John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England Nonsmooth systems: synchronization, sliding and other open problems
2 Nonsmooth Systems
3 What is a nonsmooth system? Any system with sudden changes in position or velocity (charge or current) or where a threshold (or boundary) exists can be called nonsmooth. Application areas Genetic regulatory networks, neuronal firing Electrical circuits with switches, diodes, DC/DC converters, Σ-Δ modulators Mechanical systems with impact, friction, backlash, freeplay (gears, rocking blocks) Control engineering, digital control
4 Unique dynamics of nonsmooth systems (1) Grazing Solution trajectory tangent to boundary between regions Hogan Proc. Roy. Soc. Lond. A. 425, 441-476 (1989) Period adding bifurcation sequences Can have bifurcation sequences such as 1 2 3 4 Piassi et al Chaos 14, 477-486 (2004)
5 Unique dynamics of nonsmooth systems (2) Zenoness (or chatter) Infinite number of boundary crossings in a finite time (safety issue) Heymann et al IEEE Trans. Automatic Control 50, 376-383 (2005) Robust chaos No periodic windows Homer et al IEEE Trans. Circuits & Systems II 51, 511-516 (2004)
6 Unique dynamics of nonsmooth systems (3) Big bang bifurcation collision of an accumulation point with a boundary
7 Nonsmooth Maps (including synchronization)
8 Period adding in a simple nonsmooth map Di Bernardo et al. Chaos, Solitons & Fractals 10, 1881-1908 (1999) α = 1/5 β = -10 + A A 2 B
9 Labelling of orbits A, a denotes a stable, unstable period 1 solution > 0 B, b denotes a stable, unstable period 1 solution < 0 AB, ab denotes a stable, unstable period 2 solution, with one iterate > 0 and the other < 0 A k-p B p, a k-p b p denotes a stable, unstable period k solution, with k-p iterates > 0 and p iterates < 0 Hogan et al Proc. Roy. Soc. Lond. A. 463, 49-65 (2007)
10 Higher order solutions (γ = 0, µ > 0) + +
Synchronization of coupled nonsmooth maps 11 Polynikis, di Bernardo & Hogan Chaos, Solitons and Fractals 41 (2009) 1353 1367 x 1 (t + 1) = (1 ɛ)f(x 1 (t)) + ɛf(x 2 (t)) x 2 (t + 1) = ɛf(x 1 (t)) + (1 ɛ)f(x 2 (t)) where ɛ is the coupling f(x) = { αx µ, x 0 βx µ γ, x 0 α =0.4,β = 8,µ =0.1,γ =0 +
12 Bifurcation diagram of the coupled system of nonsmooth maps (note critical value of coupling and periodic windows)
13 Basin of attraction of the period-four attractor %). ( %)- %), + %)+! ' #$ % & %)* %)' %)" % + + +!%)"(!%)" % %)" %)' %)* %)+ %), %)- %).! #$ & " %
14 Critical value of the coupling ε (Pikovsky et al. 1991) ɛ c = 1 e λ 2 where λ is the Lyapunov exponent of the uncoupled chaotic system!%!'!)!+ a b d! "#% "#$+ "#$) "#$' "!$" "#$%!$%!$'!$)!$+!%"! "#$ "#% "#& "#' "#( "#) "#* "#+ "#, c! "#$ "#"+ "#") "#"' "#"%
15 α =0.4,β = 8,µ =0.1,γ =0
16 Nonsmooth Flows (including sliding)
17 Nonsmooth flows Described by a finite set of ODEs dx/dt = f (i) (x,µ), x Si R n where Si, i = 1, 2,..., m are open, nonoverlapping regions, such that i Si R n, separated by (n-1)-dimensional submanifolds (boundaries, discontinuity sets),. Assume that f (i) smooth, continuous.
18 Geometry of phase space We distinguish 3 distinct behaviours on the discontinuity set: crossing (or sewing) escaping (or unstable sliding) sliding (or stable sliding)
19 For sliding sections, use Filippov convex method Associate the convex function g(x) with each nonsingular point in
20 Dry friction oscillator (Guardia, Hogan & Seara 2009)
21 Phase space structure (note sliding occurs here when the block sticks!!)
22 (Known) sliding bifurcations
23 Kowalczyk & Piiroinen (2008)
24 More detail near point B B
25 Point A definitions
26 Trajectory along γ 1 (crossing-sliding)
27 Curve γ 1 (cross-sliding) On γ 1, So if then { x(π/ω) = x 0 y(π/ω) =0 x(t) =F cos t + F sin(π/ω) 1 + cos(π/ω) sin t F + 1 1 ω 2 sin(ωt + s 0) y(t) = F sin t + F sin(π/ω) 1 + cos(π/ω) cos t + ω 1 ω 2 cos(ωt + s 0) s(t) =s 0 + ωt ( s 0 =2π arccos F ω2 1 sin(π/ω) ) ω 1 + cos(π/ω) x(0) sin s(0) = F ɛ = ω 1 2 γ 1 (ɛ) = 1 ( 13 ) 3 4 9 ɛ+ 27 π2 ɛ 2 + 1 ( 7 96 6 48 π2 80 ) ( 121 ɛ 3 + 27 243 127 3456 π2 23 18432 π4) ɛ 4 +O 5 (ɛ)
28 Theory vs. numerics at point A A
29 Point B definitions B
30 Theory vs. numerics at B B
31 Other open problems
32 Sliding bifurcations - are there any more? See talk by Mike Jeffrey (Tues 1140) for the answer!!
33 Other issues General theory of nonsmooth bifurcations Global problems: what is the real question? Nonsmooth Melnikov method Grazing of invariant tori Differential inclusions Hybrid systems Complementarity framework Hysteresis, delays, noise Large dimensional nonsmooth systems Homer & Hogan Chaos Solitons & Fractals 10 (1999), 1869-1880 Homer & Hogan Int. J. Bif. Chaos 17, (2007) 561-573
34 Coarsening/smoothing/pinching/regularisation What is the best approximation of a nonsmooth system?
35 Numerical detection of bifurcations Smoothing is an obvious way to do (approximate) bifurcation continuation of nonsmooth systems. But this creates problems i) the resulting system is stiff ii) sliding disappears. Special algorithms have been developed for planar Filippov systems (Kuznetzov et al 2003). For systems of higher dimension, a lot of work needs to be done.
36 Thanks