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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.