw. 41 1 9 12 1177.78041 10.1016/j.advengsoft.2008.12.012
    • (2010) Adv. Eng. Softw. , vol.41 , Issue.1 , pp. 9-12
    • Oldham, K.B.1
  • 37
    • 79955117342 scopus 로고    scopus 로고
    • Fractional central pattern generators for bipedal locomotion
    • 2736974 1204.92007 10.1007/s11071-010-9696-4
    • C.M.A. Pinto J.A. Tenreiro Machado 2010 Fractional central pattern generators for bipedal locomotion Nonlinear Dyn. 62 1-2 27 37 2736974 1204.92007 10.1007/s11071-010-9696-4
    • (2010) Nonlinear Dyn. , vol.62 , Issue.12 , pp. 27-37
    • Pinto, C.M.A.1    Tenreiro MacHado, J.A.2
  • 39
    • 0032653954 scopus 로고    scopus 로고
    • Fractional-order systems and PID-controllers
    • 1666937 1056.93542 10.1109/9.739144
    • I. Podlubny 1999 Fractional-order systems and PID-controllers IEEE Trans. Automat. Control 44 1 208 213 1666937 1056.93542 10.1109/9.739144
    • (1999) IEEE Trans. Automat. Control , vol.44 , Issue.1 , pp. 208-213
    • Podlubny, I.1
  • 40
    • 66249115324 scopus 로고    scopus 로고
    • More details on analysis of fractional-order Van der Pol oscillator
    • 2528195 10.1177/1077546308096101
    • M.S. Tavazoei M. Haeri M. Attari S. Bolouki M. Siami 2009 More details on analysis of fractional-order Van der Pol oscillator J. Vib. Control 15 6 803 819 2528195 10.1177/1077546308096101
    • (2009) J. Vib. Control , vol.15 , Issue.6 , pp. 803-819
    • Tavazoei, M.S.1    Haeri, M.2    Attari, M.3    Bolouki, S.4    Siami, M.5
  • 41
    • 0001805106 scopus 로고    scopus 로고
    • Discrete-time fractional-order controllers
    • 1815334 1111.93307
    • J. Tenreiro Machado 2001 Discrete-time fractional-order controllers J. Fract. Calc. Appl. Anal. 4 47 66 1815334 1111.93307
    • (2001) J. Fract. Calc. Appl. Anal. , vol.4 , pp. 47-66
    • Tenreiro MacHado, J.1
  • 42
    • 63449116783 scopus 로고    scopus 로고
    • Fractional derivatives: Probability interpretation and frequency response of rational approximations
    • 10.1016/j.cnsns.2009.02.004
    • J.A. Tenreiro Machado 2009 Fractional derivatives: probability interpretation and frequency response of rational approximations Commun. Nonlinear Sci. Numer. Simul. 14 9-10 3492 3497 10.1016/j.cnsns.2009.02.004
    • (2009) Commun. Nonlinear Sci. Numer. Simul. , vol.14 , Issue.910 , pp. 3492-3497
    • Tenreiro MacHado, J.A.1
  • 44
    • 0000873906 scopus 로고
    • On relaxation-oscillations
    • B. van der Pol 1926 On relaxation-oscillations Philos. Mag. 7 2 978 992
    • (1926) Philos. Mag. , vol.7 , Issue.2 , pp. 978-992
    • Van Der Pol, B.1
  • 45
    • 0001874281 scopus 로고
    • Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode)
    • B. van der Pol 1927 Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode) Philos. Mag. 3 65 80
    • (1927) Philos. Mag. , vol.3 , pp. 65-80
    • Van Der Pol, B.1
  • 46
    • 30244449696 scopus 로고
    • Frequency demultiplication
    • 10.1038/120363a0
    • B. van der Pol J. van der Mark 1927 Frequency demultiplication Nature 120 363 364 10.1038/120363a0
    • (1927) Nature , vol.120 , pp. 363-364
    • Van Der Pol, B.1    Van Der Mark, J.2


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