Hopf bifurcation - UCL

1 hopf bifurcationHopf bifurcation for flowsThe term hopf bifurcation (also sometimes called Poincar e-Andronov- hopf bifurcation ) refers to thelocal birth or death of a periodic solution (self-excited oscillation) from an equilibrium as a parametercrosses a critical value. It is the simplest bifurcation not just involving equilibria and therefore belongsto what is sometimes calleddynamic(as opposed tostatic) bifurcation theory. In a differential equationa hopf bifurcation typically occurs when a complex conjugate pair of eigenvalues of the linearised flowat a fixed point becomes purely imaginary. This implies that a hopf bifurcation can only occur insystems of dimension two or a periodic solution should be generated in this event is intuitively clear from Fig. 1. Whenthe real parts of the eigenvalues are negative the fixed point is a stable focus (Fig.)

2 1a); when they crosszero and become positive the fixed point becomes an unstable focus, with orbits spiralling out. Butthis change of stability is a local change and the phase portrait sufficiently far from the fixed pointwill be qualitatively unaffected: if the nonlinearity makes the far flow contracting then orbits will stillbe coming in and we expect a periodic orbit to appear where the near and far flow find a balance (asin Fig. 1b).The hopf bifurcation theorem makes the above precise. Consider the planar system x=f (x, y), y=g (x, y),(1)where is a parameter. Suppose it has a fixed point (x, y) = (x0, y0), which may depend on . Letthe eigenvalues of the linearised system about this fixed point be given by ( ), ( ) = ( ) i ( ).Suppose further that for a certain value of , say = 0, the following conditions are satisfied:1.

3 ( 0) = 0, ( 0) = 6= 0, where sgn( ) = sgn[( g / x)| = 0(x0, y0)](non-hyperbolicity condition: conjugate pair of imaginary eigenvalues) ( )d = 0=d6= 0(transversality condition: the eigenvalues cross the imaginary axis with non-zero speed) 0, wherea=116(fxxx+fxyy+gxxy+gyyy) +116 (fxy(fxx+fyy) gxy(gxx+gyy) fxxgxx+fyygyy),withfxy= ( 2f / x y) = 0(x0, y0), etc.(genericity condition)Then a unique curve of periodic solutions bifurcates from the fixed point into the region > 0ifad <0 or < 0ifad >0. The fixed point is stable for > 0(resp. < 0) and unstable for < 0(resp. > 0) ifd <0 ( >0) whilst the periodic solutions are stable (resp. unstable)if the fixed point is unstable (resp. stable) on the side of = 0where the periodic solutions amplitude of the periodic orbits grows like | 0|whilst their periods tend to 2 /| |as tends to 0.

4 The bifurcation is calledsupercriticalif the bifurcating periodic solutions are stable, andsubcriticalif they are 2D version of the hopf bifurcation theorem was known to Andronov and his co-workers fromaround 1930 [1], and had been suggested by Poincar e [5] in the early 1890s. hopf [2], in 1942, provedthe result for arbitrary (finite) dimensions. Throughcentre manifold reductionthe higher-dimensionalversion essentially reduces to the planar one provided that apart from the two purely imaginary eigen-values no other eigenvalues have zero real part. In his proof (which predates the centre manifold1(a)(b)-3-2-1 0 1 2 3-3-2-1 0 1 2 3vu-3-2-1 0 1 2 3-3-2-1 0 1 2 3vuFigure 1: Phase portraits of (2) for (a) = , (b) = There is a supercritical hopf bifurcationat = ), hopf assumes the functionsf andg to be analytic, butC5differentiability is sufficient (aproof can be found in [3]).

5 Extensions exist to infinite-dimensional problems such as differential delayequations and certain classes of partial differential equations (including the Navier-Stokes equations)[3].Example:Consider the oscillator x ( x2) x+x= 0 (an example of a so-called Li enard system), which, withu=x,v= x, we can write as the first-order system u=v, v= u+ ( u2)v.(2)The origin (u, v) = (0,0) is a fixed point for each , with eigenvalues ( ), ( ) =12( i 4 2).The system has a hopf bifurcation at = 0. We have = 1,d=12anda= 18, so the bifurcationis supercritical and there is a stable isolated periodic orbit (limit cycle) if >0 for each sufficientlysmall (see Fig. 1). hopf bifurcation for mapsThere is a discrete-time counterpart of the hopf bifurcation . It occurs when a pair of complex conjugateeigenvalues of a map crosses the unit circle.

