Abstract
Paper 185 wrote that the single word “stable” points at two things. This paper adds four more and shows numerically that what separates all six is what you perturb. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, staying near and coming back are different. Around a centre the orbit holds its radius to one part in $10^{14}$ and stays as close as one likes, and it never once gets closer (Section 2).
Second, coming back and coming back at a rate are also different. For $\dot x=-x^3$ each halving costs $1.50$, $6.00$, $24.00$, $96.00$ — four times the last — and the effective decay rate slides from $1.52\times10^{-1}$ at $T=10$ to $7.25\times10^{-6}$ at $T=10^{6}$ (Section 3).
Third, this is the core. Attracting and staying near are different. For $\dot\theta=1-\cos\theta$ every orbit converges to the fixed point, and yet an orbit starting $0.001$ away first travels $3.141380$ away before returning — the closer the start, the closer the excursion comes to $\pi$ (Section 4).
Fourth, perturbing the state and perturbing the equation are different. Adding $\varepsilon$ to a centre, a perturbation of size $\pm10^{-6}$ decides by sign alone whether the orbit spirals in or out (Section 5).
Fifth, the six are mutually independent. A saddle is structurally stable and Lyapunov unstable; a centre is Lyapunov stable and structurally unstable — neither direction of implication holds (Section 6).
Sixth, the fence around linearisation is hyperbolicity. $\dot x=-x^3$ and $\dot x=+x^3$ have the same linearisation (identically zero), and one converges while the other reaches infinity in finite time (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — Lyapunov stability, asymptotic stability, exponential stability, structural stability and the Hartman–Grobman theorem are all standard. No test for stability is proposed — this paper only separates and lines up six existing definitions. Infinite-dimensional systems are not treated — every number here comes from a one- or two-dimensional ordinary differential equation. Structural stability is not decided in general — what is shown here are counterexamples, not a classification of which systems are structurally stable. Relation to Paper 185: 185 treated the classification of defects and noted along the way that “stable” points at two things, topological and energetic. This paper adds Lyapunov, asymptotic, exponential and structural to those two and lines all six up under one separator. Relation to Paper 161: 161 took a census of “fixed point” and found it is not one theorem — that was a census of whether a fixed point exists, and this is a census of in what sense that fixed point is stable. What is added is putting a number on the difference between attracting and staying near, by way of an orbit that travels almost $\pi$ away, measuring the decay rate sliding to zero across six decades, confirming over two thousand perturbations that a saddle and a centre imply nothing about each other, and showing that the fence is hyperbolicity by way of two systems with identical linearisations.
1. Notation and Position
Consider an ordinary differential equation $\dot y=f(y)$ and a fixed point $y_{*}$ of it. Every integration below uses a fourth-order Runge–Kutta scheme written here, and is checked against the closed form wherever one exists.
Position: Paper 185, in the course of classifying defects, recorded that “stable” points at two things — topologically stable, meaning it cannot be removed by continuous deformation, and energetically stable, meaning it sits at a minimum. This paper continues from there. On the dynamical side there are four more, and what separates all six is the single question of what is being perturbed.
2. Staying Near and Coming Back Are Different
Check 2.1: integrate the centre $\dot x=y$, $\dot y=-x$ to $T=200$ and measure the radius.
| Initial radius | Largest radius | Smallest radius | Drift from the start |
|---|---|---|---|
| $1.0$ | $1.000000000000$ | $1.000000000000$ | $2.82\times10^{-14}$ |
| $0.1$ | $0.100000000000$ | $0.100000000000$ | $1.78\times10^{-15}$ |
| $0.01$ | $0.010000000000$ | $0.010000000000$ | $3.52\times10^{-16}$ |
The radius does not move, and the drift is the size of the integrator's rounding.
Observation 2.2 This is Lyapunov stable and not asymptotically stable. Whatever $\varepsilon$ is demanded, starting closer than that keeps the orbit inside $\varepsilon$ forever — and yet it never once approaches the fixed point. “A disturbance does no harm” and “a disturbance is undone” are separate claims.
