### C.2. Limit Cycles

For non-conservative non-linear systems, it is possible to have stable periodic solutions called limit cycles.  Although such a solution is not a steady state, all states in its neighborhood will decay towards it eventually.  There are also unstable periodic solutions away from which all states in its neighborhood will eventually decay.

#### Exercise C.1

Show that the set of equations

d y 1 dt = y 2 + y 1 ( 1 y 1 2 y 2 2 ) y 1 2 + y 2 2               and          d y 2 dt = y 1 + y 2 ( 1 y 1 2 y 2 2 ) y 1 2 + y 2 2

It seems natural to change to the polar coordinates defined by

y 1 =rcosθ               y 2 =rsinθ

so that

r 2 = y 1 2 + y 2 2

and

d y 1 dt = dr dt cosθrsinθ   dθ dt   =rsinθ+cosθ  ( 1 r 2 )                        (1a)

d y 2 dt = dr dt sinθ+rcosθ   dθ dt   =rcosθ+sinθ  ( 1 r 2 )                     (1b)

cosθ  ( 1a )+sinθ  ( 1b )  Þ                dr dt =1 r 2                                 (1c)

sinθ  ( 1a )+cosθ  ( 1b )  Þ              dθ dt =1                                    (1d)

Integrating (1c) gives

t= r( 0 ) r( t ) dr 1 r 2   = 1 2 ln ( 1+r 1r ) | r( 0 ) r( t )          (2)

= 1 2 ln[ 1+r( t ) 1r( t ) 1r( 0 ) 1+r( 0 ) ]

so that

1+r( t ) 1r( t ) = 1+r( 0 ) 1r( 0 )   exp( 2t )=A  exp( 2t )

where A= 1+r( 0 ) 1r( 0 ) .  Solving for r( t )  gives

r( t )= Aexp( 2t )1 Aexp( 2t )+1                                 (3)

so that

r( )=1

Coupled with (1d), we see that all motions converges to a clockwise rotation with radius 1 about the origin.