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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

* Corresponding author: Jonathan E. Rubin

Received  April 2019 Revised  December 2019 Published  April 2020

Fund Project: The authors received partial support from NSF award DMS 1612913.

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.

Citation: Benjamin Letson, Jonathan E. Rubin. Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020088
##### References:
 [1] E. Beno\^it, M. Br{\o}ns, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] W. Kühnel, Differential Geometry: Curves, Surfaces, Manifolds, 2$^nd$ edition, Student Mathematical Library: Vol. 16, American Mathematical Society, 2005. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] S. Shirasaka, W. Kurebayashi and H. Nakao, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, \emphChaos, 27 (2017), 023119, 7pp. doi: 10.1063/1.4977195.  Google Scholar [13] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.  doi: 10.1006/jdeq.2001.4001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] D. Wilson and J. Moehlis, Isostable reduction of periodic orbits, Phys. Rev. E, 94 (2016), 052213. doi: 10.1103/PhysRevE.94.052213.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] E. Beno\^it, M. Br{\o}ns, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] W. Kühnel, Differential Geometry: Curves, Surfaces, Manifolds, 2$^nd$ edition, Student Mathematical Library: Vol. 16, American Mathematical Society, 2005. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] S. Shirasaka, W. Kurebayashi and H. Nakao, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, \emphChaos, 27 (2017), 023119, 7pp. doi: 10.1063/1.4977195.  Google Scholar [13] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.  doi: 10.1006/jdeq.2001.4001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] D. Wilson and J. Moehlis, Isostable reduction of periodic orbits, Phys. Rev. E, 94 (2016), 052213. doi: 10.1103/PhysRevE.94.052213.  Google Scholar [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.  Google Scholar [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.  Google Scholar
The geometric setup for Local Orthogonal Rectification. We consider an inital condition $x_0$ near a given manifold, and decompose the trajectory through $x_0$, denoted by $\phi$, into a curve on the manifold and a curve in the normal bundle to the manifold
(Left) The dynamics on $\left\{ \hat{\xi }=0 \right\}$ of system (40) for $\mu=3$. Note how the approximate trajectories organize around the orange curve, called $\gamma$ in the text. (Right) A plot of the rivers of (35), three of which cross the fold of the critical manifold. The orange curve, $\gamma(z)$, is indistinguishable from the identified approximate canard solution $\Psi_2(\gamma(t),0)$
The dynamics near the critical manifold for $\mu=3$. (Left) The trapping region detailed in Prop. 3, between $\left\{ \xi =0 \right\}$ and the correction $\Xi$. Note how trajectories with $\eta(t)\in\mathcal{F}$ cannot escape, as they are bounded above by $\Xi$. (Right) The full trapping region, where the $\mathcal{O}( \varepsilon^2)$ term of (39) is negative. Note the twisting of orbits as they escape from the influence of the correction
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