Normal form of the Hopf bifurcationThese notes review the derivation of the normal form of the Hopf bifurcation by meansof near-identity changes of variables, on the example of the following system:dxdt= µx + y(1)dydt= −x + µy − x2yThe origin is a fixed point of (1), and it undergoes a Hopf bifurcation at µ = 0. Sincethe Jacobian of (1) at the origin, evaluated at µ = 0, is anti-diagonal, we can immediatelyintroduce a complex variablez = x + iy,and derive an equation for the dynamics of z. This equation readsdzdt=dxdt+ idydt= µx + y + i(−x + µy − x2y) = µ(x + iy) − i(x + iy) − ix2y= µz − iz − iz + ¯z22z − ¯z2= µz − iz −18z2+ 2|z|2+ ¯z2(z − ¯z)= µz − iz −18z3+ 2|z|2z + |z|2¯z − |z|2z − 2|z|2¯z − ¯z3= µz − iz −18z3+ |z|2z − |z|2¯z − ¯z3,where we have used the fact thatx =z + ¯z2and y =z − ¯z2i.The above equation contains linear and cubic terms in z and ¯z. We are going to make anear-identity change of variables to “remove” some of the cubic terms. This transformationwill generate nonlinear terms of order higher than 3. In what follows, we illustrate thisprocess by eliminating only one of the terms, of the form zp¯zq, with p + q = 3. To do so,we first make the near-identity change of variablez = ˜z + α ˜zp¯˜zq, (2)where α is an unknown, a priori complex coefficient, to be determined later. This relationcan be inverted as follows:˜z = z − α ˜zp¯˜zq= z − αz − α ˜zp¯˜zqpz − α ˜zp¯˜zqq= z − αzp¯zq+ h.o.t., (3)where h.o.t. stands for terms of order 4 and higher in z and ¯z. Then,d˜zdt=dzdt− α p zp−1dzdt¯zq− α zpq ¯zq−1d¯zdt+ddt(h.o.t.)= µz − iz −18z3+ |z|2z − |z|2¯z − ¯z3−α p zp−1(µz − iz + O(3)) ¯zq− α zpq ¯zq−1(µ¯z + i¯z + O(3)) + h.o.t.= µz − iz −18z3+ |z|2z − |z|2¯z − ¯z3− α(µ − i)p zp¯zq− α(µ + i)q zp¯zq+ h.o.t.= µ˜z + α ˜zp¯˜zq− i˜z + α ˜zp¯˜zq−18˜z3+ |˜z|2˜z − |˜z|2¯˜z −¯˜z3−α (p(µ − i) + q(µ + i)) ˜zp¯˜zq+ h.o.t.= (µ − i)˜z −18˜z3+ |˜z|2˜z − |˜z|2¯˜z −¯˜z3− α (−µ + i + p(µ − i) + q(µ + i)) ˜zp¯˜zq+h.o.t.In the above calculation, O(3) denotes cubic terms in z and ¯z, and h.o.t. refers to termsof order 4 and higher, in z and ¯z or ˜z and¯˜z. Also note that we first expressed d˜z/dt interms of z and ¯z, and then used the expression of z as a function of ˜z and¯˜z, to obtain anequation, correct to order 3, that only involves ˜z and¯˜z.We now see that in order to get an equation that does not contain the nonlinear term˜zp¯˜zq, with p + q = 3, we need to choose α so that this term cancels out from the equationfor ˜z. You can check that all of the cubic terms can be removed, except the term in |˜z|2˜z.Indeed, to remove such a term, we would have to choose α such that−α (−µ + i + p(µ − i) + q(µ + i)) −18= 0,with p = 2 and q = 1. This reads−α (−µ + i + 2µ − 2i + µ + i) −18= 0,i.e.−2µα −18= 0,whose solution, α =−116µdiverges at the bifurcation, when µ = 0. In this case, the near-identity change of variable necessary to remove the term in |z|2z is not defined across thebifurcation, and therefore cannot be done.Once we have removed the 3 other cubic terms, we are left with the normal form forthe Hopf bifurcation of system (1), which readsd˜zdt= (µ − i)˜z −18|˜z|2˜z + h.o.t. (4)2Equation (4) shows that the bifurcation is s upercritical (since the coefficient of the cubicterm is negative), and that the frequency of oscillations of the limit cycle that exists abovethe bifurcation does not depend on its amplitude (since the imaginary part of the coefficientof the cubic term is zero).By letting u =1√8˜zeit, one obtainsdudt= µu − |u|2u + h.o.t.,which is the scaled normal form for a supercritical Hopf