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September  2021, 26(9): 4839-4865. doi: 10.3934/dcdsb.2020315

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

Peter Giesl

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

Received  June 2020 Revised  September 2020 Published  September 2021 Early access  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 [7]. In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.

Keywords: Periodic orbit, stability, contraction metric, converse theorem, matrix-valued partial differential equation, existence, uniqueness.
Mathematics Subject Classification: Primary: 34C25, 34D20; Secondary: 37C27.
Citation: Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4839-4865. doi: 10.3934/dcdsb.2020315
References:
[1]

V. A. Boĭchenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10.

[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, Texts in Applied Mathematics, 34. Springer, New York, 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, 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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[11]

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.

[12]

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. 

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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

show all references

References:
[1]

V. A. Boĭchenko and G. A. Leonov, Lyapunov orbital exponents of autonomous systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 3 (1988), 7–10.

[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, Texts in Applied Mathematics, 34. Springer, New York, 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, 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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[11]

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.

[12]

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. 

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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

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