3. Coming Back and Coming Back at a Rate Are Also Different
Check 3.1: integrate $\dot x=-x^3$ from $x(0)=1$. The closed form is $x(t)=1/\sqrt{1+2t}$.
| $t$ | Numeric | Closed form | $e^{-t}$ for comparison |
|---|---|---|---|
| $1$ | $0.577350269190$ | $0.577350269190$ | $3.679\times10^{-1}$ |
| $10$ | $0.218217890236$ | $0.218217890236$ | $4.540\times10^{-5}$ |
| $100$ | $0.070534561586$ | $0.070534561586$ | $3.720\times10^{-44}$ |
| $1000$ | $0.022355091700$ | $0.022355091700$ | $0$ in double precision |
Check 3.2: the time to halve is $3/(2x^2)$ in closed form.
| Current value | Time to halve | Times the row above |
|---|---|---|
| $1$ | $1.50$ | ── |
| $0.5$ | $6.00$ | $4$ |
| $0.25$ | $24.00$ | $4$ |
| $0.125$ | $96.00$ | $4$ |
Check 3.3: exponential stability requires $|x(t)|\le C|x(0)|e^{-\alpha t}$ for some fixed $\alpha>0$. Measure the effective decay rate $-\log x(t)/t$.
| $T$ | $x(T)$ | Effective decay rate |
|---|---|---|
| $10$ | $0.218217890235992$ | $1.5223\times10^{-1}$ |
| $100$ | $0.070534561585860$ | $2.6517\times10^{-2}$ |
| $1000$ | $0.022355091700495$ | $3.8007\times10^{-3}$ |
| $10^{4}$ | $0.007070891041799$ | $4.9518\times10^{-4}$ |
| $10^{5}$ | $0.002236062387351$ | $6.1030\times10^{-5}$ |
| $10^{6}$ | $0.000707106604410$ | $7.2543\times10^{-6}$ |
Observation 3.4 The rate slides to zero. Whatever $\alpha>0$ is offered, a large enough $T$ falls below it. So this system is asymptotically stable and not exponentially stable — “it comes back” can be said, and “how fast” cannot. The numeric and the closed form differ by only $2.3\times10^{-16}$ at $T=1000$ — the sliding rate is not integration error.
4. Attracting and Staying Near Are Different Too
Consider $\dot\theta=1-\cos\theta$ on the circle. The right-hand side is positive except at $\theta=0$, which is the only fixed point. Since $\theta$ can only increase, a point starting just to the right of $0$ must travel all the way round before it returns.
Check 4.1: measure the largest distance from the fixed point, taken along the circle.
| Start | Largest distance reached | Distance at the end |
|---|---|---|
| $+0.1$ | $3.108379$ | $0.000010$ |
| $+0.01$ | $3.138380$ | $0.000010$ |
| $+0.001$ | $3.141380$ | $0.000010$ |
| $-0.1$ | $0.100000$ | $0.009092$ |
| $-0.01$ | $0.010000$ | $0.005000$ |
(The largest possible distance on the circle is $\pi=3.141593$.)
Observation 4.2 Every orbit converges to the fixed point, and yet the fixed point is not Lyapunov stable. It fails even the loose demand $\varepsilon=1$ — because an orbit starting $0.001$ away goes $3.14$ away. And the closer the start, the larger the excursion, tending to $\pi$.
Observation 4.3 Attracting does not imply staying near. An orbit starting on the left stays inside $0.1$ from beginning to end — for the same fixed point, the answer depends on which side the approach comes from. The word “every” in the definition of Lyapunov stability is there for exactly this.
5. Perturbing the State and Perturbing the Equation Are Different
Sections 2 to 4 perturbed the initial state. Here the perturbation is applied to the equation itself.
Check 5.1: perturb the centre to $\dot x=y+\varepsilon x$, $\dot y=-x+\varepsilon y$. The radius obeys $r(t)=r(0)e^{\varepsilon t}$.
| $\varepsilon$ | Numeric $r(200)$ | $e^{200\varepsilon}$ | Fate |
|---|---|---|---|
| $-10^{-2}$ | $0.135335283236625$ | $0.135335283236613$ | spirals in |
| $-10^{-4}$ | $0.980198673306768$ | $0.980198673306755$ | spirals in |
| $-10^{-6}$ | $0.999800019998671$ | $0.999800019998667$ | spirals in |
| $0$ | $1.000000000000012$ | $1$ | closed orbit |
| $+10^{-6}$ | $1.000200020001309$ | $1.000200020001333$ | spirals out |
| $+10^{-4}$ | $1.020201340026755$ | $1.020201340026756$ | spirals out |
| $+10^{-2}$ | $7.389056098929894$ | $7.389056098930650$ | spirals out |
The trailing $12$ in the $\varepsilon=0$ row is the integrator's rounding ($1.2\times10^{-14}$); that orbit is closed.
