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No-oscillation theorem for the transient dynamics of the linear signal transduction pathway and beyond
Interior structural bifurcation of 2D symmetric incompressible flows
1. | Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey |
2. | Department of Mathematics, Marmara University, 34722 Istanbul, Turkey |
The structural bifurcation of a 2D divergence free vector field $ \mathbf{u}(\cdot, t) $ when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{x}_0 $ of zero index has been studied by Ma and Wang [
References:
[1] |
A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New
York-Toronto, Ont., 1973. |
[2] |
P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3512-2. |
[3] |
A. Bisgaard, M. Brøns and J. N. Sørensen,
Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.
doi: 10.1007/s00707-006-0367-y. |
[4] |
M. Brøns and A. V. Bisgaard,
Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.
doi: 10.1017/S0022112006002424. |
[5] |
M. Brøns and J. N. Hartnack,
Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.
doi: 10.1063/1.869881. |
[6] |
M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt,
Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.
doi: 10.1017/S0022112007006234. |
[7] |
M. Brøns, L. K. Voigt and J. N. Sørensen,
Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.
doi: 10.1017/S0022112000002512. |
[8] |
C. H. Chan, M. Czubak and T. Yoneda,
An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.
doi: 10.1016/j.physd.2014.05.004. |
[9] |
M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017).
doi: 10.1063/1.4993613. |
[10] |
P. Gaskell, M. Savage and M. Wilson,
Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.
doi: 10.1017/S0022112097005028. |
[11] |
M. Ghil, T. Ma and S. Wang,
Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.
doi: 10.1512/iumj.2001.50.2183. |
[12] |
F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp.
doi: 10.1063/1.2055527. |
[13] |
F. Gürcan and A. Deliceoğlu,
Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.
doi: 10.1016/j.amc.2005.03.012. |
[14] |
F. Gürcan, A. Deliceoğlu and P. Bakker,
Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.
doi: 10.1017/S0022112005005689. |
[15] |
F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. |
[16] |
J. N. Hartnack,
Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.
doi: 10.1007/BF01292298. |
[17] |
M. Heil, J. Rosso, A. L. Hazel and M. Brøns,
Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.
doi: 10.1017/jfm.2016.792. |
[18] |
C.-H. Hsia, J.-G. Liu and C. Wang et al.,
Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.
doi: 10.4310/MAA.2008.v15.n4.a6. |
[19] |
D. D. Joseph and L. Sturges,
The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.
doi: 10.1137/0134002. |
[20] |
R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. |
[21] |
H. Luo, Q. Wang and T. Ma,
A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.
doi: 10.1016/j.nonrwa.2014.09.007. |
[22] |
T. Ma and S. Wang,
Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.
doi: 10.1016/S0167-2789(02)00587-0. |
[23] |
T. Ma and S. Wang,
Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.
doi: 10.1063/1.1689005. |
[24] |
T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/surv/119. |
[25] |
T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 |
[26] |
M. M. Peixoto,
Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.
doi: 10.1016/0040-9383(65)90018-2. |
[27] |
A. Perry, M. Chong and T. Lim,
The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.
doi: 10.1017/S0022112082000378. |
[28] |
R. C. T. Smith,
The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.
|
[29] |
L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp.
doi: 10.1063/1.2337660. |
[30] |
Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682.
doi: 10.3934/dcdsb.2015.20.675. |
[31] |
J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152. |
show all references
References:
[1] |
A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New
York-Toronto, Ont., 1973. |
[2] |
P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3512-2. |
[3] |
A. Bisgaard, M. Brøns and J. N. Sørensen,
Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.
doi: 10.1007/s00707-006-0367-y. |
[4] |
M. Brøns and A. V. Bisgaard,
Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.
doi: 10.1017/S0022112006002424. |
[5] |
M. Brøns and J. N. Hartnack,
Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.
doi: 10.1063/1.869881. |
[6] |
M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt,
Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.
doi: 10.1017/S0022112007006234. |
[7] |
M. Brøns, L. K. Voigt and J. N. Sørensen,
Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.
doi: 10.1017/S0022112000002512. |
[8] |
C. H. Chan, M. Czubak and T. Yoneda,
An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.
doi: 10.1016/j.physd.2014.05.004. |
[9] |
M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017).
doi: 10.1063/1.4993613. |
[10] |
P. Gaskell, M. Savage and M. Wilson,
Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.
doi: 10.1017/S0022112097005028. |
[11] |
M. Ghil, T. Ma and S. Wang,
Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.
doi: 10.1512/iumj.2001.50.2183. |
[12] |
F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp.
doi: 10.1063/1.2055527. |
[13] |
F. Gürcan and A. Deliceoğlu,
Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.
doi: 10.1016/j.amc.2005.03.012. |
[14] |
F. Gürcan, A. Deliceoğlu and P. Bakker,
Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.
doi: 10.1017/S0022112005005689. |
[15] |
F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. |
[16] |
J. N. Hartnack,
Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.
doi: 10.1007/BF01292298. |
[17] |
M. Heil, J. Rosso, A. L. Hazel and M. Brøns,
Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.
doi: 10.1017/jfm.2016.792. |
[18] |
C.-H. Hsia, J.-G. Liu and C. Wang et al.,
Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.
doi: 10.4310/MAA.2008.v15.n4.a6. |
[19] |
D. D. Joseph and L. Sturges,
The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.
doi: 10.1137/0134002. |
[20] |
R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. |
[21] |
H. Luo, Q. Wang and T. Ma,
A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.
doi: 10.1016/j.nonrwa.2014.09.007. |
[22] |
T. Ma and S. Wang,
Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.
doi: 10.1016/S0167-2789(02)00587-0. |
[23] |
T. Ma and S. Wang,
Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.
doi: 10.1063/1.1689005. |
[24] |
T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/surv/119. |
[25] |
T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 |
[26] |
M. M. Peixoto,
Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.
doi: 10.1016/0040-9383(65)90018-2. |
[27] |
A. Perry, M. Chong and T. Lim,
The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.
doi: 10.1017/S0022112082000378. |
[28] |
R. C. T. Smith,
The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237.
|
[29] |
L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp.
doi: 10.1063/1.2337660. |
[30] |
Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682.
doi: 10.3934/dcdsb.2015.20.675. |
[31] |
J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152. |







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