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March  2022, 21(3): 999-1025. doi: 10.3934/cpaa.2022007

## Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

 1 Laboratoire M2N, CNAM, 292 rue Saint-Martin, 75003 Paris, France 2 Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B. P. 51, 2070 La Marsa, Tunisia 3 Laboratoire Équations aux Dérivées Partielles, LR03ES04, Faculté des sciences de Tunis, Université Tunis El Manar, 2092 El Manar, Tunisia

*Corresponding author

Received  August 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

Fund Project: The first author wishes to thanks the organizers of ICAAM 2019 in Hammamet, Tunisia, during which mathematical discussions led to working on the subject of this paper. The second author wishes to thank the department of mathematics and statistics EPN6 and the research department M2N (EA7340) of the CNAM where this work has been initiated

In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.

Citation: Thierry Horsin, Mohamed Ali Jendoubi. Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 999-1025. doi: 10.3934/cpaa.2022007
##### References:
 [1] P. A. Absil., R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.  doi: 10.1137/040605266. [2] N. Alaa and M. Pierre, Convergence to equilibrium for discretized gradient-like systems with analytic features, IMA J. Numer. Anal., 33 (2013), 1291-1321.  doi: 10.1093/imanum/drs042. [3] F. Aloui, I. Ben Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equations and applications, J. Math. Pures Appl., 100 (2013), 295-326.  doi: 10.1016/j.matpur.2013.01.002. [4] H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program. Ser. B, 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5. [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differ. Equ., 20 (2008), 643-652.  doi: 10.1007/s10884-007-9099-5. [6] R. Chill, A. Haraux and M. A. Jendoubi, Applications of the Lojasiewicz-Simon, gradient inequality to gradient-like evolution equations, Anal. Appl. Singap., 7 (2009), 351-372.  doi: 10.1142/S0219530509001438. [7] M. Crouzeix and A. Mignot, Analyse numérique des équations différentielles, in Collection Mathématiques Appliquées pour la Maîitrise, Paris, Masson, 1984. [8] D. D'Acunto and K. Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Pol. Math., 87 (2005), 51-61.  doi: 10.4064/ap87-0-5. [9] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems, Commun. Pure Appl. Anal., 11 (2012), 2393-2416.  doi: 10.3934/cpaa.2012.11.2393. [10] A. Haraux and M. A. Jendoubi, The convergence problem for dissipative autonomous systems - classical methods and recent advances, in Springer Briefs in Mathematics, Springer, Bcam, 2015. doi: 10.1007/978-3-319-23407-6. [11] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differ. Equ., 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393. [12] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asympt. Anal., 26 (2001), 21-36. [13] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evol. Equ. Contr. Theor., 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461. [14] T. Horsin and M. A. Jendoubi, Non-genericity of initial data with punctual $\omega$-limit set, Arch. Math., 114 (2020), 185-193.  doi: 10.1007/s00013-019-01377-8. [15] M. A. Jendoubi and P. Polacik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. R. Soc. Edinb., Sect. A, Math., 133 (2003), 1137-1153.  doi: 10.1017/S0308210500002845. [16] S. Lojasiewicz, Une propriété topologique des sous ensembles analytiques réels, Les équations aux dérivées partielles, (1963), 87–89. [17] S. Lojasiewicz, Ensembles semi-analytiques, I. H. E. S., notes, 1965. [18] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685. [19] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New-York, 1982.

