Boundary Noise in Chua's Circuitmaths.nuigalway.ie/~pearcehn/Pearce/GroupPresentations/... · 2019....

Boundary Noise in Chua's Circuit

Eoghan J. Staunton,Petri T. Piiroinen

1st March 2019

Chua’s Circuit

Figure: Chua’s Circuit

Figure: The V − Icharacteristic of Chua’sDiode.

Created with the aim of being the simplestautonomous circuit capable of generatingchaos [Mat84, Chu92].

First physical system for which the presenceof chaos was shown experimentally,numerically and mathematically [CKM86].

Contains four linear elements and onenonlinear resistor known as a Chua’s diode.

Easily and cheaply constructed usingstadard electronic components [Ken92].

Eoghan Staunton Postgraduate Modelling Group 1st March 2019 1 / 11

System equationsThe system equations of Chua’s circuit are

dV1dt

=1

C1(G(V2 − V1)− f(V1)),

dV2dt

=1

C2(G(V1 − V2) + I3)

dI3dt

= − 1

L(V2 +R0I3), (1)

where

G =1

Rand f(V1) = GbV1 +

1

2(Ga −Gb)(|V1 + E| − |V1 − E|). (2)

Nondimensionalising gives

dx

dt′= α(y − x− g(x)),

dy

dt′= x− y + z,

dz

dt′= −(βy + γz), (3)

where

g(x) = m1x+1

2(m0 −m1)(|x+ 1| − |x− 1|). (4)


Complicated Dynamics

Figure: A Zoo of Attractors Produced by Chua’s Circuit [BP08]


Hidden and Self-Excited Attractors

Hidden attractors: have basins of attraction that do not intersectwith small neighborhoods of equilibria.

Self-excited attractors: Can be found by following trajectories fromthe neighbourhoods of unstable equilibria until the end of a transientprocess [LK13].

Most classical attractors are self-excited attractors and easily found. Onthe other hand the search for hidden attractors can be difficult. The firstchaotic hidden attractor was found in Chua’s circuit in 2010 [KLV10].

-5 0 5

x

-10

-5

0

5

10

y


Noisy SaltationIn order to deal the with stochastically moving boundaries when linearisingwe generalise the concept of a saltation matrix, the matrix that allows usto deal with nonsmoothness when linearising deterministic systems.We extend the state space, such that the state vector is x = (x, t,∆tref)

T .Here ∆tref is the random quantity which represents the difference in thehitting time of the reference trajectory in the stochastic system comparedto the corresponding deterministic system.We calculate the saltation matrix in this extended state space beforeprojecting back. As a result, in the original state space we find that

φ(x0, t)−φ(xref0 , t) ≈ φx(xref

0 , t)(x0−xref0 )+φ2,x(xout, t−tref)(fin−fout)∆tref,

(5)where

φx(xref0 , t) = φ2,x(xout, t− tref)D∗x(xin)φ1,x(xin, tref) (6)

and

D∗x(xin) = I +(fout − fin)hx(xin)

hx(xin)fin − v(tref)− V (tref|P (tref) = 0)(7)


A Discontinuous Model

We note that in the case of continuous piecewise-smooth systems thatfin = fout and so (5) collapses to

φ(x0, t)− φ(xref0 , t) ≈ φ2,x(xout, t− tref)φ1,x(xin, tref)(x0 − xref

0 ) (8)

meaning that neither the noise nor the nonsmoothness has a linear effect.As a result we will consider an discontinuous model of Chua’s circuitwhere the function describing the V − I characteristic of Chua’s diodegiven in (4) is replaced by a discontinuous one

g(x) =

m1x+m1 −m0 if x < −1,(m0 − ε)x if |x| ≤ 1,m1x+m0 −m1 if x > 1.

(9)


A Discontinuous ModelProvided the magnitude of ε is not too large the hidden attractors shownin Figure 4 continue to exist and can be easily found by numericalcontinuation.They are destroyed in saddle-bifurcations if the magnitude of ε grows toolarge.

Figure: Bifurcation diagram showing the saddle bifurcations of C− as themagnitude of ε grows. Here α = 8.4, β = 12, γ = −0.005, m0 = −1.2 andm1 = 0.145.


Steady-State Distributions

-0.18 -0.17 -0.16 -0.15 -0.14

y

0.24

0.25

0.26

0.27

0.28

z

Figure: Steady state distribution of orbiterrors on the discontinuity boundary D−

for trajectories with initial condition onthe periodic orbit C−.

0 10 20 30 40 50

n

2

2.5

3

3.5

z

10-3

Figure: Convergence of σz to its steadystate value for the distribution shown onthe left.


The Next Steps

Figure: 5 standard deviation ellipses forincreasing values of noise amplitude.

Use our method to predictchanges in qualitativebehaviour.

I Destruction of attractorsI Merging of attractorsI Switching/FlickeringI Multi/Monostability

Interaction of lineariseddistributions with features of thedeterministic system.

Generalise our methodI Second order terms for continuous systemsI Non-identity boundary mappingsI Dealing with non-transversal intersections


Eleonora Bilotta and Pietro Pantano, A gallery of chua attractors,vol. 61, World Scientific, 2008.

Leon O Chua, The genesis of chua’s circuit, International Journal ofElectronis Communication 46 (1992), no. 4, 250–257.

Leon O Chua, Motomasa Komuro, and Takashi Matsumoto, Thedouble scroll family, IEEE transactions on circuits and systems 33(1986), no. 11, 1072–1118.

Michael Peter Kennedy, Robust op amp realization of chua’s circuit,Frequenz 46 (1992), no. 3-4, 66–80.

Nikolay V Kuznetsov, Gennady A Leonov, and Vladimir I Vagaitsev,Analytical-numerical method for attractor localization of generalizedchua’s system, IFAC Proceedings Volumes 43 (2010), no. 11, 29–33.


Gennady A Leonov and Nikolay V Kuznetsov, Hidden attractors indynamical systems. from hidden oscillations in hilbert–kolmogorov,aizerman, and kalman problems to hidden chaotic attractor in chuacircuits, International Journal of Bifurcation and Chaos 23 (2013),no. 01, 1330002.

Takashi Matsumoto, A chaotic attractor from chua’s circuit, IEEETransactions on Circuits and Systems 31 (1984), no. 12, 1055–1058.


Boundary Noise in Chua's Circuitmaths.nuigalway.ie/~pearcehn/Pearce/GroupPresentations/... · 2019....

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