Finite-time tangent growth rates and Lyapunov exponents of the model's flow: the rate per unit time of a transported deviation, with its existence convention and its limits

Let an autonomous stability model be fixed with force map of class $C^1$ in the state (The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clauses (i)-(ii)), so that the associated field $G$ is $C^1$ on the state domain $S_0$ and its local flow $\mathrm{\Phi }:\mathrm{\Omega }\to S_0$ is $C^1$ on the open flow domain $\mathrm{\Omega }$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it).

(i) The finite-time rate. Let $z\in S_0$, let $t>0$ with $(t,z)\in \mathrm{\Omega }$, and let $u\in {\mathbb{R}}^{6N}\setminus \{0\}$ satisfy $D_z{\mathrm{\Phi }}_t(z)u\mathrm{\ne }0$, where ${\mathrm{\Phi }}_t:=\mathrm{\Phi }(t,\cdot )$ and $D_z{\mathrm{\Phi }}_t(z)$ is its total derivative (The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (ii), The flow domain of a C¹ autonomous field is open and the flow is C¹ on it). The finite-time tangent growth rate of $(z,u)$ over the time $t$ is $r(t,z,u):=\frac{1}{t}\log \text{\hspace{0.17em}⁣}\left(\frac{\parallel D_z{\mathrm{\Phi }}_t(z)u{\parallel }_2}{\parallel u{\parallel }_2}\right)\in \mathbb{R},$ with the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and the natural logarithm of The natural logarithm as the inverse of the exponential function; this is the definition of Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow applied to the model's field $G$. The rate has the dimension of inverse time, $[r]={\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): it is a logarithm per unit of the frame time.

(ii) The Lyapunov exponent. Require the maximal forward lifetime of the reference state $z$ to be unbounded above, equivalently $(t,z)\in \mathrm{\Omega }$ for every $t\ge 0$. Under this condition, if the limit $\lambda (z,u):={\lim }_{\begin{array}{c}t\to \mathrm{\infty }\\ (t,z)\in \mathrm{\Omega }\end{array}}r(t,z,u)$ exists in $\mathbb{R}$, it is the Lyapunov exponent of $(z,u)$; if the forward lifetime is bounded above or the limit does not exist, no exponent is assigned and no value $+\mathrm{\infty }$ or $-\mathrm{\infty }$ is named. In particular the definition does not assert that the limit exists, and it does not average over an invariant measure or over initial data.

(iii) Reading and chart dependence. (a) A positive rate at time $t$ says that the deviation $u$ has grown by a factor $e^{r(t,z,u)t}$ in the Euclidean state norm on $[0,t]$; a negative rate says it has shrunk. (b) The rate depends on the norm and on the state chart: scaling the state coordinates by positive constants multiplies distances by a bounded factor and changes $r(t,z,u)$ by at most $t^{-1}\log (\text{scale ratio})$, so the large-time exponent is unchanged by fixed rescaling but need not be preserved by an unbounded chart change (Stability vocabulary of the model: Lyapunov, asymptotic and orbital stability, linear stability of equilibria and map fixed points, and hyperbolicity of equilibria and fixed points clause (vi)(b)). (c) The rate is attached to a state and a tangent vector and requires the reference state to remain in the flow domain for the whole time $t$; if the maximal solution through $z$ is not defined on $[0,t]$, no rate is assigned at $(z,t)$.

(iv) What the rate does not assert. A positive finite-time rate is not a definition of chaos, not a claim that the model's flow has a chaotic invariant set, and not an instability statement about the reference motion; it is a property of the linearised flow over the stated time. In particular, a hyperbolic equilibrium of the linear models used in the page's checks has positive rates along its unstable directions while its compact invariant sets need not be chaotic (see the page's counterexample on the repulsive linear model). No probabilistic, almost-everywhere or empirical claim is made by this definition.

• Exponents versus the chaos list. The page's chaos list is a list of topological properties of a map on an invariant set; it does not assert that any Lyapunov exponent exists, and no clause of this definition asserts any of the topological properties.