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The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems. (English) Zbl 1405.37054
Summary: The asymptotic phase \(\theta\) of an initial point \(x\) in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow \(\phi_t(x)\) converges as \(t\to\infty\). The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient \(\nabla_x(\theta)\) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, \(M\), can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34A36 Discontinuous ordinary differential equations
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[1] Acary, V.; De Jong, H.; Brogliato, B., Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems., Physica D: Nonlinear Phenom., 269, 103-119, (2014) · Zbl 1402.92194
[2] Aizerman, M.; Gantmakher, F., On the stability of periodic motions, J. Appl. Math. Mech., 22, 1065-1078, (1958) · Zbl 0086.29102
[3] Bizzarri, F.; Linaro, D.; Storace, M., (2007)
[4] Brown, E.; Moehlis, J.; Holmes, P., On the phase reduction and response dynamics of neural oscillator populations, Neural Comput., 16, 673-715, (2004) · Zbl 1054.92006
[5] Carmona, V.; Fernández-García, S.; Freire, E.; Torres, F., Melnikov theory for a class of planar hybrid systems, Physica D: Nonlinear Phenom., 248, 44-54, (2013) · Zbl 1337.37021
[6] Cheng, Y., Bifurcation of limit cycles of a class of piecewise linear differential systems in with three zones, Discrete Dyn. Nat. Soc., 2013, (2013)
[7] Coombes, S., Phase locking in networks of synaptically coupled McKean relaxation oscillators, Physica D: Nonlinear Phenom., 160, 173-188, (2001) · Zbl 1013.92005
[8] Coombes, S., Neuronal networks with gap junctions: A study of piecewise linear planar neuron models, SIAM J. Appl. Dyn. Syst., 7, 1101-1129, (2008) · Zbl 1159.92008
[9] Coombes, S.; Thul, R., Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function, Eur. J. Appl. Math., 27, 904-922, (2016) · Zbl 1384.34064
[10] Coombes, S.; Thul, R.; Wedgwood, K., Nonsmooth dynamics in spiking neuron models, Physica D: Nonlinear Phenom., 241, 2042-2057, (2012)
[11] Di Bernardo, M.; Budd, C.; Champneys, A. R.; Kowalczyk, P., Piecewise-Smooth Dynamical Systems: Theory and Applications, (2008), Springer Science & Business Media: Springer Science & Business Media, London · Zbl 1146.37003
[12] Edwards, R.; Gill, P., On synchronization and cross-talk in parallel networks., Dynamics of Continuous Discrete and Impulsive Systems Series B, 10, 287-300, (2003) · Zbl 1043.34051
[13] Ermentrout, B., Type I membranes, phase resetting curves, and synchrony, Neural Comput., 8, 979-1001, (1996)
[14] Ermentrout, G. B.; Terman, D. H., Foundations of Mathematical Neuroscience, (2010), Springer: Springer, Berlin, Germany · Zbl 1320.92002
[15] Ermentrout, G.; Kopell, N., Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29, (1991) · Zbl 0718.92004
[16] Field, R. J.; Noyes, R. M., Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60, 1877-1884, (1974)
[17] Filippov, A. F., Differential Equations with Discontinuous Righthand Sides, (1988), Kluwer Academic Publishers: Kluwer Academic Publishers, Dordrecht, The Netherlands
[18] Fruth, F.; Jülicher, F.; Lindner, B., An active oscillator model describes the statistics of spontaneous otoacoustic emissions, Biophys. J., 107, 815-824, (2014)
