Bifurcation, Chaos and Catastrophes in Dynamical Systems

Abstract

Consider the dynamical system (DS)

$$\mathop {x = f(x,\mu ) \equiv {f_\mu }(x)}\limits^. $$

((10.1))

where x ∈ Rⁿ is a vector of n state or internal variables and μ ∈ R^r is a vector of r parameters, f(x.μ): Rⁿ × R^r → Rⁿ or f _μ (x): Rⁿ → Rⁿ are assumed to be smooth. The DS (10.1) depends continuously on μ: starting from the same initial conditions, the flow follows a different path for each set of parameters, and to emphasize this fact, (10.1) is written as ẋ = f _μ (x): Rⁿ → Rⁿ to indicate an r-parameter-family of real differentiable functions on Rⁿ. A variation of μ, causes a change in the DS. These changes are mild in some cases, abrupt in some others, or worse still a loss of structural stability leading to chaotic and erratic flows which do not fit any conventional types, can result from some infinitesimally small and seemingly innocuous variations of some parameter. These are covered under the headings of Bifurcation Theory, Catastrophe Theory and Chaos which will be briefly introduced in this chapter. A common thread running through these fields is the presence of singularities which causes a failure of the Implicit Function theorem (IFT) and destroys the structural stability of the DS, invalidates forecasts and undermines Comparative Statics analysis. One major problem encountered here is the large number of mathematical tools needed and their advanced nature. We shall present a plain, non technical, account of these theories, emphasizing concepts, meanings and applicability rather than formal definitions and proofs.