Calculus and Analysis
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In mathematics , catastrophe theory is a branch of bifurcation theory in the study of dynamical systems ; it is also a particular special case of more general singularity theory in geometry . Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.
In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos . It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold ). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system.
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Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties. The easiest way to see what a function looks like is to use a computer as a graphing tool.
Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere".
Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used.