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

admits a limit cycle.

Answer

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.