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Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA |
We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.
References:
[1] |
E. Benoît, M. Brøns, M. Desroches and M. Krupa,
Extending the zero-derivative principle for slow-fast dynamical systems, Z. Angew. Math. Phys., 66 (2015), 2255-2270.
doi: 10.1007/s00033-015-0552-8. |
[2] |
O. Castejón, A. Guillamon and G. Huguet, Phase-amplitude response functions for transient-state stimuli, J. Math. Neurosci., 3 (2013), Art. 13, 26 pp.
doi: 10.1186/2190-8567-3-13. |
[3] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[4] |
E. Friere, A. Gasull and A. Guillamon,
Limit cycles and Lie symmetries, Bull. Sci. Math, 131 (2007), 501-517.
doi: 10.1016/j.bulsci.2006.03.015. |
[5] |
R. A. Garcia, A. Gasull and A. Guillamon,
Geometric conditions for the stability of orbits in planar systems, Math. Proc. Cambridge. Phil. Soc., 120 (1996), 499-519.
doi: 10.1017/S0305004100075046. |
[6] |
M. Krupa and M. Wechselberger,
Local analysis near a folded saddle-node singularity, J. Differential Equations, 248 (2010), 2841-2888.
doi: 10.1016/j.jde.2010.02.006. |
[7] |
W. Kühnel, Differential Geometry: Curves, Surfaces, Manifolds, 2$^nd$ edition, Student Mathematical Library: Vol. 16, American Mathematical Society, 2005. |
[8] |
S. H. Lam and D. A. Goussis,
Understanding complex chemical kinetics with computational singular perturbation, Symposium (International) on Combustion, 22 (1989), 931-941.
doi: 10.1016/S0082-0784(89)80102-X. |
[9] |
B. Letson and J. Rubin,
A new frame for an old (phase) portrait: Finding rivers and other flow features in the plane, SIAM J. Appl. Dyn. Syst., 17 (2018), 2414-2445.
doi: 10.1137/18M1186617. |
[10] |
A. Mauroy, I. Mezi\'c and J. Moehlis,
Isostables isoschrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics, Phys. D, 261 (2013), 19-30.
doi: 10.1016/j.physd.2013.06.004. |
[11] |
S. Revzen and J. M. Guckenheimer, Estimating the phase of synchronized oscillators, Phys. Rev. E, 78 (2008), 051907, 12pp.
doi: 10.1103/PhysRevE.78.051907. |
[12] |
S. Shirasaka, W. Kurebayashi and H. Nakao, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, Chaos, 27 (2017), 023119, 7pp.
doi: 10.1063/1.4977195. |
[13] |
P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[14] |
T. Vo and M. Wechselberger,
Canards of folded saddle-node type {I}, SIAM J. Math. Anal., 47 (2015), 3235-3283.
doi: 10.1137/140965818. |
[15] |
K. C. A. Wedgwood, K. K. Lin, R. Thul and S. Coombes, Phase-amplitude descriptions of neural oscillator models, J. Math. Neurosci., 3 (2013), Art. 2, 22 pp.
doi: 10.1186/2190-8567-3-2. |
[16] |
D. Wilson and J. Moehlis, Isostable reduction of periodic orbits, Phys. Rev. E, 94 (2016), 052213.
doi: 10.1103/PhysRevE.94.052213. |
[17] |
D. Wilson and B. Ermentrout,
Greater accuracy and broadened applicability of phase reduction using isostable coordinates, J. Math. Biol., 76 (2018), 37-66.
doi: 10.1007/s00285-017-1141-6. |
[18] |
A. Zagaris, C. Vandekerckhove, W. C. Gear, T. J. Kaper, I. G. Kevrekidis and G. Ioannis,
Stability and stabilization of the contrained runs schemes for equation-free projection to a slow manifold, Discrete Contin. Dyn. Syst., 32 (2012), 2759-2803.
