    ## On a matrix-valued PDE characterizing a contraction metric for a periodic orbit

 Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom

Received  June 2020 Revised  September 2020 Published  October 2020

The stability and the basin of attraction of a periodic orbit can be determined using a contraction metric, i.e., a Riemannian metric with respect to which adjacent solutions contract. A contraction metric does not require knowledge of the position of the periodic orbit and is robust to perturbations.

In this paper we characterize such a Riemannian contraction metric as matrix-valued solution of a linear first-order Partial Differential Equation. This enables the explicit construction of a contraction metric by numerically solving this equation in . In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.

Citation: Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020315
##### References:
  V. A. Boĭchenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10. Google Scholar  G. Borg, A condition for the existence of orbitally stable solutions of dynamical systems, Kungl. Tekn. Högsk. Handl. Stockholm, 153 (1960), 12 pp. Google Scholar  C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34. Springer, New York, 2006. Google Scholar  F. Forni and R. Sepulchre, A differential Lyapunov framework for contraction analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.  doi: 10.1109/TAC.2013.2285771.  Google Scholar  P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677.  doi: 10.1016/j.na.2003.07.020.  Google Scholar  P. Giesl, Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403.  doi: 10.1016/j.jmaa.2014.12.010.  Google Scholar  P. Giesl, Computation of a contraction metric for a periodic orbit using meshfree collocation, SIAM J. Appl. Dyn. Syst., 18 (2019), 1536-1564.  doi: 10.1137/18M1220182.  Google Scholar  P. Giesl, Converse theorem on a global contraction metric for a periodic orbit, Discrete Cont. Dyn. Syst., 39 (2019), 5339-5363.  doi: 10.3934/dcds.2019218.  Google Scholar  P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.  doi: 10.1137/16M1092842.  Google Scholar  P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. Google Scholar  P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178.  doi: 10.2307/1993939.  Google Scholar  A. Yu. Kravchuk, G. A. Leonov and D. V. Ponomarenko, Criteria for strong orbital stability of trajectories of dynamical systems. I, Differentsial'nye Uravneniya, 28 (1992), 1507-1520. Google Scholar  G. A. Leonov, On stability with respect to the first approximation, Prikl. Mat. Mekh., 62 (1998), 548-555.  doi: 10.1016/S0021-8928(98)00067-7.  Google Scholar  G. A. Leonov, I. M. Burkin and A. I. Shepelyavyi, Frequency Methods in Oscillation Theory, Mathematics and its Applications, 357. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-0193-3.  Google Scholar  W. Lohmiller and J.-J. E. Slotine, On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar  Ian R. Manchester and J.-J. E. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38.  doi: 10.1016/j.sysconle.2013.10.005.  Google Scholar  G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar  B. T. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.  Google Scholar

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##### References:
  V. A. Boĭchenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10. Google Scholar  G. Borg, A condition for the existence of orbitally stable solutions of dynamical systems, Kungl. Tekn. Högsk. Handl. Stockholm, 153 (1960), 12 pp. Google Scholar  C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34. Springer, New York, 2006. Google Scholar  F. Forni and R. Sepulchre, A differential Lyapunov framework for contraction analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.  doi: 10.1109/TAC.2013.2285771.  Google Scholar  P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677.  doi: 10.1016/j.na.2003.07.020.  Google Scholar  P. Giesl, Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403.  doi: 10.1016/j.jmaa.2014.12.010.  Google Scholar  P. Giesl, Computation of a contraction metric for a periodic orbit using meshfree collocation, SIAM J. Appl. Dyn. Syst., 18 (2019), 1536-1564.  doi: 10.1137/18M1220182.  Google Scholar  P. Giesl, Converse theorem on a global contraction metric for a periodic orbit, Discrete Cont. Dyn. Syst., 39 (2019), 5339-5363.  doi: 10.3934/dcds.2019218.  Google Scholar  P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.  doi: 10.1137/16M1092842.  Google Scholar  P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. Google Scholar  P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178.  doi: 10.2307/1993939.  Google Scholar  A. Yu. Kravchuk, G. A. Leonov and D. V. Ponomarenko, Criteria for strong orbital stability of trajectories of dynamical systems. I, Differentsial'nye Uravneniya, 28 (1992), 1507-1520. Google Scholar  G. A. Leonov, On stability with respect to the first approximation, Prikl. Mat. Mekh., 62 (1998), 548-555.  doi: 10.1016/S0021-8928(98)00067-7.  Google Scholar  G. A. Leonov, I. M. Burkin and A. I. Shepelyavyi, Frequency Methods in Oscillation Theory, Mathematics and its Applications, 357. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-0193-3.  Google Scholar  W. Lohmiller and J.-J. E. Slotine, On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar  Ian R. Manchester and J.-J. E. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38.  doi: 10.1016/j.sysconle.2013.10.005.  Google Scholar  G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar  B. T. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.  Google Scholar
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