[19] Glass, L.; Pasternack, J. S., Stable oscillations in mathematical models of biological control systems., J. Math. Biol., 6, 207-223, (1978) · Zbl 0391.92001
[20] Glass, L.; Pérez, R., Limit cycle oscillations in compartmental chemical systems, J. Chem. Phys., 61, 5242-5249, (1974)
[21] Guckenheimer, J., Isochrons and phaseless sets, J. Math. Biol., 1, 259-273, (1975) · Zbl 0345.92001
[22] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (1990), Springer-Verlag: Springer-Verlag, Berlin, Germany
[23] Hawkins, J., Automatic regulators for heating apparatus., Trans. Amer. Soc. Mech. Eng., 9, 432, (1887)
[24] Huan, S.-M.; Yang, X.-S., On the number of limit cycles in general planar piecewise linear systems, Discrete Continuous Dyn. Syst., 32, 2147-2164, (2012) · Zbl 1248.34033
[25] Ijspeert, A. J., Central pattern generators for locomotion control in animals and robots: A review, Neural Netw., 21, 642, (2008)
[26] Izhikevich, E. M., Phase equations for relaxation oscillators, SIAM J. Appl. Math., 60, 1789-1804, (2000) · Zbl 1016.92001
[27] Izhikevich, E. M., Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, (2007), MIT Press: MIT Press, Cambridge, Massachusetts
[28] Izhikevich, E. M.; Ermentrout, B., Phase model, Scholarpedia, 3, 1487, (2008)
[29] Johansson, K. H.; Barabanov, A. E.; Åström, K. J., Limit cycles with chattering in relay feedback systems, IEEE Trans. Autom. Control, 47, 1414-1423, (2002) · Zbl 1364.93341
[30] Kelso, J. S.; Holt, K. G.; Rubin, P.; Kugler, P. N., Patterns of human interlimb coordination emerge from the properties of non-linear, limit cycle oscillatory processes: theory and data, J. Motor Behavior, 13, 226-261, (1981)
[31] Kuramoto, Y., Chemical Oscillations Waves and Turbulence, (2003), Dover: Dover, Mineola, New York, USA · Zbl 0558.76051
[32] Leine, R.; Nijmeijer, H., Dynamics and Bifurcations of Non-Smooth Mechanical Systems, (2004), Springer-Verlag: Springer-Verlag, Heidelberg, Germany · Zbl 1068.70003
[33] Lin, C.; Wang, Q.-G.; Lee, T. H., Local stability of limit cycles for MIMO relay feedback systems, J. Math. Anal. Appl., 288, 112-123, (2003) · Zbl 1109.93330
[34] Lin, H.; Antsaklis, P. J., Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control, 54, 308-322, (2009) · Zbl 1367.93440
[35] Lin, K. K.; Wedgwood, K. C. A.; Coombes, S.; Young, L.-S., Limitations of perturbative techniques in the analysis of rhythms and oscillations., J. Math. Biol, 66, 139-161, (2012) · Zbl 1256.92006
[36] Llibre, J.; Ponce, E., Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19, 325-335, (2012) · Zbl 1268.34061
[37] Lyttle, D. N.; Gill, J. P.; Shaw, K. M.; Thomas, P. J.; Chiel, H. J., Robustness, flexibility, and sensitivity in a multifunctional motor control model, Biol. Cybern., 111, 25-47, (2017)
[38] Ma, Y.; Yuan, R.; Li, Y.; Ao, P.; Yuan, B., (2013)
[39] Machina, A.; Ponosov, A., Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks, Nonlinear Anal.: Theor. Methods Appl., 74, 882-900, (2011) · Zbl 1213.34022
[40] Mckean, H., Nagumo’s equation., Adv. Math., 4, 209-223, (1970) · Zbl 0202.16203
[41] Modolo, J.; Henry, J.; Beuter, A., Dynamics of the subthalamo-pallidal complex in Parkinsonõs disease during deep brain stimulation, J. Biol. Phys., 34, 251-266, (2008)
[42] Morrison, K.; Degeratu, A.; Itskov, V.; Curto, C., (2016)
[43] Morse, A., (1997)
[44] Nowotny, T.; Rabinovich, M. I., Dynamical origin of independent spiking and bursting activity in neural microcircuits., Phys. Rev. Lett., 98, 128106, (2007)
