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1. | Department of Mathematics, Clarkson University, Potsdam, NY, United States |
2. | Center for Dynamics & Institute of Analysis, Department of Mathematics, TU Dresden, Dresden, Germany |
3. | Department of Probability and Statistics, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam |
4. | Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, CA, United States |
References:
[1] |
L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, vol. 146 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1995. |
[2] |
M. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids, Physica D: Nonlinear Phenomena, 241 (2012), 95-105.
doi: 10.1016/j.physd.2011.10.002. |
[3] |
H. Aref and E. P. Flinchem, Dynamics of a vortex filament in a shear-flow, Journal of Fluid Mechanics, 148 (1984), 477-497.
doi: 10.1017/S0022112084002457. |
[4] |
S. Balasuriya, Explicit invariant manifolds and specialised trajectories in a class of unsteady flows, Physics of Fluids (1994-present), 24 (2012), 127101.
doi: 10.1063/1.4769979. |
[5] |
D. Blazevski and G. Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows, Physica D: Nonlinear Phenomena, 273/274 (2014), 46-62.
doi: 10.1016/j.physd.2014.01.007. |
[6] |
P. L. Boyland, H. Aref and M. A. Stremler, Topological fluid mechanics of stirring, Journal of Fluid Mechanics, 403 (2000), 277-304.
doi: 10.1017/S0022112099007107. |
[7] |
S. L. Brunton and C. W. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 017503, 12pp.
doi: 10.1063/1.3270044. |
[8] |
M. Budišić and I. Mezić, Geometry of the ergodic quotient reveals coherent structures in flows, Physica D. Nonlinear Phenomena, 241 (2012), 1255-1269.
doi: 10.1016/j.physd.2012.04.006. |
[9] |
M. S. Chong, A. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields, Physics of Fluids A: Fluid Dynamics (1989-1993), 2 (1990), 765-777.
doi: 10.1063/1.857730. |
[10] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, {Springer-Verlag, Berlin-New York}, 1978. |
[11] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[12] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[13] |
T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr and A. M. Soward, Chaotic streamlines in the ABC flows, Journal of Fluid Mechanics, 167 (1986), 353-391.
doi: 10.1017/S0022112086002859. |
[14] |
R. Durrett, Probability: Theory and Examples, 4th edition, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511779398. |
[15] |
M. Farazmand and G. Haller, Polar rotation angle identifies elliptic islands in unsteady dynamical systems, Physica D: Nonlinear Phenomena, 315 (2016), 1-12.
doi: 10.1016/j.physd.2015.09.007. |
[16] |
A. M. Fox and J. D. Meiss, Greene's residue criterion for the breakup of invariant tori of volume-preserving maps, Physica D: Nonlinear Phenomena, 243 (2013), 45-63.
doi: 10.1016/j.physd.2012.09.005. |
[17] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863 (electronic).
doi: 10.1137/S106482750238911X. |
[18] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D: Nonlinear Phenomena, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[19] |
G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D. Nonlinear Phenomena, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[20] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 043116, 10pp.
doi: 10.1063/1.3502450. |
[21] |
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2008. |
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I. Goldhirsch, P.-L. Sulem and S. A. Orszag, Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method, Physica D: Nonlinear Phenomena, 27 (1987), 311-337.
doi: 10.1016/0167-2789(87)90034-0. |
[23] |
S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probability Theory and Related Fields, 128 (2004), 82-122.
doi: 10.1007/s00440-003-0300-4. |
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S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electronic Journal of Probability, 19 (2014), 30pp.
doi: 10.1214/EJP.v19-3427. |
[25] |
J. M. Greene, Two-dimensional measure-preserving mappings, Journal of Mathematical Physics, 9 (1968), 760-768.
doi: 10.1063/1.1664639. |
[26] |
J. M. Greene, Method for Determining a Stochastic Transition, Journal of Mathematical Physics, 20 (1979), 1183-1201.
doi: 10.1063/1.524170. |
[27] |
G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Physics of Fluids, 13 (2001), 3365-3385.
