CIASS1CA MECHANICS Co3pgpe 2)

Ihe denens io nal dy nani cal Systerms can have

called tnttCycles
I9olatod but doed stable orbits
ohen certan Conduttons are 9atr8ç1ed.
ctosod trayectory tn phase 8pace having
Limit Cycle ts a trajectorg sprals into
least one other
the property that at hehavour
abpioaches nogattve 1nfinty. Some
tt etther as tme non lnear Systons
ts exth btted tn Some
Let x Ct)= (xtt),90) closedosbt
inutial Condition X(o) t3a
Soutton xCt) oor an
n an

tor untque Vae otT,

xCE) =X(t+T)0

1S the penodic totth pertod T

perodic &oluttons s ufrCiently close to t eAst

NO Other
that the perrodic obbit ls an tso latod one In limttcycte.
So
lum1t Cycle approaches t>0 (or) t’-o
The

nearby tiayectories approach a lumit cycle as t

Lf
tt lS SaLd to be stable.

L t > - o , It ts Said
to be nstable.
pernodc attractor.
Astable tmit cy cle ts a
EXample):
ltmit ycle 1s that op the
of a
ASimple example
pendulum motton.
uwatl -clock between
the batana
Cyctte motOn 19 due to
The tastana).
-excttatten and dampung (ar
tie &els
Xamplea:
provided by tho van der pol oscllater
Another eKample 9
the Vaun der pol model 0f an electrtcal
Cohich represents
a trido Valve, the reststan co propesty of tohch
CrCut with
Changes cotth Cuwent)

tohen -0
y=0 )
Sub @en

-0

(x, y) (o,o)

b d

d dy

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--A(b-A)+1
=A-btl.

The singular potnt E: (0,0) s a tunctton of the co nbro

Parameter bin the range C-oo,o) ky anaysing the gorm
O6 tho ergen Values A4

Rererence:
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