Limit Cycle Hop f

7/24/2019 Limit Cycle Hop f

1/18

imit Cycles and Hopf

imit Cycles and Hopf

Bifurcationifurcation

Chris InabnitChris Inabnit

Brandon TurnerBrandon Turner

Thomas BuckThomas Buck


2/18

Direction Field

y

dx

dy+= 2


3/18

Let the functionsLet the functions FFandand GGhave continuoushave continuous

first partial derivatives in a domainfirst partial derivatives in a domain DDof theof the

xy-plane. A closed trajectory of the systemxy-plane. A closed trajectory of the system

must necessarily enclose at least one criticalmust necessarily enclose at least one critical

(equilibrium) point. f it encloses only one(equilibrium) point. f it encloses only one

critical point! the critical point cannot be acritical point! the critical point cannot be a

saddlesaddlepoint.point.

"heorem"heorem

),( yxFdt

dx= ),( yxG

dt

dy=


4/18

#raphical nterpretation#raphical nterpretation

ydt

dx = xdt

dy =+


5/18

#raphical nterpretation#raphical nterpretation

=

x

y

y

x=


6/18

$pecific %ase of "heorem$pecific %ase of "heorem

Find solutions for the follo&in' systemFind solutions for the follo&in' system

Do both functions have continuousDo both functions have continuousfirst order partial derivativesfirst order partial derivatives

++

++=

)(

)(22

22

yxyyx

yxxyx

y

x


7/18

$pecific %ase of "heorem$pecific %ase of "heorem

Critical point of the system is (0,0)Critical point of the system is (0,0)

Eigenvalues are found by the correspondingEigenvalues are found by the corresponding

linear systemlinear system

hich turn out to be !hich turn out to be !

=

y

x

y

x

11

11

i1


8/18

hat does this tell ushat does this tell us

"rigin is an unstable spiral point for both"rigin is an unstable spiral point for both

the linear system and the nonlinearthe linear system and the nonlinear

system!system!

Therefore, any solution that starts nearTherefore, any solution that starts near

the origin in the phase plane ill spiralthe origin in the phase plane ill spiral

aay from the origin!aay from the origin!


9/18

( )22 yxxyxdt

dx++=


10/18

( )22 yxyyxdt

dy++=


11/18

"rajectories of the $ystem

dt

dx

dt

dy#orming a system out of and yields the tra$ectories shon!


12/18

*sin' +olar %oordinates

%sing & ' r cos() y ' r sin() r

' &

* y

( )22 yxxyxdt

dx++= ( )22 yxyyx

dt

dy++=

+oes to

( )21 rrdt

dr =

Critical points ( r ' 0 , r ' - )

Thus, a circle is formed at r ' -

and a point at r ' 0!


13/18

$tability of +eriod $olutions

"rbital .tability .emi/stable

%nstable


14/18

,xample of $tability

+iven the revious E1uation

( )2

1 rrdt

dr=

If r 2 -,

Then, dr3dt 4 0 (meaning the solution moves inard)

If 0 4 r 4 -,

Then, dr3dt 2 0 (meaning the solutions movies outard)


15/18

ifurcationifurcation

Bifurcation occurs hen the solution of an e1uation reaches a critical

point here it then branches off into to simultaneous solutions!

5 simple e&ample of bifurcation is the

solution of y' & !

6hen & 4 0 , y is identical to 7ero!

8oever, hen & 0 , a second

solution (y ' *3/ &) emerges!

Combining the to solutions, e see the

bifurcation point at & ' 0 ! This type of

bifurcation is called pitchfork bifurcation!

29

y ' 0 y ' &


16/18

opf ifurcation

( )22 yxxyxdt

dx++= ( )22 yxyyx

dt

dy++=

( )2

rrdt

dr

=

Introducing the ne parameter ( : )

Converting to polar form as in previous slide yields

Critical oints are no

r / 0 and r / 1

If you notice, these solutions are

e&tremely similar to those of the

previous e&ample y' &

r / 0

r / 1


17/18

opf ifurcation

5s the parameter : increases through the value 7ero, the previously

asymptotically stable critical point at the origin loses its stability, and

simultaneously a ne asymptotically stable solution (the limit cycle)

emerges!

Thus, : ' 0 is a bifurcation point! This type of bifurcation is called 8opf

bifurcation, in honor of the 5ustrian mathematician Eberhard 8opf ho

rigorously treated these types of problems in a -;


18/18

Boyce, 6illiam, and =irima, >ichard!Boyce, 6illiam, and =irima, >ichard!

=ifferential E1uations=ifferential E1uations! 8oboken ?ohn 6iley @! 8oboken ?ohn 6iley @

.ons, Inc!

.ons, Inc!

Bronson, >ichard!Bronson, >ichard! .chaumAs "utlines.chaumAs "utlines

=ifferential E1uations=ifferential E1uations! c+ra/8ill! c+ra/8ill

Companies, Inc!, -;;evieCliffAs Duick >evie=ifferential E1uations=ifferential E1uations! 6iley ublishing, Inc!,! 6iley ublishing, Inc!,

-;;!-;;!

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Limit Cycle Hop f

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