Lyapunov-like Observable Functions

• Lyapunov-like observable functions are generalized constructs that use Whitney’s size functions to quantify the spread of orbit sets in dynamical systems.
• They apply a decreasing scalar measure along trajectories to certify stability, isolate invariant sets, and detect expansive behavior in both continuous and discrete frameworks.
• The unified approach bridges metric and topological dynamics, enabling explicit construction and analysis of complex systems through the behavior of compact or continuum sets.

A Lyapunov-like observable function is a construct that generalizes the classical Lyapunov function paradigm to certify or “observe” dynamical behaviors—such as stability, expansiveness, or attractivity—by quantifying the geometric or topological evolution of orbits in a dynamical system. These functions produce a scalar value associated with the state or a set of states, and are designed to decrease (or exhibit some monotonic property) along orbits outside a target invariant set. The theory encompasses and extends traditional Lyapunov functions, enabling their application to a wide range of phenomena, including isolated invariant sets, expansive maps, continuum-wise expansive homeomorphisms, and more general non-linear or topological dynamical structures.

1. Whitney’s Size Functions as a Foundation

Whitney’s size functions are pivotal in this framework. For a compact metric space $(X,\mathrm{dist})$, consider the hyperspace $K(X)$ of all nonempty compact subsets of $X$. A size function $p: K(X)\to\mathbb{R}$ is defined by:

• $p(A) \ge 0$ for all $A\in K(X)$, with $p(A) = 0$ if and only if $A$ is a singleton,
• For $A \subsetneq B$, $p(A) < p(B)$.

A canonical construction uses a dense sequence $\{q_i\}$ in $X$, setting

$$p_i(A) = \max_{x\in A} \mathrm{dist}(q_i, x) - \min_{x\in A} \mathrm{dist}(q_i, x),\quad p(A) = \sum_{i=1}^{\infty} \frac{1}{2^i} p_i(A).$$

This $p$ captures a “spread” or “resolution” of $A$, quantifying its nontriviality (non-collapse to a point).

2. Constructing Lyapunov-like Observable Functions

Given a dynamical system, such as a continuous flow $\varphi: \mathbb{R}\times X\to X$ (with an asymptotically stable equilibrium $p$), construct the orbit set

$$\mathcal{O}(x) = \{\varphi_t(x) : t \geq 0\} \cup \{p\}.$$

The observable function is then

$V(x) = p(\mathcal{O}(x)).$

By the properties of $p$, $V(x) = 0$ if and only if $x = p$. The crucial monotonicity arises from the inclusion relationship along orbits: if $\varphi_t(x) \neq p$ for $t > 0$ and the orbit in $[0,t]$ remains inside an appropriate neighborhood, then

$V(\varphi_t(x)) < V(x),$

since $$\mathcal{O}(\varphi_t(x)) \subsetneq \mathcal{O}(x)$$ and $p$ is strictly increasing on proper inclusions.

Similar constructions apply in discrete time with a homeomorphism $f: X\to X$. Defining $f': K(X)\to K(X)$, $f'(A) = \{f(x)\mid x\in A\}$, the function $V(A) = p(A)$ or its variants serves as a Lyapunov function on compact subsets, especially when considering sets of singletons $F_1 = \{A\in K(X) : |A|=1\}$.

3. Applications to Stability, Isolated Sets, and Expansive Dynamics

a) Asymptotically Stable Equilibria and Isolated Sets

For continuous flows, $V(x) = p(\mathcal{O}(x))$ is continuous, nonnegative, and vanishes only at the equilibrium. Its strict decrease along orbits outside $p$ provides a direct Lyapunov function certifying stability. For more general isolated invariant sets $A$, a quotient or suspension construction enables “collapsing” $A$ to a point in a suitable extended space, allowing for $V(x) = p(\mathcal{O}_A(x))$ with analogous properties: positive outside $A$, zero on $A$, and strictly decreasing elsewhere.

b) Expansive and Continuum-Wise Expansive Homeomorphisms

Given an expansive homeomorphism $f$ on $X$, the singletons $F_1\subset K(X)$ form an isolated invariant set for $f'$. The Lyapunov function $V(A)$ constructed via the size function $p$, defined on a neighborhood $U$ of $F_1$, satisfies:

• $V(A) = 0$ if and only if $A$ is a singleton,
• $V(f'(A)) < V(A)$ for all non-singleton $A\in U$.

For continuum-wise expansive homeomorphisms, the space $C(X)$ of continua replaces $K(X)$, and analogous Lyapunov functions derived from size functions on $C(X)$ serve as certificates of continuum-wise expansiveness.

4. Analytical Formulation and Key Properties

Property	Description	Mathematical Expression
Size function $p$	Measures spread; detects singleton collapse	$p(A)=0 \iff |A|=1; p(A)<p(B)$ if $A\subsetneq B$
Observable function $V$	Lyapunov-like; vanishes on invariant set, strict decrease elsewhere	$V(x)=p(\{\varphi_t(x):t\geq 0\}\cup \{p\})$
Decrease condition	Strict decrease along trajectories outside the invariant set	$V(\varphi_t(x))<V(x)$ for $t>0$ (when orbit remains in $U$)

This formalism unifies classical Lyapunov analysis (cf. Massera, Lewowicz) and topological approaches via the hyperspace of compacts or continua and the use of size functions.

5. Generality and Unification Across Dynamical Contexts

The methodology based on Whitney’s size functions provides a universal approach: it applies not only to the standard context of equilibria but also to abstract invariant sets (including those not isolated in phase space but isolated in some extended or quotient construction) and to expansive phenomena beyond the reach of classical Lyapunov techniques. The key geometric property exploited is that the relevant “spread” (in the sense encoded by $p$) decreases strictly outside the target set, and this property is preserved under rich classes of dynamical evolutions.

By formulating Lyapunov-like observable functions in terms of the behavior of compact or continuum sets (rather than points), the framework bridges the gap between metric stability theory and more topological aspects of dynamical systems, such as those studied in continuum theory or topological entropy via expansivity.

6. Implications and Extensions

The use of Lyapunov-like observable functions grounded in Whitney’s size functions not only unifies a variety of settings—flows, discrete systems, isolated or more complex invariant sets, expansive and continuum-expansive maps—but also provides explicit, constructive methods for generating Lyapunov functions in these settings. These constructions have direct implications for:

• Certifying asymptotic stability in a geometric-topological context,
• Verifying isolation of invariant sets,
• Detecting and quantifying expansiveness (in both metric and continuum senses),
• Generalizing and recovering the results of classical theory in broader settings.

Key formulas essential in this approach include:

$$p(A) = \sum_{i=1}^\infty \frac{1}{2^i} \left( \max_{x\in A} \mathrm{dist}(q_i, x) - \min_{x\in A} \mathrm{dist}(q_i, x) \right),$$

and observable Lyapunov functions such as

$V(x) = p(\{\varphi_t(x): t \ge 0\} \cup \{p\}).$

The reduction of monotonicity or decrease properties to the set-inclusion ordering on orbit images is at the core of this unification.

7. Significance in Dynamical Systems Analysis

The Lyapunov-like observable function framework, grounded in Whitney’s size functions, extends the reach of stability, isolation, and expansiveness verification across classical, metric, and topological dynamical systems. It provides a systematic, constructive approach that is both general and sensitive to the fine structure of dynamical behavior, exemplifying a robust bridge between geometric, analytical, and topological methods in modern qualitative theory of dynamical systems (Artigue, 2014).