doi: 10.1063/1.1403336. |
[28] |
G. Haller, A variational theory of hyperbolic Lagrangian Coherent Structures, Physica D. Nonlinear Phenomena, 240 (2011), 574-598.
doi: 10.1016/j.physd.2010.11.010. |
[29] |
G. Haller, Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162.
doi: 10.1146/annurev-fluid-010313-141322. |
[30] |
G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows, Physica D: Nonlinear Phenomena, 241 (2012), 1680-1702.
doi: 10.1016/j.physd.2012.06.012. |
[31] |
G. Haller and A. C. Poje, Finite time transport in aperiodic flows, Physica D. Nonlinear Phenomena, 119 (1998), 352-380.
doi: 10.1016/S0167-2789(98)00091-8. |
[32] |
G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D. Nonlinear Phenomena, 147 (2000), 352-370.
doi: 10.1016/S0167-2789(00)00142-1. |
[33] |
R. S. Irving, Integers, Polynomials, and Rings, Undergraduate Texts in Mathematics, {Springer-Verlag}, New York, 2004. |
[34] |
B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proceedings of National Academy of Sciences, 17 (1931), 315-318.
doi: 10.1073/pnas.17.5.315. |
[35] |
Z. Levnajić and I. Mezić, Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 033114, 19 pp.
doi: 10.1063/1.3458896. |
[36] |
T. Ma and E. M. Bollt, Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1106-1136.
doi: 10.1137/130940633. |
[37] |
T. Ma and E. M. Bollt, Shape coherence and finite-time curvature evolution, International Journal of Bifurcation and Chaos, 25 (2015), 1550076, 10pp.
doi: 10.1142/S0218127415500765. |
[38] |
T. Ma, N. T. Ouellette and E. M. Bollt, Stretching and folding in finite time, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 023112, 11pp.
doi: 10.1063/1.4941256. |
[39] |
J. A. J. Madrid and A. M. Mancho, Distinguished trajectories in time dependent vector fields, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 013111, 18pp.
doi: 10.1063/1.3056050. |
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N. Malhotra, I. Mezić and S. Wiggins, Patchiness: A new diagnostic for lagrangian trajectory analysis in time-dependent fluid flows, International Journal of Bifurcation and Chaos, 8 (1998), 1053-1093.
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A. M. Mancho, S. Wiggins, J. Curbelo and C. Mendoza, Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3530-3557.
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doi: 10.1007/s11071-005-2824-x. |
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I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D. Nonlinear Phenomena, 197 (2004), 101-133.
doi: 10.1016/j.physd.2004.06.015. |
[45] |
I. Mezić, S. Loire, V. A. Fonoberov and P. J. Hogan, A new mixing diagnostic and Gulf oil spill movement, Science Magazine, 330 (2010), 486-489. |
[46] |
I. Mezić and F. Sotiropoulos, Ergodic theory and experimental visualization of invariant sets in chaotically advected flows, Physics of Fluids, 14 (2002), 2235-2243.
doi: 10.1063/1.1480266. |
[47] |
I. Mezić and S. Wiggins, A method for visualization of invariant sets of dynamical systems based on the ergodic partition, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 213-218.
doi: 10.1063/1.166399. |
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B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 35-65.
doi: 10.1137/100794110. |
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J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. |
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A. Okubo, Horizontal dispersion of floatable particles in vicinity of velocity singularities such as convergences, Deep-Sea Research, 17 (1970), 445-454.
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J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. |
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K. J. Palmer, A finite-time condition for exponential dichotomy, Journal of Difference Equations and Applications, 17 (2011), 221-234.
doi: 10.1080/10236198.2010.549005. |
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A. D. Perry and S. Wiggins, KAM tori are very sticky: Rigorous lower bounds on the time to move away from an invariant Lagrangian torus with linear flow, Physica D. Nonlinear Phenomena, 71 (1994), 102-121.
