Discontinuity, Nonlinearity, and Complexity
Evolution Towards the Steady State in a Hopf Bifurcation: A Scaling Investigation
Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 6779  DOI:10.5890/DNC.2018.03.006
Vinícius B. da Silva, Edson D. Leonel
Departamento de F´ısica, UNESP  Univ Estadual Paulista, Av. 24A, 1515  Bela Vista, 13506900, Rio Claro, SP, BRA
Download Full Text PDF
Abstract
Some scaling properties describing the convergence for the steady state in a Hopf bifurcation are discussed. Two different procedures are considered in the investigation: (i) a phenomenological description obtained from time series coming from the numerical integration of the system, leading to a set of critical exponents and hence to scaling laws; (ii) a direct solution of the differential equations, which is possible only in the normal form. At the bifurcation, the convergence to the stationary state obeys a generalized
and homogeneous function. For short time, the dynamics giving by the distance from the fixed point is mostly constant when a critical time is reached hence changing the dynamics to a convergence for the steady state given by a power law. Both the size of the constant plateau and the characteristic crossover time depend on the initial distance from the fixed point. Near the bifurcation, the convergence is described by an exponential decay with a relaxation time given by a power law.
Acknowledgments
V.B.S. thanks to FAPESP (2015/231420). E.D.L acknowledges support from FAPESP (2012/236885) and CNPq (303707/20151) Brazilian agencies.
References

[1]  Strogatz, S.H. (2015), Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering, Westview Press. 

[2]  Bashkirtseva, I. and Ryashko, L. (2011), Sensitivity analysis of stochastic attractors and noiseinduced transitions for population model with Allee effect, Chaos, 21, 047514. 

[3]  Strogatz, S.H. (2000), From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143, 1. 

[4]  Georgiou, I.T. and Romeo, F. (2015),Multiphysics dynamics of a mechanical oscillator coupled to an electromagnetic circuit, Int. J. Nonlinear. Mech., 70, 153. 

[5]  Gardine, L., FournierPrunaret, D., and Charge, P. (2011), Border collision bifurcations in a Twodimensional PWS map from a simple circuit, Chaos, 21, 023106. 

[6]  Inarrea, M., Palacian, J.F., and Pascual, A.I. (2011), Bifurcations of dividing surfaces in chemical reactions, J. Chem. Phys., 135, 014110. 

[7]  Bakes, D., Schreiberova, L., and Schreiber, I. (2008),Mixedmode oscillations in a homogeneous pHoscillatory chemical reaction system, Chaos 18, 015102. 

[8]  Guckenheimer, J. (2008), Return maps of folded nodes and folded saddlenodes, Chaos, 18, 015108. 

[9]  Philominathan, P., Santhiah, M., Mohamed, IR., Murali, K., and Rajasekar, S. (2011), Chaotic dynamics of a simple parametrically driven dissipative circuit, Int. J. Bifurcations and Chaos, 21, 1927. 

[10]  Virte, M., Panajotov, K., and Thienpont, H. (2013), Deterministic polarization chaos from a laser diode, Nat. Photonics, 7, 60. 

[11]  Doedel, EJ. and PandoL, CL. (2011), Isolas of periodic passive Qswitching selfpulsations in the threelevel:twolevel model for a laser with a saturable absorber, Phys. Rev. E., 84, 056207. 

[12]  Cavalcante, H.L.D.S. and LeiteRios, J.R. (2008), Experimental bifurcations and homoclinic chaos in a laser with a saturable absorber, Chaos, 18, 023107. 

[13]  Guo, Y. and Luo, A.C.J. (2012), Parametric analysis of bifurcation and chaos in a periodically driven horizontal impact, Int. J. Bifurc. Chaos, 22, 1250268. 

[14]  Luo, A.C.J. and O’Connor, D. (2009), Periodic motions and chaos with impacting chatter with stick in a gear transmission system, Int. J. Bifurc. Chaos, 19, 2093. 

[15]  Yang, J.H., Sanjuan, M.A.F., Liu, H.G., and Cheng G. (2015), Bifurcation Transition and Nonlinear Response in a FractionalOrder System, J. Comput. Nonlinear. Dyn., 10, 061017. 

[16]  Grebogi, C., Ott, E., and Yorke, J.A. (1983), Chaotic attractors in crisis, Phys. Rev. Lett., 48, 1507. 

[17]  Grebogi, C., Ott, E., and Yorke, J.A. (1983), Crises, sudden changes in chaotic attractors, and trasient chaos, Physica D, 7, 181. 

[18]  Leonel, E.D. andMcClintock, P.V.E. (2005), A crisis in the dissipative Fermi accelerator model, J. Phys. A:Math. Gen, 38, L425. 

[19]  Oliveira, D.F.M., Leonel, E.D., and Robnik, M. (2011), Boundary crisis and transient in a dissipative relativistic standard map, Phys. Lett. A, 375, 3365. 

[20]  Helrich, C.S. (2009), Modern thermodynamics with statistical mechanics, Heidelberg: SpringerVerlag. 

[21]  Reif, F. (1965), Fundamentals of statistical and thermal physics, New York: McGrawHill. 

[22]  Pottier, N. (2010), Nonequilibrium statistical physics, linear irreversible processes, Oxford: Oxford University Press. 

[23]  Cardy, J. (1996), Scaling and renormalization in statistical physics, Cambridge: Cambridge University Press. 

[24]  Kadanoff, L.P. (1999), Statistical physics: statics, dynamics and renormalization, Singapore: World Scientific. 

[25]  Leonel, E.D., da Silva, J.K.I., and Kamphorst, S.O. (2002), Relaxation and transients in a timedependent logistic map, Int. J. Bifurc. Chaos, 12, 1667. 

[26]  Teixeira, R.M.N., Rando, D.S., Geraldo, F.C., Costa Filho, R.N., Oliveira, J.A., and Leonel, E.D. (2015), Convergence towards asymptotic state 1D mappings: A scaling investigation, Phys. Lett. A, 379, 1246 

[27]  Teixeira, R.M.N., Rando, D.S., Geraldo, F.C., Costa Filho, R.N., Oliveira, J.A., and Leonel, E.D. (2015), Addendum to: “Convergence towards asymptotic state 1D mappings: A scaling investigation”, Phys. Lett. A, 379, 1246, 1796. 

[28]  Leonel, E.D. (2016),Defining universality classes for three different local bifurcations, Commun. Nonlinear Sci. Numer. Simulat., 39, 520. 

[29]  Kuznetsov, Y.A. (1998), Elements of applied bifurcation theory, New York: Springer Science & Business Media, 112. 