Hopf bifurcation

In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed point, and instead become attracted to (or repelled by) an oscillatory, periodic solution. The Hopf bifurcation is a two-dimensional analog of the pitchfork bifurcation. Many different kinds of systems exhibit Hopf bifurcations, from radio oscillators to railroad bogies. Trailers towed behind automobiles become infamously unstable if loaded incorrectly, or if designed with the wrong geometry. This offers an intuitive example of a Hopf bifurcation in the ordinary world, where stable motion becomes unstable and oscillatory as a parameter is varied. Fluid flows also exhibit Hopf bifurcation behavior when the transition from steady to unsteady laminar flow occurs. The general theory of how the solution sets of dynamical systems change in response to changes of parameters is called bifurcation theory; the term bifurcation arises, as the set of solutions typically split into several classes. Stability theory pursues the general theory of stability in mechanical, electronic and biological systems. The conventional approach to locating Hopf bifurcations is to work with the Jacobian matrix associated with the system of differential equations. When this matrix has a pair of complex-conjugate eigenvalues that cross the imaginary axis as a parameter is varied, that point is the bifurcation. That crossing is associated with a stable fixed point "bifurcating" into a limit cycle. A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf.

Overview

Normal form Hopf bifurcations occur in a large variety of dynamical systems described by differential equations. Near such a bifurcation, a two-dimensional subset of the dynamical system is approximated by a normal form, canonically expressed as the following time-dependent differential equation:

d z d t = z ( ( λ + i ) + b | z | 2 ) {\displaystyle {\frac {dz}{dt}}=z\left((\lambda +i)+b|z|^{2}\right)}

Here z {\displaystyle z} is the dynamical variable; it is a complex number. The parameter λ {\displaystyle \lambda } is real, and b = α + i β {\displaystyle b=\alpha +i\beta } is a complex parameter. The number α {\displaystyle \alpha } is called the first Lyapunov coefficient. The above has a simple exact solution, given below. This solution exhibits two distinct behaviors, depending on whether λ > 0 {\displaystyle \lambda >0} or λ < 0 {\displaystyle \lambda <0} . This change of behavior, as a function of λ , {\displaystyle \lambda ,} is termed the "Hopf bifurcation". The study of Hopf bifurcations is not so much the study of the above and its solution, as it is the study of how such two-dimensional subspaces can be identified and mapped onto this normal form. One approach is to examine the eigenvalues of the Jacobian matrix of the differential equations as a parameter is varied near the bifurcation point.

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