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show all references
References:
[1] |
L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, vol. 146 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1995. |
[2] |
M. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids, Physica D: Nonlinear Phenomena, 241 (2012), 95-105.
doi: 10.1016/j.physd.2011.10.002. |
[3] |
H. Aref and E. P. Flinchem, Dynamics of a vortex filament in a shear-flow, Journal of Fluid Mechanics, 148 (1984), 477-497.
doi: 10.1017/S0022112084002457. |
[4] |
S. Balasuriya, Explicit invariant manifolds and specialised trajectories in a class of unsteady flows, Physics of Fluids (1994-present), 24 (2012), 127101.
doi: 10.1063/1.4769979. |
[5] |
D. Blazevski and G. Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows, Physica D: Nonlinear Phenomena, 273/274 (2014), 46-62.
doi: 10.1016/j.physd.2014.01.007. |
[6] |
P. L. Boyland, H. Aref and M. A. Stremler, Topological fluid mechanics of stirring, Journal of Fluid Mechanics, 403 (2000), 277-304.
doi: 10.1017/S0022112099007107. |
[7] |
S. L. Brunton and C. W. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 017503, 12pp.
doi: 10.1063/1.3270044. |
[8] |
M. Budišić and I. Mezić, Geometry of the ergodic quotient reveals coherent structures in flows, Physica D. Nonlinear Phenomena, 241 (2012), 1255-1269.
doi: 10.1016/j.physd.2012.04.006. |
[9] |
M. S. Chong, A. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields, Physics of Fluids A: Fluid Dynamics (1989-1993), 2 (1990), 765-777.
doi: 10.1063/1.857730. |
[10] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, {Springer-Verlag, Berlin-New York}, 1978. |
[11] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[12] |
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 221-264.
doi: 10.1016/S1874-575X(02)80026-1. |
[13] |
T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr and A. M. Soward, Chaotic streamlines in the ABC flows, Journal of Fluid Mechanics, 167 (1986), 353-391.
doi: 10.1017/S0022112086002859. |
[14] |
R. Durrett, Probability: Theory and Examples, 4th edition, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511779398. |
[15] |
M. Farazmand and G. Haller, Polar rotation angle identifies elliptic islands in unsteady dynamical systems, Physica D: Nonlinear Phenomena, 315 (2016), 1-12.
doi: 10.1016/j.physd.2015.09.007. |
[16] |
A. M. Fox and J. D. Meiss, Greene's residue criterion for the breakup of invariant tori of volume-preserving maps, Physica D: Nonlinear Phenomena, 243 (2013), 45-63.
doi: 10.1016/j.physd.2012.09.005. |
[17] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863 (electronic).
doi: 10.1137/S106482750238911X. |
[18] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D: Nonlinear Phenomena, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[19] |
G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D. Nonlinear Phenomena, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[20] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 043116, 10pp.
doi: 10.1063/1.3502450. |
[21] |
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2008. |
[22] |
I. Goldhirsch, P.-L. Sulem and S. A. Orszag, Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method, Physica D: Nonlinear Phenomena, 27 (1987), 311-337.
doi: 10.1016/0167-2789(87)90034-0. |
[23] |
S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probability Theory and Related Fields, 128 (2004), 82-122.
doi: 10.1007/s00440-003-0300-4. |
[24] |
S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electronic Journal of Probability, 19 (2014), 30pp.
doi: 10.1214/EJP.v19-3427. |
[25] |
J. M. Greene, Two-dimensional measure-preserving mappings, Journal of Mathematical Physics, 9 (1968), 760-768.
doi: 10.1063/1.1664639. |
[26] |
J. M. Greene, Method for Determining a Stochastic Transition, Journal of Mathematical Physics, 20 (1979), 1183-1201.
doi: 10.1063/1.524170. |
[27] |
G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Physics of Fluids, 13 (2001), 3365-3385.
doi: 10.1063/1.1403336. |
[28] |
G. Haller, A variational theory of hyperbolic Lagrangian Coherent Structures, Physica D. Nonlinear Phenomena, 240 (2011), 574-598.
doi: 10.1016/j.physd.2010.11.010. |
[29] |
G. Haller, Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162.
doi: 10.1146/annurev-fluid-010313-141322. |
[30] |
G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows, Physica D: Nonlinear Phenomena, 241 (2012), 1680-1702.
