bifurcations have also been shown to occur in fission waves. The Selkov model is The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle's stable motion at low speeds crosses over to unstable at high speeds. One aim of the nonlinear analysis of these systems is to perform an analytical investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles on a tangent track, which uses the Bogoliubov method. Definition of a Hopf bifurcation The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs. Theorem (see section 11.2 of ). Let be the Jacobian of a continuous parametric dynamical system evaluated at a steady point . Suppose that all eigenvalues of have negative real part except one conjugate nonzero purely imaginary pair . A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters. Routh–Hurwitz criterion Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea. Sturm series Let be Sturm series associated to a characteristic polynomial . They can be written in the form: The coefficients for in correspond to what is called Hurwitz determinants. Their definition is related to the associated Hurwitz matrix. Propositions Proposition 1. If all the Hurwitz determinants are positive, apart perhaps then the associated Jacobian has no pure imaginary eigenvalues. Proposition 2. If all Hurwitz determinants (for all in are positive, and then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair. The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition. Example Consider the classical Van der Pol oscillator written with ordinary differential equations: The Jacobian matrix associated to this system follows: The characteristic polynomial (in ) of the linearization at (0,0) is equal to: The coefficients are: The associated Sturm series is: The Sturm polynomials can be