American Institute of Mathematical Sciences

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May  2011, 10(3): 963-981. doi: 10.3934/cpaa.2011.10.963

On finite-time hyperbolicity

Arno Berger 1,

1. 

Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1

Received  March 2009 Revised  September 2009 Published  December 2010

A solution of a nonautonomous ordinary differential equation is finite-time hyperbolic, i.e. hyperbolic on a compact interval of time, if the linearisation along that solution exhibits a strong exponential dichotomy. As a finite-time variant and strengthening of classical asymptotic facts, it is shown that finite-time hyperbolicity guarantees the existence of stable and unstable manifolds of the appropriate dimensions. Eigenvalues and -vectors are often unsuitable for detecting hyperbolicity. A (dynamic) partition of the extended phase space is used to circumvent this difficulty. It is proved that any solution staying clear of the elliptic and degenerate parts of the partition is finite-time hyperbolic. This extends and unifies earlier partial results.
Keywords: exponential dichotomy, dynamic partition., finite-time dynamics, stable manifold, Hyperbolicity.
Mathematics Subject Classification: Primary: 34A30, 37B55, 37D05; Secondary: 34D09, 37D10, 37N1.
Citation: Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
References:
[1]

A. Berger, T. S. Doan and S. Siegmund, Nonautonomous finite-time dynamics,, Discrete Continuous Dynam. Systems - B, 9 (2008), 463.   Google Scholar

[2]

A. Berger, T. S. Doan and S. Siegmund, A remark on finite-time hyperbolicity,, PAMM Proc. Appl. Math. Mech., 8 (2008), 10917.  doi: doi:10.1002/pamm.200810917.  Google Scholar

[3]

A. Berger, T. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time,, J. Differential Equations, 246 (2009), 1098.  doi: doi:10.1016/j.jde.2008.06.036.  Google Scholar

[4]

M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces,", Springer, (1988).   Google Scholar

[5]

A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics \textbf{629}, 629 (1978).   Google Scholar

[6]

L. H. Duc and S. Siegmund, Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 641.  doi: doi:10.1142/S0218127408020562.  Google Scholar

[7]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10 (2000), 99.  doi: doi:10.1063/1.166479.  Google Scholar

[8]

G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows,, Physica D, 149 (2001), 248.  doi: doi:10.1016/S0167-2789(00)00199-8.  Google Scholar

[9]

G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence,, Physics of Fluids, 13 (2001), 3365.  doi: doi:10.1063/1.1403336.  Google Scholar

[10]

G. Haller, An objective definition of a vortex,, J. Fluid Mech., 525 (2005), 1.  doi: doi:10.1017/S0022112004002526.  Google Scholar

[11]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence,, Physica D, 147 (2000), 352.  doi: doi:10.1016/S0167-2789(00)00142-1.  Google Scholar

[12]

M. C. Irwin, "Smooth Dynamical Systems,", World Scientific, (2001).  doi: doi:10.1142/9789812810120.  Google Scholar

[13]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1980).   Google Scholar

[14]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Cambridge University Press, (1995).   Google Scholar

[15]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,", Kluwer, (2000).   Google Scholar

[16]

R. M. Samelson and S. Wiggins, "Lagrangian Transport in Geophysical Jets and Waves. The Dynamical Systems Approach,'', Springer, (2006).   Google Scholar

[17]

S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,, Physica D, 212 (2005), 271.  doi: doi:10.1016/j.physd.2005.10.007.  Google Scholar

[18]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Springer, (1990).   Google Scholar

show all references

References:
[1]

A. Berger, T. S. Doan and S. Siegmund, Nonautonomous finite-time dynamics,, Discrete Continuous Dynam. Systems - B, 9 (2008), 463.   Google Scholar

[2]

A. Berger, T. S. Doan and S. Siegmund, A remark on finite-time hyperbolicity,, PAMM Proc. Appl. Math. Mech., 8 (2008), 10917.  doi: doi:10.1002/pamm.200810917.  Google Scholar

[3]

A. Berger, T. S. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time,, J. Differential Equations, 246 (2009), 1098.  doi: doi:10.1016/j.jde.2008.06.036.  Google Scholar

[4]

M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces,", Springer, (1988).   Google Scholar

[5]

A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics \textbf{629}, 629 (1978).   Google Scholar

[6]

L. H. Duc and S. Siegmund, Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 641.  doi: doi:10.1142/S0218127408020562.  Google Scholar

[7]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10 (2000), 99.  doi: doi:10.1063/1.166479.  Google Scholar

[8]

G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows,, Physica D, 149 (2001), 248.  doi: doi:10.1016/S0167-2789(00)00199-8.  Google Scholar

[9]

G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence,, Physics of Fluids, 13 (2001), 3365.  doi: doi:10.1063/1.1403336.  Google Scholar

[10]

G. Haller, An objective definition of a vortex,, J. Fluid Mech., 525 (2005), 1.  doi: doi:10.1017/S0022112004002526.  Google Scholar

[11]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence,, Physica D, 147 (2000), 352.  doi: doi:10.1016/S0167-2789(00)00142-1.  Google Scholar

[12]

M. C. Irwin, "Smooth Dynamical Systems,", World Scientific, (2001).  doi: doi:10.1142/9789812810120.  Google Scholar

[13]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1980).   Google Scholar

[14]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Cambridge University Press, (1995).   Google Scholar

[15]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,", Kluwer, (2000).   Google Scholar

[16]

R. M. Samelson and S. Wiggins, "Lagrangian Transport in Geophysical Jets and Waves. The Dynamical Systems Approach,'', Springer, (2006).   Google Scholar

[17]

S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,, Physica D, 212 (2005), 271.  doi: doi:10.1016/j.physd.2005.10.007.  Google Scholar

[18]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Springer, (1990).   Google Scholar

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