doi: 10.1016/j.physd.2012.06.012. |
[31] |
G. Haller and A. C. Poje, Finite time transport in aperiodic flows, Physica D. Nonlinear Phenomena, 119 (1998), 352-380.
doi: 10.1016/S0167-2789(98)00091-8. |
[32] |
G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D. Nonlinear Phenomena, 147 (2000), 352-370.
doi: 10.1016/S0167-2789(00)00142-1. |
[33] |
R. S. Irving, Integers, Polynomials, and Rings, Undergraduate Texts in Mathematics, {Springer-Verlag}, New York, 2004. |
[34] |
B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proceedings of National Academy of Sciences, 17 (1931), 315-318.
doi: 10.1073/pnas.17.5.315. |
[35] |
Z. Levnajić and I. Mezić, Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 033114, 19 pp.
doi: 10.1063/1.3458896. |
[36] |
T. Ma and E. M. Bollt, Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1106-1136.
doi: 10.1137/130940633. |
[37] |
T. Ma and E. M. Bollt, Shape coherence and finite-time curvature evolution, International Journal of Bifurcation and Chaos, 25 (2015), 1550076, 10pp.
doi: 10.1142/S0218127415500765. |
[38] |
T. Ma, N. T. Ouellette and E. M. Bollt, Stretching and folding in finite time, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 023112, 11pp.
doi: 10.1063/1.4941256. |
[39] |
J. A. J. Madrid and A. M. Mancho, Distinguished trajectories in time dependent vector fields, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 013111, 18pp.
doi: 10.1063/1.3056050. |
[40] |
N. Malhotra, I. Mezić and S. Wiggins, Patchiness: A new diagnostic for lagrangian trajectory analysis in time-dependent fluid flows, International Journal of Bifurcation and Chaos, 8 (1998), 1053-1093.
doi: 10.1142/S0218127498000875. |
[41] |
A. M. Mancho, S. Wiggins, J. Curbelo and C. Mendoza, Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3530-3557.
doi: 10.1016/j.cnsns.2013.05.002. |
[42] |
I. Mezić, On the Geometrical and Statistical Properties of Dynamical Systems: Theory and Applications, Phd thesis, California Institute of Technology, 1994. |
[43] |
I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 41 (2005), 309-325.
doi: 10.1007/s11071-005-2824-x. |
[44] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D. Nonlinear Phenomena, 197 (2004), 101-133.
doi: 10.1016/j.physd.2004.06.015. |
[45] |
I. Mezić, S. Loire, V. A. Fonoberov and P. J. Hogan, A new mixing diagnostic and Gulf oil spill movement, Science Magazine, 330 (2010), 486-489. |
[46] |
I. Mezić and F. Sotiropoulos, Ergodic theory and experimental visualization of invariant sets in chaotically advected flows, Physics of Fluids, 14 (2002), 2235-2243.
doi: 10.1063/1.1480266. |
[47] |
I. Mezić and S. Wiggins, A method for visualization of invariant sets of dynamical systems based on the ergodic partition, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 213-218.
doi: 10.1063/1.166399. |
[48] |
B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 35-65.
doi: 10.1137/100794110. |
[49] |
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. |
[50] |
A. Okubo, Horizontal dispersion of floatable particles in vicinity of velocity singularities such as convergences, Deep-Sea Research, 17 (1970), 445-454.
doi: 10.1016/0011-7471(70)90059-8. |
[51] |
J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. |
[52] |
K. J. Palmer, A finite-time condition for exponential dichotomy, Journal of Difference Equations and Applications, 17 (2011), 221-234.
doi: 10.1080/10236198.2010.549005. |
[53] |
A. D. Perry and S. Wiggins, KAM tori are very sticky: Rigorous lower bounds on the time to move away from an invariant Lagrangian torus with linear flow, Physica D. Nonlinear Phenomena, 71 (1994), 102-121.
doi: 10.1016/0167-2789(94)90184-8. |
[54] |
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