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, 702701, 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), 5571. 

2 
M. Marhl, T. Haberichter, M. Brumen and R. Heinrich, Complex calcium oscillations and the role of mitochondria and cytosolic protens, BioSystems, 57 (2000), 7586. 

3 
M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons & Fractals, 27 (2006), 395403. 

4 
P. Strizhak and M. Menzinger, Slow passage through a supercritical Hopf bifurcation: Timedelayed response in the BelousovZhabotinsky reaction in a batch reactor, J. Chem. Phys., 105 (1996), 1090510910. 

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), 692696. 

8 
J. von Stamm, U. Gerdts, T. Buzug and G. Pfister, Symmetry breaking and period doubling on a torus in the VLFregime in TaylorCouette flow , Phys. Rev., E, 54 (1996), 4938. 

9 
C. S. Dutcher and S. J. Muller, Spatiotemporal mode dynamics and higher order transitions in high aspect ratio Newtonian TaylorCouette flows, J. Fluid Mech., 641 (2009), 85113. 

10 
J. Su, Persistent unstable periodic motions, I, J. Math. Analysis and Applications, 198 (1996), 796825. 

11 
J. Su, Persistent unstable periodic motions, II, J. Math. Analysis and Applications, 199 (1996), 88119. 

12 
L. Holden and T. Erneux, Slow passage through a HopfbifurcationFrom oscillatios to steadystate solutions, SIAM J. Appl. Math., 53 (1993), 10451058. 

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), 69106. 

15 
V. Booth, T. W. Carr and T. Erneux, Nearthreshold bursting is delayed by a slow passage near a limit point, SIAM J. Appl. Math., 57 (1997), 14061420. 

16 
L. Ng, R. Rand and M. O'Neil, Slow passage through resonance in Mathieu's equation, J. Vibration & Control, 9 (2003), 685707. 

17 
J. P. Denier and R. Grimshaw, Slowlyvarying bifurcation theory in dissipative systems, J. Austral. Math. Soc. Ser. B, 31 (1990), 301318. 

18 
P. Hall, On the nonlinear stability of slowly varying timedependent viscous flows, J., Fluid Mech., 126 (1983), 357368. 

19 
P. Yu, Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales, Nonlinear Dyn., 27 (2002), 1953. 

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), 263281. 

22 
J. M. Lopez, J. E. Hart, F. Marques, S. Kittelman and J. Shen, Instability and mode interactions in a differentiallydriven rotating cylinder, J. Fluid Mech., 462 (2002), 383409. 

23 
J. M. Lopez and F. Marques, Small aspect ratio TaylorCouette flow: On set of a verylowfrequency threetorus 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), 265288. 

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), 327354. 

27 
J. M. Lopez and F. Marques, Finite aspect ratio TaylorCouette flow: Shil'nikov dynamics of 2tori, Physica D, 211 (2005), 168191. 

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 timeperiodic modes in an enclosed swirling flow, Phys. Fluids, 18 (2006), 104106. 

30 
F. Marques and J. M. Lopez, Onset of threedimensional unsteady states in small aspectratio TaylorCouette flow, J. Fluid Mech., 561 (2006), 255277. 

31 
F. Marques, I. Mercader, O. Batiste and J. M. Lopez, Centrifugal effects in rotating convection: Axisymmetric states and threedimensional instabilities, J. Fluid Mech., 580 (2007), 303318. 

32 
J. M. Lopez, F. Marques, I. Mercader and O. Batiste, Onset of convection in a moderate aspectratio rotating cylinder: EckhausBenjaminFeir instability, J. Fluid Mech., 590 (2007), 187208. 

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), 269297. 

35 
J. M. Lopez and F. Marques, Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially corotating lid, Phys. Fluids, 22 (2010), 114109. 

36 
Y. Do and Y.C. Lai, Scaling laws for noiseinduced superpersistent chaotic transients, Phys. Rev. E, 71 (2005), 046208. 

37 
A. Rubio, J. M. Lopez and F. Marques, Onset of KüppersLortzlike dynamics in finite rotating thermal convection, J. Fluid Mech., 644 (2010), 337357. 

38 
M. Avila, F. Marques, J. M. Lopez and A. Meseguer, Stability control and catastrophic transition in a forced TaylorCouette system, J. Fluid Mech., 590 (2007), 471496. 

39 
M. Sinha, I. G. Kevrekidis and A. J. Smits, Experimental study of a NeimarkSacker bifurcation in axially forced TaylorCouette flow, J. Fluid Mech., 558 (2006), 132. 

Go to top
