# American Institute of Mathematical Sciences

September  2019, 39(9): 5339-5363. doi: 10.3934/dcds.2019218

## Converse theorem on a global contraction metric for a periodic orbit

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

Received  October 2018 Revised  February 2019 Published  May 2019

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable and we obtain an upper bound on the rate of exponential attraction.

In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the upper bound on the rate of exponential attraction.

Citation: Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
##### References:
 [1] V. A. Boichenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10,123.  Google Scholar [2] 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 [3] C. Chicone, Ordinary Differential Equations with Applications, New York: Springer-Verlag, 2006.  Google Scholar [4] 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 [5] P. Giesl, On a matrix-valued PDE characterizing a contraction metric for a periodic orbit, Submitted. Google Scholar [6] 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 [7] 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 [8] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.  Google Scholar [9] 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 [10] A. Y. Kravchuk, G. A. Leonov and D. V. Ponomarenko, Criteria for strong orbital stability of trajectories of dynamical systems. Ⅰ, Differentsialnye Uravneniya, 28 (1992), 1507–1520, 1652.  Google Scholar [11] G. A. Leonov, I. M. Burkin and A. I. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl.: Vol. 357, Kluwer, 1996. doi: 10.1007/978-94-009-0193-3.  Google Scholar [12] W. Lohmiller and J.-J. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar [13] B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.  Google Scholar [14] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4312-0.  Google Scholar

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##### References:
 [1] V. A. Boichenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10,123.  Google Scholar [2] 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 [3] C. Chicone, Ordinary Differential Equations with Applications, New York: Springer-Verlag, 2006.  Google Scholar [4] 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 [5] P. Giesl, On a matrix-valued PDE characterizing a contraction metric for a periodic orbit, Submitted. Google Scholar [6] 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 [7] 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 [8] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.  Google Scholar [9] 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 [10] A. Y. Kravchuk, G. A. Leonov and D. V. Ponomarenko, Criteria for strong orbital stability of trajectories of dynamical systems. Ⅰ, Differentsialnye Uravneniya, 28 (1992), 1507–1520, 1652.  Google Scholar [11] G. A. Leonov, I. M. Burkin and A. I. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl.: Vol. 357, Kluwer, 1996. doi: 10.1007/978-94-009-0193-3.  Google Scholar [12] W. Lohmiller and J.-J. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar [13] B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.  Google Scholar [14] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4312-0.  Google Scholar
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