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 and Continuous Dynamical Systems, 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.

[2]

G. Borg, A condition for the existence of orbitally stable solutions of dynamical systems, Kungl. Tekn. Högsk. Handl. Stockholm, 153 (1960), 12 pp.

[3]

C. Chicone, Ordinary Differential Equations with Applications, New York: Springer-Verlag, 2006.

[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.

[5]

P. Giesl, On a matrix-valued PDE characterizing a contraction metric for a periodic orbit, Submitted.

[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.

[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.

[8]

P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.

[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.

[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.

[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.

[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.

[13]

B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.

[14]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4312-0.

show all references

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.

[2]

G. Borg, A condition for the existence of orbitally stable solutions of dynamical systems, Kungl. Tekn. Högsk. Handl. Stockholm, 153 (1960), 12 pp.

[3]

C. Chicone, Ordinary Differential Equations with Applications, New York: Springer-Verlag, 2006.

[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.

[5]

P. Giesl, On a matrix-valued PDE characterizing a contraction metric for a periodic orbit, Submitted.

[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.

[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.

[8]

P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.

[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.

[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.

[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.

[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.

[13]

B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155.  doi: 10.7146/math.scand.a-10661.

[14]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4312-0.

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