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Smooth to discontinuous systems: A geometric and numerical method for slowfast dynamics
1.  School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA 
2.  Dipartimento di Matematica, Univ. of Bari, I70100, Bari, Italy 
We consider a smooth planar system having slowfast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous righthand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phasespace where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit RungeKutta schemes, and that we obtain much the same orbits of the original smooth system.
References:
[1] 
L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous righthand side, Journal of Computational and Applied Mathematics, 236 (2012), 39673991. doi: 10.1016/j.cam.2012.02.011. 
[2] 
A. F. Filippov, Differential Equations with Discontinuous RightHand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. 
[3] 
J. K. Hale, Ordinary Differential Equations, Krieger Publishing Co, Malabar, 1980. 
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M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS, 32 (1981), ⅵ+147 pp. 
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N. Levinson, A second order differential equation with singular solutions, Annals of Mathematics, 50 (1949), 127153. doi: 10.2307/1969357. 
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S. Natsiavas, Dynamics of piecewise linear oscillators with van der Pol type damping, International Journal of Non Linear Mechanics, 26 (1991), 349366. doi: 10.1016/00207462(91)900652. 
[7] 
A. Roberts and P. Glendinning, Canardlike phenomena in piecewisesmooth Van der Pol systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, Chaos, 24 (2014), 023138, 11pp. 
[8] 
J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207223. 
[9] 
D. W. Storti and R. H. Rand, A simplified model of coupled relaxation oscillators, Int.l J. Nonlin. Mechanics, 22 (1987), 283289. doi: 10.1016/00207462(87)900205. 
[10] 
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, 1987. 
show all references
References:
[1] 
L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous righthand side, Journal of Computational and Applied Mathematics, 236 (2012), 39673991. doi: 10.1016/j.cam.2012.02.011. 
[2] 
A. F. Filippov, Differential Equations with Discontinuous RightHand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. 
[3] 
J. K. Hale, Ordinary Differential Equations, Krieger Publishing Co, Malabar, 1980. 
[4] 
M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS, 32 (1981), ⅵ+147 pp. 
[5] 
N. Levinson, A second order differential equation with singular solutions, Annals of Mathematics, 50 (1949), 127153. doi: 10.2307/1969357. 
[6] 
S. Natsiavas, Dynamics of piecewise linear oscillators with van der Pol type damping, International Journal of Non Linear Mechanics, 26 (1991), 349366. doi: 10.1016/00207462(91)900652. 
[7] 
A. Roberts and P. Glendinning, Canardlike phenomena in piecewisesmooth Van der Pol systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, Chaos, 24 (2014), 023138, 11pp. 
[8] 
J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207223. 
[9] 
D. W. Storti and R. H. Rand, A simplified model of coupled relaxation oscillators, Int.l J. Nonlin. Mechanics, 22 (1987), 283289. doi: 10.1016/00207462(87)900205. 
[10] 
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, 1987. 
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