Flows and Cascades

1. Consider an autonomous system of differential equations where x ∈ ℝn. We assume throughout the book that the function F ∈ Cr (ℝn), r ≥ 1. Fix an arbitrary point x₀ ∈ ℝn. By the Existence and Uniqueness Theorem there exists a number h > 0 such that there is a unique solution of system (1.1) defined on (-h, h) and having the following property: The graph of the map is called the integral curve of this solution. The projection of the integral curve on the phase space ℝn, i.e. the set, is called the trajectory of system (1.1) with initial conditions (0, x₀). Throughout this book we denote this trajectory by.