Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, let $F:U\to {\mathbb{R}}^n$ be of class $C^1$ ($C^k$ Euclidean maps and diffeomorphisms), and let $\mathrm{\Phi }:\mathrm{\Omega }\to U$ be the local flow of $x^{\mathrm{\prime }}=F(x)$ (The local flow of an autonomous first-order system); recall that $\mathrm{\Omega }$ is open and $\mathrm{\Phi }$ is $C^1$ on $\mathrm{\Omega }$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it).

Let $x\in U$ and $t>0$ with $(t,x)\in \mathrm{\Omega }$, and let $u\in {\mathbb{R}}^n\setminus \{0\}$ be such that $D_x{\mathrm{\Phi }}_t(x)u\mathrm{\ne }0$, where $D_x{\mathrm{\Phi }}_t(x)$ is the derivative of the map $y\mapsto \mathrm{\Phi }(t,y)$ at $x$ (The local flow of an autonomous first-order system, $C^k$ Euclidean maps and diffeomorphisms). The finite-time tangent growth rate of $(x,u)$ over the time $t$ is

$r(t,x,u):=\frac{1}{t}\log \text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\parallel }{D}_x{\mathrm{\Phi }}_t(x)u{\mathrm{\parallel }}_2}{\mathrm{\parallel }u{\mathrm{\parallel }}_2}\right)\in \mathbb{R},$

the Euclidean norms being those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and $\log$ the natural logarithm (The natural logarithm as the inverse of the exponential function). The expression is a real number because $\mathrm{\parallel }u{\mathrm{\parallel }}_2>0$ and $\mathrm{\parallel }{D}_x{\mathrm{\Phi }}_t(x)u{\mathrm{\parallel }}_2>0$.

For the infinite-time notion require the forward lifetime $I_x=\{t:(t,x)\in \mathrm{\Omega }\}$ to be unbounded above. Since $I_x$ is an interval containing $0$, this means that $(t,x)\in \mathrm{\Omega }$ for every $t\ge 0$. Under this additional condition, if the limit

$\lambda (x,u):={\lim }_{\begin{array}{c}t\to \mathrm{\infty }\\ (t,x)\in \mathrm{\Omega }\end{array}}r(t,x,u)$

exists in $\mathbb{R}$, it is called the Lyapunov exponent of $(x,u)$. If the forward lifetime is bounded above or the limit does not exist, no Lyapunov exponent is assigned; in particular no value $+\mathrm{\infty }$ or $-\mathrm{\infty }$ is named by this definition.

• The derivative $D_x{\mathrm{\Phi }}_t(x)$ is invertible whenever it is defined: on the open set $\mathrm{\Omega }$, the maps ${\mathrm{\Phi }}_t$ and ${\mathrm{\Phi }}_{-t}$ are inverse to each other by The local flow is a partial one-parameter group: identity, composition and inverse-time laws, and the chain rule applied to ${\mathrm{\Phi }}_{-t}\circ {\mathrm{\Phi }}_t=\mathrm{i}\mathrm{d}$ gives $D{\mathrm{\Phi }}_{-t}({\mathrm{\Phi }}_t(x))\circ D_x{\mathrm{\Phi }}_t(x)=\mathrm{i}\mathrm{d}$. Hence the admissibility condition $D_x{\mathrm{\Phi }}_t(x)u\mathrm{\ne }0$ holds automatically for $u\mathrm{\ne }0$ at every point where $D_x{\mathrm{\Phi }}_t(x)$ is defined; it is recorded so that the definition does not depend on the invertibility statement.

• The rate is homogeneous in the direction: $r(t,x,cu)=r(t,x,u)$ for every real $c\mathrm{\ne }0$, since both norms scale by $\mathrm{\mid }c\mathrm{\mid }$. Thus opposite directions have the same rate, and the quantity may be negative, zero or positive, and may depend on both $x$ and the direction $u$.

• For a linear field $F(x)=Ax$ the flow is $\mathrm{\Phi }(t,x)=e^{tA}x$ wherever it is defined, so $D_x{\mathrm{\Phi }}_t(x)=e^{tA}$ is independent of $x$ and the finite-time rate reduces to $t^{-1}\log (\mathrm{\parallel }{e}^{tA}u{\mathrm{\parallel }}_2\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_2)$. Whether the limit exists is not claimed by this definition; it is a property of $A$ and $u$.