Glossary and Definitions

There are many terms used in dynamical systems; here we try to list the most important ones relevant to the topics in this text.

Often slightly different versions of these definitions are used by different authors. We will generally conform to the definitions in the book Chaos, an Introduction to Dynamical Systems by Alligood, Sauer, and Yorke.

Glossary

Bifurcation: a change in the number or stability of equilibria or periodic orbits in a dynamical system caused by a small change in a parameter.
Box-counting dimension: a bounded set $S$ has box-counting dimension $D_b = \lim_{\epsilon \rightarrow 0} \frac{ln(N(\epsilon))}{ln(1/\epsilon)}$ where $N(\epsilon)$ is the number of boxes of size $\epsilon$ in a uniform subdivision containing any elements of $S$ . This is also sometimes called the Minkowski or Minkowski-Bouligand dimension.
Contraction mapping: a map $f$ on a space with a metric $d$ is a contraction if there is a constant $k<1$ such that $d(f(x), f(y)) < k d(x,y)$ for all $x$ and $y$ .
Critical point: a function $f$ has a critical point at $x$ if the Jacobian matrix $Df$ has less than maximal rank, or if $Df$ is not defined at $x$ . For a one-dimensional function, this means that if the derivative is defined at $x$ it is a critical point if $f'(x) = 0$ .
Critical value: the value of a function at a critical point.
Diameter: the diameter of a set $S$ in a space with metric $d$ is defined as $\sup \{ d(x,y) | x,y \in S\}$ , with the empty set having a diameter of 0.
Diffeomorphism: a map $f$ is a $C^r$ -diffeomorphism if it is a homeomorphism that is $r$ -times continuously differentiable, and the inverse of $f$ is also $r$ -times continously differentiable.
Eigenvector/eigenvalue: a nonzero vector $v$ is an eigenvector of a square matrix $A$ with eigenvalue $\lambda$ if $Av = \lambda v$ .
Fixed point: $x$ is a fixed point of a map $f$ if $f(x)=x$ .
Homeomorphism: a map $f$ is a homeomorphism if it is one-to-one, onto, continuous, and has a continuous inverse.
Hyperbolic: a linear map $A$ is hyperbolic if it has no eigenvalues of norm 1. For a nonlinear map, a fixed point $p$ is hyperbolic if the Jacobian matrix $DF$ is hyperbolic. A matrix $A$ defining a system of linear differential equations $x' = Ax$ is called hyperbolic if none of its eigenvalues have a real part equal to zero.
Irreducible matrix: a square matrix $A$ is irreducible if for each matrix coordinate $(i,j)$ there is a non-negative integer $m$ such that the $(i,j)$ th entry of $A^m$ is positive. This notion is usually within the context of Markov matrices.
Iterated function system: a collection of maps $\{f_1, f_2, \ldots f_r\}$ usually on $R^m$ , together with a discrete probability distribution $(p_1, p_2, \ldots, p_r)$ .
Jacobian matrix: for a multivariate map $F = (f_1, f_2, \ldots, f_n)$ of $n$ variables $x_i$ , the Jacobian matrix $DF$ is the $n$ by $n$ matrix of partial derivatives of the $f_i$ ; i.e. the $i$ th, $j$ th entry of $DF$ is $\frac{\partial f_i}{\partial x_j}$ . The determinant of the Jacobian matrix is also sometimes called the Jacobian of $F$ .
Jordan curve (planar): the image of a 1-to-1 map from the circle into $\mathbb{R}^2$ , or alternatively a map $f$ from the interval $[0,1]$ into $\mathbb{R}^2$ with $f(0)=f(1)$ and 1-to-1 on the interior of $[0,1]$ .
Lyapunov number: if $F$ is a differentiable map, the $k$ th Lyapunov number of the orbit starting at a point $x_0$ is $L_k = \lim_{n \rightarrow \infty} (r_{k;n})^{1/n}$ , where $r_{k,n}$ is the $k$ th principal value of the Jacobian matrix of $F$ evaluated at the $n$ th point of the orbit.
Map: a function whose range is contained in (possibly equal to) its domain (so it can be iterated).
Markov matrix: a square matrix with non-negative entries whose columns add up to 1. Sometimes also called a stochastic matrix. If both the row and column sums are 1, then it is called a doubly stochastic matrix.
Matrix exponential: the exponential of a square matrix $A$ is defined by $\displaystyle e^A = \sum_{k=0}^{\infty}\frac{A^k}{k!}$ .
One-to-one/injective: a map $f$ is one-to-one, or injective, if $f(x_i) = f(x_j)$ implies that $x_i = x_j$ .
Onto/surjective: a map $f$ is onto, or surjective, if for every point $y$ in the range of $f$ there is an $x$ in the domain such that $f(x) = y$ .
Orbit: the orbit of a point $x$ under a map $f$ is the set $\{x, f(x), f^2(x), f^3(x), \ldots \}$ .
Period- $k$ point: for a map $f$ and a point $p$ , if $k$ is the smallest positive integer such that $f^k(p) = p$ then we call $p$ a period- $k$ point; equivalently $p$ is periodic with period $k$ .
Saddle: a hyperbolic fixed point of a linear map $A$ is a saddle, or saddle point, if $A$ has eigenvalues both larger and smaller than 1 in magnitude.
Sensitive dependence near a point: A map $f$ and point $x$ have sensitive dependence on initial conditions if there exists a distance $d$ such that for any neighborhood $U$ of $x$ , there is a point $y$ with $|f^k(x) - f^k(y)| > d$ for some positive integer $k$ .
Sink: also called an attracting fixed point, this is a fixed point $p$ of a map $f$ which has an open neighborhood $U$ such that for all $x \in U$ , the limit of the iterates of $x$ by $f$ converge to $p$ .
Source: a fixed point $p$ of a map $f$ which has an open neighborhood $U$ of $p$ such that for any point $x \in U$ , $x \neq p$ , there is an iterate $f^m(x) \notin U$ .
Stable manifold: the stable manifold of a point $p$ for a map $f$ is the set of points $x$ such that $|f^n(x) - f^n(p)|$ has a limit of zero as $n$ approaches infinity.
Topological conjugacy: two maps $f$ and $g$ are topologically conjugate if there exists a homeomorphism $h$ such that $h \circ f = g \circ h$ .
Topological transitivity: a map $f$ on a space $S$ is topologically transitive if for any two nonempty open sets $U$ and $V$ in $S$ there exists an integer $n$ such that $f^n(U) \cap V \neq \emptyset$ .
Transitive matrix: a square matrix $A$ is transitive if there is some power of $A$ with all positive entries. This notion is usually within the context of Markov matrices.
$\omega$ -limit set: For a map $f$ , the $\omega$ -limit set of a point $x_0$ is the set of points $x$ such that for any integer $N$ and $\epsilon > 0$ there exists an integer $n > N$ such that $|f^n(x_0) - x| < \epsilon$ . (There are several other equivalent definitions.)