Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Slow passage through multiple bifurcation points
Pages: 95 - 107, Issue 1, January 2013

doi:10.3934/dcdsb.2013.18.95      Abstract        References        Full text (546.7K)           Related Articles

Younghae Do - Department of Mathematics, Kyungpook National University, Daegu, 702-701, South Korea (email)
Juan M. Lopez - School of Mathematical and Statistical Sciences, Arizona State Univ., Tempe AZ, 85287, United States (email)

1 S. M. Baer, T. Erneux and J. Rinzel, The slow passage through a Hopf bifurcation: delay, memory effects, and resonance, SIAM J. Appl. Math., 49 (1989), 55-71.       
2 M. Marhl, T. Haberichter, M. Brumen and R. Heinrich, Complex calcium oscillations and the role of mitochondria and cytosolic protens, BioSystems, 57 (2000), 75-86.
3 M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons & Fractals, 27 (2006), 395-403.
4 P. Strizhak and M. Menzinger, Slow passage through a supercritical Hopf bifurcation: Time-delayed response in the Belousov-Zhabotinsky reaction in a batch reactor, J. Chem. Phys., 105 (1996), 10905-10910.
5 Y. Park, Y. Do, and J. M. Lopez, Slow passage through resonance, Phys. Rev., E, 84 (2011), 056604.
6 K. Park, G. L. Crawford and R. J. Donnelly, Determination of transition in Couette flow in finite geometries, Phys. Rev. Lett., 47 (1981), 1448.
7 J. E. Hart and S. Kittelman, Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder, Phys. Fluids, 8 (1996), 692-696.       
8 J. von Stamm, U. Gerdts, T. Buzug and G. Pfister, Symmetry breaking and period doubling on a torus in the VLFregime in Taylor-Couette flow , Phys. Rev., E, 54 (1996), 4938.
9 C. S. Dutcher and S. J. Muller, Spatio-temporal mode dynamics and higher order transitions in high aspect ratio Newtonian Taylor-Couette flows, J. Fluid Mech., 641 (2009), 85-113.
10 J. Su, Persistent unstable periodic motions, I, J. Math. Analysis and Applications, 198 (1996), 796-825.       
11 J. Su, Persistent unstable periodic motions, II, J. Math. Analysis and Applications, 199 (1996), 88-119.       
12 L. Holden and T. Erneux, Slow passage through a Hopf-bifurcation-From oscillatios to steady-state solutions, SIAM J. Appl. Math., 53 (1993), 1045-1058.       
13 S. M. Baer and E. M. Gaekel, Slow acceleration and deacceleration through a Hopf bifurcation: Power ramps, target nucleation, and elliptic bursting, Phys. Rev., E, 78 (2009), 036205.       
14 R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), 69-106.       
15 V. Booth, T. W. Carr and T. Erneux, Near-threshold bursting is delayed by a slow passage near a limit point, SIAM J. Appl. Math., 57 (1997), 1406-1420.       
16 L. Ng, R. Rand and M. O'Neil, Slow passage through resonance in Mathieu's equation, J. Vibration & Control, 9 (2003), 685-707.       
17 J. P. Denier and R. Grimshaw, Slowly-varying bifurcation theory in dissipative systems, J. Austral. Math. Soc. Ser. B, 31 (1990), 301-318.       
18 P. Hall, On the nonlinear stability of slowly varying time-dependent viscous flows, J., Fluid Mech., 126 (1983), 357-368.
19 P. Yu, Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales, Nonlinear Dyn., 27 (2002), 19-53.       
20 Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Springer, third edition, 2004.       
21 F. Marques, J. M. Lopez and J. Shen, Mode interactions in an enclosed swirling flow: A double Hopf bifurcation between azimuthal wavenumbers 0 and 2, J. Fluid Mech., 455 (2002), 263-281.       
22 J. M. Lopez, J. E. Hart, F. Marques, S. Kittelman and J. Shen, Instability and mode interactions in a differentially-driven rotating cylinder, J. Fluid Mech., 462 (2002), 383-409.       
23 J. M. Lopez and F. Marques, Small aspect ratio Taylor-Couette flow: On set of a very-low-frequency three-torus state, Phys. Rev. E, 68 (2003), 036302.       
24 F. Marques, A. Y. Gelfgat and J. M. Lopez, Tangent double Hopf bifurcation in a differentially rotating cylinder flow, Phys. Rev. E, 68 (2003), 016310.       
25 J. M. Lopez and F. Marques, Mode competition between rotating waves in a swirling flow with reflection symmetry, J. Fluid Mech., 507 (2004), 265-288.       
26 J. M. Lopez, F. Marques and J. Shen, Complex dynamics in a short annular container with rotating bottom and inner cylinder, J. Fluid Mech., 51 (2004), 327-354.       
27 J. M. Lopez and F. Marques, Finite aspect ratio Taylor-Couette flow: Shil'nikov dynamics of 2-tori, Physica D, 211 (2005), 168-191.       
28 M. Avila, A. Meseguer and F. Marques Double Hopf bifurcation in corotating spiral Poiseuille flow, Phys. Fluids, 18 (2006), 064101.       
29 J. M. Lopez, Y. D. Cui and T. T. Lim, An experimental and numerical investigation of the competition between axisymmetric time-periodic modes in an enclosed swirling flow, Phys. Fluids, 18 (2006), 104106.
30 F. Marques and J. M. Lopez, Onset of three-dimensional unsteady states in small aspect-ratio Taylor-Couette flow, J. Fluid Mech., 561 (2006), 255-277.       
31 F. Marques, I. Mercader, O. Batiste and J. M. Lopez, Centrifugal effects in rotating convection: Axisymmetric states and three-dimensional instabilities, J. Fluid Mech., 580 (2007), 303-318.       
32 J. M. Lopez, F. Marques, I. Mercader and O. Batiste, Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus-Benjamin-Feir instability, J. Fluid Mech., 590 (2007), 187-208.
33 M. Avila, M. Grimes, J. M. Lopez and F. Marques, Global endwall effects on centrifugally stable flows, Phys. Fluids, 20 (2008), 104104.
34 J. M. Lopez and F. Marques, Centrifugal effects in rotating convection: Nonlinear dynamics, J. Fluid Mech., 628 (2009), 269-297.       
35 J. M. Lopez and F. Marques, Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially co-rotating lid, Phys. Fluids, 22 (2010), 114109.
36 Y. Do and Y.-C. Lai, Scaling laws for noise-induced superpersistent chaotic transients, Phys. Rev. E, 71 (2005), 046208.
37 A. Rubio, J. M. Lopez and F. Marques, Onset of K├╝ppers-Lortz-like dynamics in finite rotating thermal convection, J. Fluid Mech., 644 (2010), 337-357.       
38 M. Avila, F. Marques, J. M. Lopez and A. Meseguer, Stability control and catastrophic transition in a forced Taylor-Couette system, J. Fluid Mech., 590 (2007), 471-496.
39 M. Sinha, I. G. Kevrekidis and A. J. Smits, Experimental study of a Neimark-Sacker bifurcation in axially forced Taylor-Couette flow, J. Fluid Mech., 558 (2006), 1-32.

Go to top