# American Institute of Mathematical Sciences

September  2018, 23(7): 2743-2762. doi: 10.3934/dcdsb.2018103

## Periodic orbits of planar discontinuous system under discretization

 1 School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA 2 Dept. of Mathematics and Systems Analysis, Aalto University, Espoo, FI-00076, Finland 3 Dipartimento di Matematica, Univ. of Bari, I-70100, Bari, Italy

Received  February 2017 Revised  November 2017 Published  September 2018 Early access  March 2018

Fund Project: Discussion on this work begun while the last two authors were visiting the School of Mathematics of Georgia Tech in Spring 2015, and it is dedicated to the memory of our beloved friend and co-author Timo Eirola who unfortunately never saw the end of this work.

We consider a model planar system with discontinuous right-hand side possessing an attracting periodic orbit, and we investigate what happens to a Euler discretization with stepsize $τ$ of this system. We show that, in general, the resulting discrete dynamical system does not possess an invariant curve, in sharp contrast to what happens for smooth problems. In our context, we show that the numerical trajectories are forced to remain inside a band, whose width is proportional to the discretization stepsize $τ$. We further show that if we consider an event-driven discretization of the model problem, whereby the solution is forced to step exactly on the discontinuity line, then there is a discrete periodic solution near the one of the original problem (for sufficiently small $τ$). Finally, we consider what happens to the Euler discretization of the regularized system rewritten in polar coordinates, and give numerical evidence that the discrete solution now undergoes a period doubling cascade with respect to the regularization parameter $\epsilon$, for fixed $τ$.

Citation: Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103
##### References:
 [1] K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos, Springer-Verlag, New York, 1997. [2] W.-J. Beyn, On invariant closed curves for one-step methods, Numerische Mathematik, 51 (1987), 103-122.  doi: 10.1007/BF01399697. [3] L. Dieci, C. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, DCDS-B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165. [4] L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth dynamical systems with a line of discontinuity, Journal of Dynamics and Differential Equations, 26 (2104), 1049-1078.  doi: 10.1007/s10884-014-9380-3. [5] L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous right-hand side, Journal of Computational and Applied Mathematics, 236 (2012), 3967-3991.  doi: 10.1016/j.cam.2012.02.011. [6] T. Eirola, Invariant curves of one-step methods, BIT, 28 (1988), 113-122.  doi: 10.1007/BF01934699. [7] J. Sotomayor and A. L. Machado, Structurally stable discontinuous vector fields on the plane, Qual.Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339. [8] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207-223. [9] A. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.

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##### References:
 [1] K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos, Springer-Verlag, New York, 1997. [2] W.-J. Beyn, On invariant closed curves for one-step methods, Numerische Mathematik, 51 (1987), 103-122.  doi: 10.1007/BF01399697. [3] L. Dieci, C. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, DCDS-B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165. [4] L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth dynamical systems with a line of discontinuity, Journal of Dynamics and Differential Equations, 26 (2104), 1049-1078.  doi: 10.1007/s10884-014-9380-3. [5] L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous right-hand side, Journal of Computational and Applied Mathematics, 236 (2012), 3967-3991.  doi: 10.1016/j.cam.2012.02.011. [6] T. Eirola, Invariant curves of one-step methods, BIT, 28 (1988), 113-122.  doi: 10.1007/BF01934699. [7] J. Sotomayor and A. L. Machado, Structurally stable discontinuous vector fields on the plane, Qual.Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339. [8] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207-223. [9] A. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.
Euler trajectory on (1) and blow-up.
Exchange mechanism of Euler trajectories on (1); few iterates on the left figure, and many iterates on the right figure.
Example 7. Blow-up of the asymptotically stable periodic orbit of period $2N$ and unstable periodic orbit of period $N$ for $\epsilon = 0.0055$.
Example 7. Plot of the periodic orbit obtained with Euler's method versus the solution obtained with $\tt ode45$; $\epsilon = 0.0065$.
In support of the proof of Claim 1. The red segment contains the band.
In support of Lemma 15
Example 18: Euler approximation for $\tau = 0.03$.
Period doubling bifurcation values for Example 7. $N = 512$.
 $\epsilon$ Period Ratio 0.00621151300 2 0.005097818578 4 0.00488819181 8 5.312749... 0.00482735428 16 3.4456818... 0.00481444430 32 4.7124418... 0.00481166749 64 4.6492126... 0.00481107261 128 4.66784898... 0.00481094519 256 4.668691...
 $\epsilon$ Period Ratio 0.00621151300 2 0.005097818578 4 0.00488819181 8 5.312749... 0.00482735428 16 3.4456818... 0.00481444430 32 4.7124418... 0.00481166749 64 4.6492126... 0.00481107261 128 4.66784898... 0.00481094519 256 4.668691...
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