doi: 10.3934/dcds.2012.32.2759. |
show all references
References:
[1] |
E. Benoît, M. Brøns, M. Desroches and M. Krupa,
Extending the zero-derivative principle for slow-fast dynamical systems, Z. Angew. Math. Phys., 66 (2015), 2255-2270.
doi: 10.1007/s00033-015-0552-8. |
[2] |
O. Castejón, A. Guillamon and G. Huguet, Phase-amplitude response functions for transient-state stimuli, J. Math. Neurosci., 3 (2013), Art. 13, 26 pp.
doi: 10.1186/2190-8567-3-13. |
[3] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[4] |
E. Friere, A. Gasull and A. Guillamon,
Limit cycles and Lie symmetries, Bull. Sci. Math, 131 (2007), 501-517.
doi: 10.1016/j.bulsci.2006.03.015. |
[5] |
R. A. Garcia, A. Gasull and A. Guillamon,
Geometric conditions for the stability of orbits in planar systems, Math. Proc. Cambridge. Phil. Soc., 120 (1996), 499-519.
doi: 10.1017/S0305004100075046. |
[6] |
M. Krupa and M. Wechselberger,
Local analysis near a folded saddle-node singularity, J. Differential Equations, 248 (2010), 2841-2888.
doi: 10.1016/j.jde.2010.02.006. |
[7] |
W. Kühnel, Differential Geometry: Curves, Surfaces, Manifolds, 2$^nd$ edition, Student Mathematical Library: Vol. 16, American Mathematical Society, 2005. |
[8] |
S. H. Lam and D. A. Goussis,
Understanding complex chemical kinetics with computational singular perturbation, Symposium (International) on Combustion, 22 (1989), 931-941.
doi: 10.1016/S0082-0784(89)80102-X. |
[9] |
B. Letson and J. Rubin,
A new frame for an old (phase) portrait: Finding rivers and other flow features in the plane, SIAM J. Appl. Dyn. Syst., 17 (2018), 2414-2445.
doi: 10.1137/18M1186617. |
[10] |
A. Mauroy, I. Mezi\'c and J. Moehlis,
Isostables isoschrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics, Phys. D, 261 (2013), 19-30.
doi: 10.1016/j.physd.2013.06.004. |
[11] |
S. Revzen and J. M. Guckenheimer, Estimating the phase of synchronized oscillators, Phys. Rev. E, 78 (2008), 051907, 12pp.
doi: 10.1103/PhysRevE.78.051907. |
[12] |
S. Shirasaka, W. Kurebayashi and H. Nakao, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, Chaos, 27 (2017), 023119, 7pp.
doi: 10.1063/1.4977195. |
[13] |
P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[14] |
T. Vo and M. Wechselberger,
Canards of folded saddle-node type {I}, SIAM J. Math. Anal., 47 (2015), 3235-3283.
doi: 10.1137/140965818. |
[15] |
K. C. A. Wedgwood, K. K. Lin, R. Thul and S. Coombes, Phase-amplitude descriptions of neural oscillator models, J. Math. Neurosci., 3 (2013), Art. 2, 22 pp.
doi: 10.1186/2190-8567-3-2. |
[16] |
D. Wilson and J. Moehlis, Isostable reduction of periodic orbits, Phys. Rev. E, 94 (2016), 052213.
doi: 10.1103/PhysRevE.94.052213. |
[17] |
D. Wilson and B. Ermentrout,
Greater accuracy and broadened applicability of phase reduction using isostable coordinates, J. Math. Biol., 76 (2018), 37-66.
doi: 10.1007/s00285-017-1141-6. |
[18] |
A. Zagaris, C. Vandekerckhove, W. C. Gear, T. J. Kaper, I. G. Kevrekidis and G. Ioannis,
Stability and stabilization of the contrained runs schemes for equation-free projection to a slow manifold, Discrete Contin. Dyn. Syst., 32 (2012), 2759-2803.
doi: 10.3934/dcds.2012.32.2759. |



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