6 It is slightly more complicated than the version for corresponding theorem was first proved independently by Naimark [4] and Sacker [6] and thebifurcation is therefore sometimes called the Naimark-Sacker bifurcation . A proof can again be foundin [3].Consider the planar mapF = (f , g ) : IR2 IR2, with parameter , and suppose it has a fixedpoint (x, y) = (x0, y0), which may depend on . Suppose further that at this fixed pointDF has acomplex conjugate pair of eigenvalues ( ), ( ) =| ( )|e i ( ), and that for a certain value of , say = 0, the following conditions are satisfied:1.| ( 0)|= 1(non-hyperbolicity condition: eigenvalues on the unit circle)22. k( 0)6= 1 fork= 1,2,3,4(non-strong-resonance condition) | ( )|d = 0=d6= 0(transversality condition) 0, wherea= Re[(1 2eic)e 2ic1 eicc11c20] 12|c11|2 |c02|2+ Re(e icc21),c= ( 0),sgn( ( 0)) = sgn[( g / x)| = 0(x0, y0)]andc20=18[(fxx fyy+ 2gxy) +i(gxx gyy 2fxy)],c11=14[(fxx+fyy) +i(gxx+gyy)],c02=18[(fxx fyy 2gxy) +i(gxx gyy+ 2fxy)],c21=116[(fxxx+fxyy+gxxy+gyyy) +i(gxxx+gxyy fxxy fyyy)].

7 (genericity condition)Then an invariant simple closed curve bifurcates into either > 0or < 0, depending on the signsofdanda. This invariant circle is attracting if it bifurcates into the region of where the origin isunstable (a supercritical bifurcation ) and repelling if it bifurcates into the region where the origin isstable (a subcritical bifurcation ).Note that this result says nothing about the dynamicsonthe invariant circle. In fact, the dynam-ics on the circle has the full complexity of so-calledcircle maps(including the possibility of havingattracting periodic orbits on the invariant circle) and depends sensitively on any perturbation (see theexample below). Consequently, unlike the hopf bifurcation for flows, the hopf bifurcation for maps isnot structurally :Consider the following family of maps:F (xy)= (1 +d +a(x2+y2))(cos(c+b(x2+y2)) sin(c+b(x2+y2))sin(c+b(x2+y2)) cos(c+b(x2+y2)))(xy).

8 (3)The origin is a fixed point for each . The Jacobian matrix ofF at this fixed point isDF (0,0) = (1 +d )(cosc sincsinccosc)(4)and the eigenvalues are ( ), ( ) = (1 +d )e ic. The map takes a simpler, semi-decoupled, form inpolar co-ordinatesr= x2+y2, = arctan(y/x):(r )7 (r(1 +d +ar2) +c+br2).(5)This 5-parameter map is in fact thenormal formfor the hopf bifurcation up to cubic terms ( , bya smooth change of co-ordinates we can bring anyF into this form (plus higher-order terms)). Theparametersa,canddin (3) and (5) are precisely those defined in the conditions above. We choosea= ,b=c= ,d= The map then undergoes a supercritical hopf bifurcation at = 0,as can be confirmed by a simple graphical analysis of the decoupledrmap (fora >0 it would be3(a)(b)-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4yx-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4yxFigure 2: Phase portraits of (3) for (a) = , (b) = There is a supercritical hopf bifurcationat = 0.)

9 (a= ,b=c= ,d= )subcritical). For sufficiently small >0 we have an attracting invariant circle given byr= d /a(see Fig. 2). On the circle the map is given by 7 +c bd /a. This is simply a rotation through afixed angle =c bd /a, giving periodic orbits if 2 / IQ, or dense (irrational) orbits if 2 / IR\ the hopf bifurcation occurs in a map associated with the return map (Poincar e map) near a periodicorbit of an autonomous flow then the bifurcation is often called asecondary hopf bifurcation . In thiscase the invariant curve corresponds to an invariant torus for the flow and attracting periodic orbitson the circle correspond tomode-lockedperiodic motion on the torus, whilst dense orbits hopf bifurcationsIf one or more of the listed conditions for a hopf bifurcation are not satisfied (for instance becauseof symmetry) one may still have the emergence of a periodic orbit but some of the conclusions of thetheorem may cease to hold true.

10 The bifurcation is then called adegenerate hopf bifurcation . Forinstance, if the transversality condition is not fulfilled the fixed point may not change stability, ormultiple periodic solutions may bifurcate. An important case is provided by a Hamiltonian system forwhich complex eigenvalues come in symmetric quadruples and therefore the transversality conditioncannot be satisfied. This is why the analogous bifurcation in Hamiltonian systems (the so-calledHamiltonian- hopf bifurcation [7]) is much more complicated. For one thing, it needs a 4-dimensionalphase balance between local excitation and global damping mentioned at the beginning occurs com-monly in physical systems and the hopf bifurcation underlies many spontaneous oscillations such asairfoil flutter and other wind-induced oscillations ( , those that caused the Tacoma-Narrows bridgecollapse) in structural engineering systems, vortex shedding in fluid flow around a solid body at suf-ficiently high stream velocity, LCR oscillations in electrical circuits, relaxation oscillations ( , asdescribed by the Van der Pol oscillator), the periodic firing of neurons in nervous systems ( , in theFitzHugh-Nagumo equation modelling these phenomena), oscillations in autocatalytic chemical reac-4tions ( , the Belousov-Zhabotinsky reaction)