noise & multistability in the square root mapeoghan/ul_siam_presentation.pdfodelling research...
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Modelling
Research
GroupNoise & Multistability in the Square Root Map
Eoghan Staunton, Petri T. Piiroinen
7 December, 2016
Eoghan Staunton SIAM Student Conference 2016 1 / 14
Noise and Nonsmoothness in Dynamical SystemsBoth noise and nonsmoothness have been shown to independently be thedrivers of significant changes in qualitative behaviour.
Nonsmooth systems - qualitative changes in the behavior of thesystem under parameter variation that do not occur in the smoothsetting.
Adding noise to (smooth) systems - does more than just blur theoutcome of the system in the absence of noise
Figure: From Chin et. al, [CONG94]. Figure: From Linz and Lucke, [LL86].
Eoghan Staunton SIAM Student Conference 2016 2 / 14
The Square Root Map
Many impacting systems, including impactoscillators, are described by a 1-D mapknown as the square root map.
xn+1 = S(xn) =
{µ+ bxn if xn < 0µ− a√xn if xn ≥ 0
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Eoghan Staunton SIAM Student Conference 2016 3 / 14
The Square Root Map
This continuous, nonsmooth map can be derived as an approximation forsolutions of piecewise smooth differential equations near certain types ofgrazing bifurcation.
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Eoghan Staunton SIAM Student Conference 2016 4 / 14
Multistability In the Square Root Map
If 0 < b < 14 there are values of µ > 0 for which
a single stable periodic orbit of period m, with code (RLm−1)∞,exists for each m = 2, 3, . . .
two stable periodic orbits, one of period m, with code (RLm−1)∞,and the other of period m+ 1, with code (RLm)∞, exist for eachm = 2, 3, . . .
These are the only possible attractors of the system except at bifurcationpoints.
Eoghan Staunton SIAM Student Conference 2016 5 / 14
Types of Noise
In two separate papers Simpson, Hogan and Kuske and Simpson andKuske make a careful analysis of how noise in impacting systems manifestsin the map. They conclude that there are several different models. Wefocus on two of the simpler models with Gaussian white noise.
1 Additive Noise
xn+1 = Sa(xn) =
{µ+ bxn + ξn if xn < 0µ− a√xn + ξn if xn ≥ 0
(1)
2 Parametric Noise
xn+1 = Sp(xn) =
{µ+ bxn if xn < 0µ−
(a+ 1
2ξn)√
xn if xn ≥ 0(2)
Eoghan Staunton SIAM Student Conference 2016 6 / 14
The Effect of NoiseMy work thus far has focused on phase space sensitivity for period m andm+ 1 coexistence, investigating a shift of the proportion of points goingto one behaviour or the other, for both parametric and additive noise.
The results have not been entirely as we had expected. The relationshipbetween noise amplitude and the proportion of points going to each of thecoexisting attractors is not monotonic for µ in a neighbourhood of µsm.
Eoghan Staunton SIAM Student Conference 2016 7 / 14
Proportions
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Eoghan Staunton SIAM Student Conference 2016 8 / 14
The Transition Mechanism
Perhaps the most interesting phenomenon that we have observed is thepotential for repeated intervals of persistent RL dynamics in a noisysystem with µ < µs2.In the case of both additive and parametric noise, we have observed thatthe noise-induced transition between RLL and RL behaviour in this casetakes the following symbolic form
RLLRLL . . . RLLRLLRLRRLRL . . . RL. (3)
The most significant feature of the transitiongiven in (3) is the repeated R, corresponding torepeated low velocity impacts.
These repeated low velocity impacts allow thedynamics to be pushed into the region of phasespace with slow dynamics, in the vicinity of theunstable (RL)∞ orbit of the deterministic system.
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Eoghan Staunton SIAM Student Conference 2016 9 / 14
The Transition Mechanism
Let AX1,X2,...Xm with Xi ∈ {L,R} for i ∈ {1, 2, . . .m} denote the set ofvalues x1 such that the sequence x1, x2, . . . xm generated under iterationhas the symbolic representation X1, X2, . . . Xm.
Now since ARR =(0, (µ/a)2
)and L2
3, the second left iterate of thedeterministic period-3 orbit, is close to 0 for µ in a neighbourhood of µs2,it is easy to see how a small error due to noise could push the dynamics ofa settled RLL orbit into ARR.
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Eoghan Staunton SIAM Student Conference 2016 10 / 14
The Transition Mechanism
Let z1 = x1, zn+1 = S(zn) and εn = xn − zn for n ∈ {2, 3, . . .}.Given x1 ≈ R3, the right iterate of the stable deterministic period-3 orbit,and the noise terms are such that |ξi|≪ 1 for all i, it is most likely thatthe driving force behind such a transition is the error ε4.
ε6 = ab(√z4 −
√z4 + ε4) + bξ4 + ξ5 and ε4 = b2ξ1 + bξ2 + ξ3.
For ε4 to contribute positively to the transition we must have that ε4 < 0.
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Eoghan Staunton SIAM Student Conference 2016 11 / 14
Holding the unstable stable
Not only can noise cause us to transition from RLL behaviour to RLbehaviour in a situation where only the (RLL)∞ attractor is stable in thedeterministic system, but it can also cause orbits to remain in this RLbehaviour for longer periods of time than they would in the correspondingdeterministic system.
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Eoghan Staunton SIAM Student Conference 2016 12 / 14
Generalising
In general we see that these features are repeated as we look at thecoexistence of attractors (RLm−1)∞ and (RLm)∞ for increasing m. Inparticular we observe transitions of the following form for µ in aneighbourhood of µsm such that µ < µsm.
RLmRLm . . . RLmRLm−1RLk−2RLm−1RLm−1 . . . RLm−1 (4)
for k ∈ {2, 3, . . . ,m}.
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Eoghan Staunton SIAM Student Conference 2016 13 / 14
W. Chin, E. Ott, H. E. Nusse, and C. Grebogi, Grazing bifurcations inimpact oscillators, Physical Review E 50 (1994), no. 6, 4427–4444.
S. J. Linz and M. Lucke, Effect of additive and multiplicative noise onthe first bifurcations of the logistic model, Physical Review A 33(1986), no. 4, 2694–2703.
A.B. Nordmark, Non-periodic motion caused by grazing incidence inan impact oscillator, J. Sound Vib. 145 (1991), 279–297.
D.J.W. Simpson, S.J. Hogan, and R. Kuske, Stochastic regulargrazing bifurcations, SIADS 12 (2013), 533–559.
D.J.W. Simpson and R. Kuske, The influence of localised randomnesson regular grazing bifurcations with applications to impact dynamics,Journal of Vibration and Control 12 (2016), 1077546316642054.
J.B.W. Webber, A bi-symmetric log transformation for wide-rangedata, Measurement Science and Technology 24 (2013), 027001.
Eoghan Staunton SIAM Student Conference 2016 14 / 14