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

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  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.

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. KravchukG. 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. KravchukG. 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.

[1]

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

[2]

Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310

[3]

Demetris Hadjiloucas. Stochastic matrix-valued cocycles and non-homogeneous Markov chains. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 731-738. doi: 10.3934/dcds.2007.17.731

[4]

Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

[5]

Daniel Alpay, Eduard Tsekanovskiĭ. Subclasses of Herglotz-Nevanlinna matrix-valued functtons and linear systems. Conference Publications, 2001, 2001 (Special) : 1-13. doi: 10.3934/proc.2001.2001.1

[6]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793

[7]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657

[8]

Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409

[9]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252

[10]

Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3345-3374. doi: 10.3934/dcdsb.2021188

[11]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[12]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385

[13]

Peter Giesl, Holger Wendland. Construction of a contraction metric by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3843-3863. doi: 10.3934/dcdsb.2018333

[14]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[15]

Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations and Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185

[16]

Jaime Angulo Pava, Borys Alvarez Samaniego. Existence and stability of periodic travelling-wavesolutions of the Benjamin equation. Communications on Pure and Applied Analysis, 2005, 4 (2) : 367-388. doi: 10.3934/cpaa.2005.4.367

[17]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[18]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

[19]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[20]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (153)
  • HTML views (260)
  • Cited by (0)

Other articles
by authors

[Back to Top]