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##### References:
 [1] P. A. Absil., R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.  doi: 10.1137/040605266. [2] N. Alaa and M. Pierre, Convergence to equilibrium for discretized gradient-like systems with analytic features, IMA J. Numer. Anal., 33 (2013), 1291-1321.  doi: 10.1093/imanum/drs042. [3] F. Aloui, I. Ben Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equations and applications, J. Math. Pures Appl., 100 (2013), 295-326.  doi: 10.1016/j.matpur.2013.01.002. [4] H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program. Ser. B, 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5. [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differ. Equ., 20 (2008), 643-652.  doi: 10.1007/s10884-007-9099-5. [6] R. Chill, A. Haraux and M. A. Jendoubi, Applications of the Lojasiewicz-Simon, gradient inequality to gradient-like evolution equations, Anal. Appl. Singap., 7 (2009), 351-372.  doi: 10.1142/S0219530509001438. [7] M. Crouzeix and A. Mignot, Analyse numérique des équations différentielles, in Collection Mathématiques Appliquées pour la Maîitrise, Paris, Masson, 1984. [8] D. D'Acunto and K. Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Pol. Math., 87 (2005), 51-61.  doi: 10.4064/ap87-0-5. [9] M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems, Commun. Pure Appl. Anal., 11 (2012), 2393-2416.  doi: 10.3934/cpaa.2012.11.2393. [10] A. Haraux and M. A. Jendoubi, The convergence problem for dissipative autonomous systems - classical methods and recent advances, in Springer Briefs in Mathematics, Springer, Bcam, 2015. doi: 10.1007/978-3-319-23407-6. [11] A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differ. Equ., 144 (1998), 313-320.  doi: 10.1006/jdeq.1997.3393. [12] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asympt. Anal., 26 (2001), 21-36. [13] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evol. Equ. Contr. Theor., 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461. [14] T. Horsin and M. A. Jendoubi, Non-genericity of initial data with punctual $\omega$-limit set, Arch. Math., 114 (2020), 185-193.  doi: 10.1007/s00013-019-01377-8. [15] M. A. Jendoubi and P. Polacik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. R. Soc. Edinb., Sect. A, Math., 133 (2003), 1137-1153.  doi: 10.1017/S0308210500002845. [16] S. Lojasiewicz, Une propriété topologique des sous ensembles analytiques réels, Les équations aux dérivées partielles, (1963), 87–89. [17] S. Lojasiewicz, Ensembles semi-analytiques, I. H. E. S., notes, 1965. [18] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685. [19] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New-York, 1982.
Simulation for (5.1). Here $\alpha = 0.4$ and $\Delta t = 0.01$, the dotted curve is the apparently decreasing relative energy $E(u_n,v_n)/E(u_0,v_0)$. The solid curve is the boundary of the zero level of $F$ (which is in its interior). Convergence occurs to a critical point of $F$ which may be in the interior of the zero level set of $F$
$\alpha = 0.4$ and $\Delta t = 0.01$, the dotted curve is the value of the velocity $v_n$. The solid curve describes the coordinates of $u_n$. The dash-dotted curve corresponds to the sequence $(u_n,F(u_n,v_n))$
Simulation for (5.1). Here $\alpha = 0.4$ and $\Delta t = 0.1$, the dotted curve is the relative energy $E(u_n,v_n)/E(u_0,v_0)$. The solid curve is the boundary of the zero level of $F$ (which is in its interior). Convergence occurs to a critical point of $F$ which seems to be on the boundary of the zero level of $F$
Simulation for (5.1). Here $\alpha = 0.4$ and $\Delta t = 0.1$, the dotted curve is the value of the velocity $v_n$. The solid curve describes the coordinates of $u_n$. The dash-dotted curve corresponds to the sequence $(u_n,F(u_n,v_n))$
Simulation for (5.2). Here $\alpha = 0.4$ and $\Delta t = 0.01$, the dotted curve is the relative energy $E(u_n,v_n)/E(u_0,v_0)$ it is apparently decreasing. The solid curve is the boundary of the zero level of $F$ (which is in its interior). Convergence occurs to a critical point of $F$ which may be in the interior the zero level of $F$
$\alpha = 0.4$ and $\Delta t = 0.01$, the dotted curve is the value of the velocity $v_n$. The solid curve describes the coordinates of $u_n$. The dash-dotted curve corresponds to the sequence $(u_n,F(u_n,v_n))$
Simulation for (5.2). Here $\alpha = 0.4$ and $\Delta t = 0.1$, the dotted curve is the relative energy $E(u_n,v_n)/E(u_0,v_0)$. The solid curve is the boundary of the zero level of $F$ (which is in its interior). Convergence occurs to a critical point of $F$ which seems to be on the boundary of the zero level of $F$
Simulation for (5.2). Here $\alpha = 0.4$ and $\Delta t = 0.1$, the dotted curve is the value of the velocity $v_n$. The solid curve describes the coordinates of $u_n$. The dash-dotted curve corresponds to the sequence $(u_n,F(u_n,v_n))$
$\alpha = 0.4$. We again observe the numerical convergence, but have no proof yet
$\alpha = 0.4$
$\alpha = 0.5$
$\alpha = 1.5$
$\alpha = 1.5$
$\alpha = 0.5$, $T = 9/16$, $f(u) = u-\sin(u)$
$\alpha = 0.5$, $T = 9/8$, $f(u) = u-\sin(u)$
$\alpha = 0.5$, $T = 45/8$, $f(u) = u-\sin(u)$. Here we can see some oscillations appearing, though there seem to have convergence to $0$
$\alpha = 0.5$, $T = 9/16$, $f(u) = u-\sin(u)$
$\alpha = 0.5$, $T = 45/8$, $f(u) = u-\sin(u)$
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