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 [
Citation: |
[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.![]() ![]() ![]() |