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John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England

John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England. Piecewise smooth systems: a personal view Soto70, São Paulo, Brazil, 30 May - 1 June 2012. References at http://www.enm.bris.ac.uk/anm/staff/sjh.html. What is a piecewise smooth system?.

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John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England

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  1. John HoganBristol Centre for Applied Nonlinear Mathematics,University of Bristol, England Piecewise smooth systems: a personal view Soto70, São Paulo, Brazil, 30 May - 1 June 2012

  2. References athttp://www.enm.bris.ac.uk/anm/staff/sjh.html

  3. What is a piecewise smooth system? Physical: Any system with sudden changes in a variable (e.g. position, velocity) or where there is a threshold or boundary. Virtuous circle? Mathematical: Any dynamical system whose phase space is partitioned into different regions by a switching manifold, each region associated with a different smooth vector field.

  4. Application areas • Cartoons… • Genetic regulatory networks • Switches, diodes • DC/DC converters, Σ-Δ modulators • Impact, friction, backlash, freeplay • Gears, rocking blocks • Control engineering, digital control • Global circulation model • Economics • Valve generators • Automatic pilots • Steam-engines

  5. Theory Arch. Math. 7, 148 (1956)

  6. Application areas Tellus 13, 224 (1961)529 citations (31 May 2012)

  7. Application areas J. Acoustic Soc. Am. 47, 1390 (1969)

  8. Application areas

  9. Books Chapter VIII

  10. Unique dynamics (1) • Grazing • Solution trajectory tangent to boundary between regions • Period adding bifurcation sequences • Can have bifurcation sequences such as 1 → 2→ 3→ 4→ … Hogan (1989) Piassi et al(2004)

  11. Unique dynamics (2) • Zenoness (or chatter) • Infinite number of boundary crossings in a finite time (safety issue) • Sliding • System gets stuck on switching manifold Heymann et al (2005) Filippov (1988)

  12. Piecewise linear mapsRegularisationPiecewise smooth flowsMelnikov methodNetworks What is in this talk?

  13. Unsolved problems* Colombo, A., Di Bernardo, M., Hogan, S.J. and Jeffrey, M.Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems. Physica D (to appear) Sliding problemsSee next talk by Mike Jeffrey…* including agreed definition of bifurcation…. What is NOT in this talk?

  14. Piecewise linear mapsor how Russian conferences can be good for you…

  15. Period adding in piecewise linear map Di Bernardo, Feigin, Hogan, & Homer (1999)

  16. A 1D piecewise linear map with a gap Hogan Higham & Griffin (2007)

  17. Fixed points as μ varies γ = 1 γ = -1 A A A,B B B

  18. Higher order solutions (γ= 0, μ> 0)

  19. Higher order solutions (γ= -1, μ= 1/6)

  20. Comparsion with work on DC/DC converters Banerjee et al.(2004) xn Note coexistence of stable period 1 solution with stable period 2 solution β = selected values of β

  21. β = - 1/4

  22. (a) β = - 1/3, (b) β = - 1/2, (c) β = - 1, (d) β = - 3/2

  23. Regularisationor surely you can get all this by smoothing? Sotomayor J. and Teixeira M.A. (1995)Sotomayor J. and Machado A.L.F. (2002)

  24. What is the ‘best’ smooth approximation of a piecewise smooth system (analytic)? • If a piecewise smooth system is replaced by a smooth one, what happens to the unique dynamics? • What part of sliding survives smoothing?

  25. Singular limit?

  26. Piecewise smooth flows: the rocking blockor how in 1988, I said ‘I’ll get an answer to you by 4pm…’

  27. Rocking block • West’s formula:block topples when aH > (B/H) g aH

  28. Simple example: rocking block • Fallen Japanese tombstones used to predict historic earthquake magnitudes (Popular Science, Apr 1936)

  29. Simple example: rocking block • Significant problem in AGR nuclear reactors (stacked graphite rods can fracture) • Ikyushima (1982)

  30. Or this….

  31. Rocking block equations Housner (1963), Hogan (1989), ..., (2000)

  32. Phase portrait (undamped, r =1)

  33. Period doubling (damped, r <1)

  34. Excellent agreement with experiment Amplitude Frequency

  35. Melnikovmethod (1) Granados, Hogan, Seara SIADS (to appear 2012)

  36. Melnikovmethod (3)

  37. Theorem

  38. Proof outline

  39. Rocking block

  40. Results (1)

  41. Results (2)

  42. Networksor how a heat exchanger is like a cornfield

  43. Networks • Glass (1975) networks have been proposed as a way to model gene regulation, chemical kinetics, neural networks. • But what about very large networks of piecewise smooth systems? • Wind over cornfield • Heat exchangers innuclear reactor Moon, Kuroda (2001)

  44. Experimental evidence

  45. Model

  46. Numerical results r =0.8,100 masses

  47. Piecewise smooth systems and graph theory • Impacts are vertices of a directed graph. • The evolution of the system between two impacts is an edge of the graph. • Any periodic orbit of the piecewise smooth system is a circuit of the graph.

  48. ‘Simple circuit’ (3 masses) L, R = left, right wall, mass j = 1,2,3, L-1 is impact of left wall with mass 1, etc

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