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

Thierry Horsin; Mohamed Ali Jendoubi

doi:10.3934/cpaa.2022007

Article Contents

2022, Volume 21, Issue 3: 999-1025. Doi: 10.3934/cpaa.2022007

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

Thierry Horsin^1, , and
Mohamed Ali Jendoubi^2,3,

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
^*Corresponding author

Received: July 31, 2021

Revised: October 31, 2021

Early access: December 2021

Published: March 2022

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

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Abstract

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.

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Mathematics Subject Classification: 34E10, 37N30, 40A05, 35L05, 35L70, 65L07.

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Figure 1. 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 $

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Figure 2. $ \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)) $

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Figure 3. 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 $

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Figure 4. 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)) $

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Figure 5. 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 $

Download: Full-size image PowerPoint slide

Figure 6. $ \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)) $

Download: Full-size image PowerPoint slide

Figure 7. 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 $

Download: Full-size image PowerPoint slide

Figure 8. 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)) $

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Figure 9. $ \alpha = 0.4 $. We again observe the numerical convergence, but have no proof yet

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Figure 10. $ \alpha = 0.4 $

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Figure 11. $ \alpha = 0.5 $

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Figure 12. $ \alpha = 1.5 $

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Figure 13. $ \alpha = 1.5 $

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Figure 14. $ \alpha = 0.5 $, $ T = 9/16 $, $ f(u) = u-\sin(u) $

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Figure 15. $ \alpha = 0.5 $, $ T = 9/8 $, $ f(u) = u-\sin(u) $

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Figure 16. $ \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 $

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Figure 17. $ \alpha = 0.5 $, $ T = 9/16 $, $ f(u) = u-\sin(u) $

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Figure 18. $ \alpha = 0.5 $, $ T = 45/8 $, $ f(u) = u-\sin(u) $

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