AP-001: Prescribed Lyapunov Dynamics in Autonomous Analog Systems

Austurusa · problem statement v0.2 · 18 August 2026 · status: open, active

Problem. Can a continuous-time autonomous analog system be synthesized whose largest Lyapunov exponent follows a prescribed monotone transfer law over a robust chaotic operating interval?

Target	Construct a bounded autonomous analog flow with control voltage $V_C$ such that $\lambda_1(V_C)=aV_C+b$ to a stated tolerance over a nonzero interval, with $\lambda_1>0$ throughout.
Constraint	Periodic windows should be excluded by construction, not avoided by choosing convenient sweep points after the fact.
Physical test	The transfer law should survive a realizable circuit model, finite bandwidth, component tolerances, and device variation.
Failure condition	If the prescribed law cannot survive a stated perturbation class or converged numerical integration, the proposed architecture fails.

1. Formal objective

Let

$$\dot{\mathbf{x}}=f(\mathbf{x},V_C), \qquad \mathbf{x}\in\mathbb{R}^n.$$

The desired operating interval $V_C\in[V_1,V_2]$ supports a bounded chaotic attractor with

$$\lambda_1(V_C)=aV_C+b,\qquad a\neq0,\qquad \lambda_1(V_C)>0.$$

Exact equality is an ideal target. A physical realization would state an explicit error bound and perturbation model.

2. Design issue

In a conventional tunable chaotic oscillator, changing a parameter usually changes both local tangent-space expansion and the invariant measure of the attractor. The largest exponent can therefore be written schematically as

$$\lambda_1(V_C)=\int \rho(\mathbf{x};V_C)\,q(\mathbf{x},V_C)\,d\mathbf{x},$$

so a simple control voltage need not produce a simple $\lambda_1$ law. The constructive problem is to separate control of local expansion from reinjection geometry, residence time, and invariant measure strongly enough that the global exponent becomes an engineered transfer characteristic.

3. Structural boundary at zero

A bounded non-equilibrium trajectory of an autonomous continuous-time flow has a neutral direction associated with time translation. Thus a sustained autonomous oscillatory attractor cannot have a strictly negative largest Lyapunov exponent. This places a structural boundary at $\lambda_1=0$ for any controller that attempts to regulate the largest exponent while maintaining oscillation.

4. Candidate constructive route

One route is to seek an autonomous flow with a controlled Poincaré return map $F$ and nearly fixed return time $T_0$. If the return map has exponent $\Lambda(V_C)$, then the flow suggests

$$\lambda_1(V_C)=\frac{\Lambda(V_C)}{T_0}.$$

The resulting circuit problem is to realize stretching, folding/reinjection, and autonomous timing in a way that preserves the intended exponent law rather than merely producing visually chaotic trajectories.

5. Validation plan

Derive a candidate system and its expected Lyapunov law.
Validate with independent ODE solvers and timestep convergence.
Implement a behavioral LTspice model and check solver sensitivity.
Replace behavioral blocks with realizable analog circuitry.
Build a PCB only after the mathematical and numerical model survives falsification attempts.

6. Status

No novelty claim is made on this page. AP-001 remains an open research problem until the prior-art search and constructive analysis are complete.