Seminar: Introduction to Bifurcation Theory

Dr. habil. Irina Kmit

Time & Place

Wednesday: 15:00-17:00 (weekly)

Rudower Chaussee 25, room 1.011

Short description

The term “bifurcation” is used to describe  qualitative changes  in the behaviour of time-varying systems  (describing natural and social processes or phenomena) when certain parameters vary.  The concept of bifurcation  was  introduced by the French mathematician Henri Poincaré at the end of the 19th century. 

Classical (finite-dimensional)  bifurcation theory focuses on autonomous parametrised differential and difference equations of the form

(*)                         dx/dt = f (x , λ)     or    xt+1 = f(xt , λ) 

for a smooth function  f  and a bifurcation  parameter λ .  It is important to note that the (vector valued) parameter λ  is external to the dynamical system, meaning that it does not  depend  on the state variable  $x=x(t)$. Consequently, one can analyze different dynamical systems for various values of λ. We refer to system (*) at  λ= λ0   as undergoing a bifurcation if there exists a value   λ1  arbitrarily close to  λ0  such that the dynamics at  λ= λ1 is not topologically equivalent to that at  λ= λ0,  i.e., there is no homeomorphism between  system (*) with   λ= λ0 and  system (*) with  λ= λ1 .

The term “bifurcation” exhibits two qualitatively different dynamic states before and after the critical (bifurcation) value  λ0 .  Bifurcations can lead to the creation or disappearance of equilibrium points, periodic orbits, homo-/heteroclinic cycles, quasi-periodic and chaotic trajectories, as well as in changes in the structure and stability of the se dynamic objects. A central question in bifurcation analysis is how the stability, attractiveness, and multiplicity properties of invariant sets in system (*) change under variations of  λ. In the simplest case, these invariant sets correspond to equilibria (fixed points) or periodic solutions of (*).

In addition to ordinary differential and difference equations (mentioned above), bifurcations are also studied in other types of dynamical systems, including partial differential equations, delay equations, impulsive equations, integral equations, integro-differential equations, stochastic differential equations,  functional differential equations, and various other types of equations.

Bifurcation theory is closely related to other mathematical fields, including oscillations, stability, averaging, resonances, chaos, fractals, synchronization, symmetries, and catastrophe theory. It is widely applied in natural and engineering sciences, as well as in economics and sociology.

Prerequisites

Analysis  III; Linear Algebra II. 

Seminar talks

Every participant will be asked to give a 70-minute presentation.

References

• • Steven H. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Taylor & Francis, 2018.

  • Paul Glendinning. Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge Texts in Applied Mathematics, 1994.

  •  Kuznetsov Y. Elements of applied bifurcation theory, Springer–Verlag, 1995.

  •  Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, 2002.