American Institute of Mathematical Sciences

Advanced
January  2013, 18(1): 95-107. doi: 10.3934/dcdsb.2013.18.95

Slow passage through multiple bifurcation points

Younghae Do 1, and  Juan M. Lopez 2,

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

Received  April 2012 Revised  May 2012 Published  September 2012

The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations. The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp--a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly. We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter. These delay and memory effects are independent of the ramp rate, for small enough ramp rates. The slow passage through the torus break-up bifurcation is qualitatively different. It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate. This is typical of transient behavior. We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used. The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.
Keywords: stability., Slow passage problem, Hopf bifurcation, delay effect, Neimark-Sacker bifurcation.
Mathematics Subject Classification: Primary: 37B55, 74H60; Secondary: 70K7.
Citation: Younghae Do, Juan M. Lopez. Slow passage through multiple bifurcation points. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 95-107. doi: 10.3934/dcdsb.2013.18.95
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.

[1]

Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019
[2]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[3]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[4]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[5]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[6]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[7]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[8]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[9]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[10]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[11]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[12]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[13]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[14]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[15]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[16]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[17]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[18]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[19]

Shu Li, Zhenzhen Li, Binxiang Dai. Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022025
[20]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy