Genre: Information/Explanation

Bifurcation Theory In a dynamical system , a bifurcation is a qualitative change in the behaviour of the system caused by a small smooth change in the value of a system parameter . Bifurcations are responsible for the creation and annihilation of equilibria and periodic solutions in a given system and their occurrence is accompanied by a change of solution stability . The knowledge of such qualitative changes is paramount to understanding the long-run behaviour of any dynamical system . The occurrence of bifurcations can be investigated geometrically by analysing the phase space for different parameter values of a given system . Geometric representations of bifurcation structures have the advantage of allowing a visual interpretation of how the parameters of a system affect its behaviour . A bifurcation diagram is a plot of the solutions ( or some feature of the solutions such as the maximum or L2 norm ) of a given system against a single system parameter . These plots are commonly used to display how the stability of solutions changes as the given parameter is varied . In the analysis of dynamical systems with more than one parameter , it is important to determine not only which parameters will cause bifurcations when their values are changed , but also how the bifurcation structure for a particular parameter is affected when changing the value of another . This is known as two-parameter bifurcation analysis . Generally , two-parameter bifurcation diagrams include curves of codimension-1 bifurcations , as well as specific points on these curves defining codimension-2 bifurcations . In order to characterise a particular bifurcation , the behaviour of the system within the region of a local bifurcation point can often be reduced to a simplified normal form . The normal form is the system with a minimum number of parameters and nonlinear terms whereby all systems undergoing the same bifurcation are topologically equivalent . A system that fulfils these requirements is known as the topological normal form of a given bifurcation . The Hartman-Grobmann theorem states that within the local region of a hyperbolic equilibrium , the behaviour of system is topologically equivalent to a linear system . However , at a bifurcation point , the number of eigenvalues with zero real parts is non-zero and thus the point is always non-hyperbolic . Consequentially , the topological normal forms of bifurcation points are always nonlinear .