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8/7/2019 Chaotic systems and Chuas Circuit
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ByARKA CHAKRABORTY
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` Nonlinearsystems Local behavior
Global behavior
` Chaos and Chuas Circuit Periodicorbits
Strangeattractors
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` Nonlineardynamic systems contain products or
functions of thedependent variable.
Nonlinearsystems
Local behavior Global
behavior
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` to study nonlinear systems, locally, around somespecial sets, a technique knownas local
linearization.
x Linearizationaroundequilibrium points.
x Linearizationaround periodicorbits.
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` Autonomous circuit consisting twocapacitors,
inductor, resistor, andnonlinearresistor.
` Exhibits avariety ofchaotic phenomenaexhibited
by morecomplex circuits` Readily constructedat lowcost using standard
electroniccomponents
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` Chuas circuit is simpleandexibits avariety of
phenomena:
x
Equilibrium points, periodicorbitsx Chaos andlimit cycles
` Signs ofchaos:
x
Sensitivity toinitialdatax Strangeattractors
x Unpredictability
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Applying KCL &KVL the following
equations
can be obtained:-
2
2122
1121
1
)(
)()(
Vdt
di
L
iR
VV
dt
dVC
VfR
VVdtdVC
L
L
!
!
!
Chuas diode
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L= 18mH R4=R5=22k
R1=R2=220kC2=100uF
C1=10uF R3=2.2 k
R6=3.3k
|}||){|(5.0 1111 EVEVGGVGVf bab !
mSG
mSG
b
a
5.0
8.0
!
!
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Dimensionless form:-
|)1||1)(|(5.0)( ! xxbabxxxh
The equation becomes:
Where,
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` Rateofevolutionof the systemis zero
0
.
!X
If Re() Stable
If Re()>0=> Unstable
Stability
Stability oflinearsystems is determined by
eigenvalues of Jacobian matrix.
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Thecircuit can bedividedinto threeregions :
Thus wehave threeequilibrium points :
P+=(k,0,k)
P=(0,0,0)
P-=(-k,0,-k) where k=(b-a)/(1+b)
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-
00
111
1)(
)(
F
EE xh
xJ
The Jacobian matrix is given by:
Theeigen values for = 10and = 14.3 foreacheqilibrium
point is
E F
71.297.0,22.205.319.0,94.3
0jjp
s!
s!
P
P
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Theeigen spaces at theoriginis given by:
Theeigen spaceat P is given by
0)(:)0(
:)0(
2
2
!
!!
zyxE
zyxE
v
u
EEKFKK
FKFKK
0)())((:)(
:)(
2
2
!s
s!!
kzykxPE
kzykxPE
PPP
v
PPP
u
EEKFKKFKFKK
O
O
s
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An attractor is a set towards whicha dynamicalsystem evolves over time.
1. Fixed points: stationary solutions
2. Limit cycles: periodic solutions
3. Quasiperiodic orbits: periodic solutions withat least
twoincommensurable frequencies
4. Chaoticorbits: boundednon-periodic solutions
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Thevalueof the
Resistanceis varied
from
1.85k to 10k
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` http://sprott.physics.wisc.edu/chaos/lespec.htm` http://sprott.physics.wisc.edu/chaos/lyapexp.htm` http://cse.ucdavis.edu/~chaos/courses/nlp/Software/part7.htm` http://cse.ucdavis.edu/~chaos/courses/nlp/Software/Part7
Code/LorenzODELCE.py` http://adsabs.harvard.edu/abs/1985PhyD...16..285W` Oancea , S. Synchronizationof chaoticelectronicchuas circuits.
Journalof Optoelectronics and Advanced Materials Vol. 7, No. 6,December2005, p.2919 2923
` Almeida, D.I. R., Alvarez, J., & Barajas, J.G. Robust synchronizationofSprott circuits using slidingmodecontrol. Chaos Solitons & Fractals,Vol. 30(1), 2005,p.1118.
` Pan, L.,Zhou, W., Fang J.Dynamics analysis ofanew simplechaoticattractor.Inrenational Journalof Control, Automationand Systems .Volume 8, No.2, 2010, p. 468-472.