bordercollision

Border collision, in the context of dynamical systems, refers to a border-collision bifurcation that occurs in piecewise-smooth systems when an invariant set such as a fixed point or a periodic orbit collides with a switching boundary that separates regions governed by different rules. At the collision, the orbit lies on the boundary, and as a parameter is varied the set can be created, destroyed, or experience an abrupt change in stability. Because the governing equations are not smooth across the boundary, standard smooth-bifurcation theory does not apply and specialized analysis is used.

Mathematical framework typically involves a map or flow that is piecewise defined by distinct expressions on either side of a switching manifold S = { x : h(x) = 0 }. A border collision occurs whenever a fixed point or a member of a periodic orbit meets S as a parameter μ passes through a critical value μ*, or equivalently when part of the orbit crosses the boundary.

A common way to describe local behavior is through a piecewise-affine or piecewise-linear normal form near the boundary. In one dimension, for example, the map may be given by x_{n+1} = f_L(x_n; μ) for x_n < 0 and x_{n+1} = f_R(x_n; μ) for x_n > 0, with f_L(0; μ) = f_R(0; μ) = 0 at the collision. The stability and existence of the collided orbit depend on the left and right slopes, and the bifurcation can cause abrupt changes in attractors without a smooth transition.

Consequences and applications: Border collisions can create or annihilate fixed points, trigger period-adding sequences, or lead to complex dynamics in piecewise maps. They are common in electronic switching circuits, mechanical systems with impacts or friction, and control systems with hysteresis. They are a key feature of non-smooth dynamical systems and require specialized mathematical techniques to analyze.