Multiple cycles and the Bautin bifurcation in the Goodwin model of a class struggle

In the field of economic growth, a huge interest has been devoted to explain the mechanisms through which a nonlinear system looses its stability and starts to oscillate around the steady state. Since the seminal Kalekian analysis, limit cycles represent the theoretical way of characterizing the emergence of persistent oscillations of macro-economic variables, namely the rise of indeterminacy problems (see [1]). In the quest for explaining the emergence of periodic fluctuations in an economic system the powerful tool of the Andronov–Hopf bifurcation theorem has been employed to derive the set of necessary and sufficient conditions for the rise of limit cycles (see, for example, [2]). Unfortunately, this theorem is not able to tell the full story of the global behavior of the dynamical system, which can exhibit a more complicated picture in the large, when some degenerate conditions occur (see, for example, [3–5]). That is, depending on the parameters configuration, we may obtain both (i) uniqueness of stable limit cycles (i.e., Hopf bifurcation), or (ii) multiple limit cycles of opposite stability (i.e., Bautin bifurcation), that eventually collide and disappear. An interesting problem to be investigated occurs, in fact, when the first Lyapunov coefficient vanishes, thus letting cycles of different orientation to coexist. In this case, a second order genericity condition, the second Lyapunov coefficient, must be computed, whose control leaves the door open to unexpected outcomes, which is commonly known as a Bautin bifurcation.