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Weak KAM theory for nonregular commuting Hamiltonians
Slow passage through multiple bifurcation points
1. | Department of Mathematics, Kyungpook National University, Daegu, 702-701 |
2. | School of Mathematical and Statistical Sciences, Arizona State Univ., Tempe AZ, 85287, United States |
References:
[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.
doi: 10.1137/0149003. |
[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.
doi: 10.1016/S0303-2647(00)00090-3. |
[3] |
M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons & Fractals, 27 (2006), 395-403.
doi: 10.1016/j.chaos.2005.03.045. |
[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.
doi: 10.1063/1.472860. |
[5] |
Y. Park, Y. Do, and J. M. Lopez, Slow passage through resonance, Phys. Rev., E, 84 (2011), 056604.
doi: 10.1103/PhysRevE.84.056604. |
[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.
doi: 10.1103/PhysRevLett.47.1448. |
[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.
doi: 10.1063/1.868854. |
[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.
doi: 10.1103/PhysRevE.54.4938. |
[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.
doi: 10.1017/S0022112009991431. |
[10] |
J. Su, Persistent unstable periodic motions, I, J. Math. Analysis and Applications, 198 (1996), 796-825.
doi: 10.1006/jmaa.1996.0113. |
[11] |
J. Su, Persistent unstable periodic motions, II, J. Math. Analysis and Applications, 199 (1996), 88-119.
doi: 10.1006/jmaa.1996.0128. |
[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.
doi: 10.1137/0153052. |
[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.
doi: 10.1103/PhysRevE.78.036205. |
[14] |
R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), 69-106.
doi: 10.1137/0137006. |
[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.
doi: 10.1137/S0036139995295104. |
[16] |
L. Ng, R. Rand and M. O'Neil, Slow passage through resonance in Mathieu's equation, J. Vibration & Control, 9 (2003), 685-707.
doi: 10.1177/1077546303009006004. |
[17] |
J. P. Denier and R. Grimshaw, Slowly-varying bifurcation theory in dissipative systems, J. Austral. Math. Soc. Ser. B, 31 (1990), 301-318.
doi: 10.1017/S0334270000006664. |
[18] |
P. Hall, On the nonlinear stability of slowly varying time-dependent viscous flows, J., Fluid Mech., 126 (1983), 357-368.
doi: 10.1017/S0022112083000208. |
[19] |
P. Yu, Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales, Nonlinear Dyn., 27 (2002), 19-53.
doi: 10.1023/A:1017993026651. |
[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.
doi: 10.1017/S0022112001007285. |
[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.
doi: 10.1017/S0022112002008649. |
[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.
doi: 10.1103/PhysRevE.68.036302. |
[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.
doi: 10.1103/PhysRevE.68.016310. |
[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.
doi: 10.1017/S002211200400864X. |
[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.
doi: 10.1017/S0022112003007493. |
[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.
doi: 10.1016/j.physd.2005.08.011. |
[28] |
M. Avila, A. Meseguer and F. Marques, Double Hopf bifurcation in corotating spiral Poiseuille flow, Phys. Fluids, 18 (2006), 064101.
doi: 10.1063/1.2204967. |
[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.
doi: 10.1063/1.2362782. |
[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.
doi: 10.1017/S0022112006000681. |
[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.
doi: 10.1017/S0022112007005447. |
[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.
doi: 10.1017/S0022112007008038. |
[33] |
M. Avila, M. Grimes, J. M. Lopez and F. Marques, Global endwall effects on centrifugally stable flows, Phys. Fluids, 20 (2008), 104104.
doi: 10.1063/1.2996326. |
[34] |
J. M. Lopez and F. Marques, Centrifugal effects in rotating convection: Nonlinear dynamics, J. Fluid Mech., 628 (2009), 269-297.
doi: 10.1017/S0022112009006193. |
[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.
doi: 10.1063/1.3517292. |
[36] |
Y. Do and Y.-C. Lai, Scaling laws for noise-induced superpersistent chaotic transients, Phys. Rev. E, 71 (2005), 046208.
doi: 10.1103/PhysRevE.71.046208. |
[37] |
A. Rubio, J. M. Lopez and F. Marques, Onset of Küppers-Lortz-like dynamics in finite rotating thermal convection, J. Fluid Mech., 644 (2010), 337-357.