[45] Osinga, H. M.; Moehlis, J., Continuation-based computation of global isochrons, SIAM J. Appl. Dyn. Syst., 9, 1201-1228, (2010) · Zbl 1232.37014
[46] Park, Y., (2013)
[47] Park, Y.; Ermentrout, B., Weakly coupled oscillators in a slowly varying world, J. Comput. Neurosci., 40, 269-281, (2016) · Zbl 1382.93017
[48] Pettit, N. B., Analysis of Piecewise Linear Dynamical Systems, (1996), John Wiley & Sons: John Wiley & Sons, New York, New York, USA · Zbl 0875.93202
[49] Pettit, N. B.; Wellstead, P. E., Analyzing piecewise linear dynamical systems, Control Systems, IEEE, 15, 43-50, (1995)
[50] Poggi, T.; Sciutto, A.; Storace, M., Piecewise linear implementation of nonlinear dynamical systems: From theory to practice, Electron. Lett., 45, 966-967, (2009)
[51] Ponce, E.; Ros, J.; Vela, E.; Ibáñez, S.; Pérez Del Río, J. S.; Pumariño, A.; Rodríguez, J. Á., Progress and Challenges in Dynamical Systems, The focus-center-limit cycle bifurcation in discontinuous planar piecewise linear systems without sliding, 335-349, (2013), Springer: Springer, Berlin Heidelberg
[52] Rohden, M.; Sorge, A.; Timme, M.; Witthaut, D., Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109, 064101, (2012)
[53] Schwemmer, M. A.; Lewis, T. J., Phase Response Curves in Neuroscience Theory, Experiment, and Analysis, The Theory of Weakly Coupled Oscillators, 3-31, (2012), Springer
[54] Shaw, K. M.; Lyttle, D. N.; Gill, J. P.; Cullins, M. J.; Mcmanus, J. M.; Lu, H.; Thomas, P. J.; Chiel, H. J., The significance of dynamical architecture for adaptive responses to mechanical loads during rhythmic behavior, J. Comput. Neurosci., 38, 25-51, (2015)
[55] Shaw, K. M.; Park, Y.; Chiel, H. J.; Thomas, P. J., Phase resetting in an asymptotically phaseless system: On the phase response of limit cycles verging on a heteroclinic orbit, SIAM J. Appl. Dyn. Syst., 11, 350-391, (2012) · Zbl 1242.34088
[56] Shirasaka, S.; Kurebayashi, W.; Nakao, H., Phase reduction theory for hybrid nonlinear oscillators, Phys. Rev. E, 95, (2017)
[57] Simpson, D. J. W.; Jeffrey, M. R., Fast phase randomization via two-folds, Proc. R. Soc. A: Math. Phys. Eng. Sci., 472, 20150782, (2016) · Zbl 1371.37106
[58] So, P.; Francis, J. T.; Netoff, T.; Schiff, S. J., Periodic orbits: a new language for neuronal dynamics, Biophys. J., 74, 2776-2785, (1998)
[59] Stensby, J. L., Phase-Locked Loops: Theory and Applications, (1997), CRC Press: CRC Press, Boca Raton, Florida
[60] Storace, M.; De Feo, O., Piecewise-linear approximation of nonlinear dynamical systems, IEEE Trans. Circuits Syst. I: Regular Papers, 51, 830-842, (2004) · Zbl 1374.37069
[61] Tsypkin, I. Z., Relay Control Systems, (1984), Cambridge University Press: Cambridge University Press, Cambridge, UK · Zbl 0571.93001
[62] G. Aiko, V. A.; Van Horssen, W. T., Global analysis of a piecewise linear Liénard-type dynamical system, Int. J. Dyn. Syst. Differ. Equ., 2, 115-128, (2009) · Zbl 1198.34045
[63] Walsh, J.; Widiasih, E.; Hahn, J.; Mcgehee, R., Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles, Nonlinearity, 29, 1843, (2016) · Zbl 1364.37172
[64] Zinovik, I.; Chebiryak, Y.; Kroening, D., Periodic orbits and equilibria in Glass models for gene regulatory networks, IEEE Trans. Inform. Theory, 56, 805-820, (2010) · Zbl 1366.92052
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