doi: 10.1017/S0022112009992400. |
[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.
doi: 10.1017/S0022112007008105. |
[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.
doi: 10.1017/S0022112006009207. |
show all references
References:
[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.
doi: 10.1137/0149003. |
[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.
doi: 10.1016/S0303-2647(00)00090-3. |
[3] |
M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Solitons & Fractals, 27 (2006), 395-403.
doi: 10.1016/j.chaos.2005.03.045. |
[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.
doi: 10.1063/1.472860. |
[5] |
Y. Park, Y. Do, and J. M. Lopez, Slow passage through resonance, Phys. Rev., E, 84 (2011), 056604.
doi: 10.1103/PhysRevE.84.056604. |
[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.
doi: 10.1103/PhysRevLett.47.1448. |
[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.
doi: 10.1063/1.868854. |
[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.
doi: 10.1103/PhysRevE.54.4938. |
[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.
doi: 10.1017/S0022112009991431. |
[10] |
J. Su, Persistent unstable periodic motions, I, J. Math. Analysis and Applications, 198 (1996), 796-825.
doi: 10.1006/jmaa.1996.0113. |
[11] |
J. Su, Persistent unstable periodic motions, II, J. Math. Analysis and Applications, 199 (1996), 88-119.
doi: 10.1006/jmaa.1996.0128. |
[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.
doi: 10.1137/0153052. |
[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.
doi: 10.1103/PhysRevE.78.036205. |
[14] |
R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), 69-106.
doi: 10.1137/0137006. |
[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.
doi: 10.1137/S0036139995295104. |
[16] |
L. Ng, R. Rand and M. O'Neil, Slow passage through resonance in Mathieu's equation, J. Vibration & Control, 9 (2003), 685-707.
doi: 10.1177/1077546303009006004. |
[17] |
J. P. Denier and R. Grimshaw, Slowly-varying bifurcation theory in dissipative systems, J. Austral. Math. Soc. Ser. B, 31 (1990), 301-318.
doi: 10.1017/S0334270000006664. |
[18] |
P. Hall, On the nonlinear stability of slowly varying time-dependent viscous flows, J., Fluid Mech., 126 (1983), 357-368.
doi: 10.1017/S0022112083000208. |
[19] |
P. Yu, Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales, Nonlinear Dyn., 27 (2002), 19-53.
doi: 10.1023/A:1017993026651. |
[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.
doi: 10.1017/S0022112001007285. |
[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.
doi: 10.1017/S0022112002008649. |
[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.
doi: 10.1103/PhysRevE.68.036302. |
[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.
doi: 10.1103/PhysRevE.68.016310. |
[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.
doi: 10.1017/S002211200400864X. |
[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.
doi: 10.1017/S0022112003007493. |
[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.
doi: 10.1016/j.physd.2005.08.011. |
[28] |
M. Avila, A. Meseguer and F. Marques, Double Hopf bifurcation in corotating spiral Poiseuille flow, Phys. Fluids, 18 (2006), 064101.
doi: 10.1063/1.2204967. |
[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.
doi: 10.1063/1.2362782. |
[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.
doi: 10.1017/S0022112006000681. |
[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.
doi: 10.1017/S0022112007005447. |
[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.
doi: 10.1017/S0022112007008038. |
[33] |
M. Avila, M. Grimes, J. M. Lopez and F. Marques, Global endwall effects on centrifugally stable flows, Phys. Fluids, 20 (2008), 104104.
doi: 10.1063/1.2996326. |
[34] |
J. M. Lopez and F. Marques, Centrifugal effects in rotating convection: Nonlinear dynamics, J. Fluid Mech., 628 (2009), 269-297.
doi: 10.1017/S0022112009006193. |
[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.
doi: 10.1063/1.3517292. |
[36] |
Y. Do and Y.-C. Lai, Scaling laws for noise-induced superpersistent chaotic transients, Phys. Rev. E, 71 (2005), 046208.
doi: 10.1103/PhysRevE.71.046208. |
[37] |
A. Rubio, J. M. Lopez and F. Marques, Onset of Küppers-Lortz-like dynamics in finite rotating thermal convection, J. Fluid Mech., 644 (2010), 337-357.
doi: 10.1017/S0022112009992400. |
[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.
doi: 10.1017/S0022112007008105. |
[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.
doi: 10.1017/S0022112006009207. |
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