Observation 5.2 It is the sign, not the size, that decides. Even a perturbation of $10^{-6}$ sends the orbit outward if positive and inward if negative. So a centre is not structurally stable — however small a neighbourhood is taken, it contains systems with a different phase portrait.
Check 5.3: the same happens to the system of Section 3. Taking $\dot x=-x^3+\varepsilon x$, the linearisation at the origin becomes $\varepsilon$, and for $\varepsilon>0$ new fixed points appear at $\pm\sqrt{\varepsilon}$.
| $\varepsilon$ | $T$ | $x(T)$ from $x(0)=10^{-3}$ | Fate |
|---|---|---|---|
| $-10^{-2}$ | $2000$ | $0.000000000$ | toward $0$ |
| $-10^{-4}$ | $20000$ | $0.000134676$ | toward $0$ |
| $0$ | $20000$ | $0.000980581$ | toward $0$, but slowly |
| $+10^{-4}$ | $20000$ | $0.005962055$ | toward $\sqrt{\varepsilon}=0.0100$ |
| $+10^{-2}$ | $2000$ | $0.100000000$ | toward $\sqrt{\varepsilon}=0.1000$ |
Observation 5.4 Asymptotic stability does not imply structural stability. The origin of $\dot x=-x^3$ is asymptotically stable, and yet adding $\varepsilon x$ with any positive $\varepsilon$, however small, makes the origin unstable.
6. The Six Are Mutually Independent
Check 6.1: add random perturbations of size $10^{-2}$ to the coefficient matrix of the saddle $\dot x=x$, $\dot y=-y$, two thousand times, and measure the eigenvalues.
Result: in all two thousand the eigenvalues stayed real and kept one positive and one negative sign (the largest imaginary part seen was $0$). The phase portrait does not change — a saddle is structurally stable.
Check 6.2: for the same saddle, shrink the initial value of $x$ and run to $T=30$.
| $x(0)$ | $x(30)$ |
|---|---|
| $10^{-3}$ | $1.068647\times10^{10}$ |
| $10^{-6}$ | $1.068647\times10^{7}$ |
| $10^{-9}$ | $1.068647\times10^{4}$ |
However close the start, the orbit leaves every neighbourhood — a saddle is Lyapunov unstable.
Check 6.3 (the independence table):
| System | Lyapunov stable | Asymptotically stable | Structurally stable |
|---|---|---|---|
| Centre $\dot x=y$, $\dot y=-x$ | yes | no | no |
| Stable node $\dot x=-x$, $\dot y=-y$ | yes | yes | yes |
| Saddle $\dot x=x$, $\dot y=-y$ | no | no | yes |
| $\dot x=-x^3$ | yes | yes | no |
Observation 6.4 The saddle and the centre break the implication in both directions. The saddle is structurally stable and Lyapunov unstable; the centre is Lyapunov stable and structurally unstable. So “structurally stable implies stable” and “stable implies structurally stable” are both false. The two are not stronger and weaker versions of one thing: they are different questions.
7. The Fence Is Hyperbolicity
Check 7.1: $\dot x=-x^3$ and $\dot x=+x^3$ have linearisations at the origin that are both identically zero. Measure where the orbits go.
| System | Linearisation at $0$ | Fate from $x(0)=0.1$ |
|---|---|---|
| $\dot x=-x^3$ | $0$ | $0.0224$ at $t=1000$, slowly to $0$ |
| $\dot x=+x^3$ | $0$ | reaches infinity at $t=50$ |
| $\dot x=-x$ | $-1$ | $2.061154\times10^{-10}$ at $t=20$ |
Check 7.2: the closed form for $\dot x=+x^3$ is $x(t)=x_0/\sqrt{1-2x_0^2t}$, which for $x_0=0.1$ diverges at $t=50$.
| $t$ | Numeric | Closed form |
|---|---|---|
| $20$ | $0.129099445$ | $0.129099445$ |
| $40$ | $0.223606798$ | $0.223606798$ |
| $49$ | $0.707106781$ | $0.707106781$ |
| $49.9$ | $2.236067978$ | $2.236067977$ |
Observation 7.3 (the address of the fence) The fence around “the eigenvalues decide the stability” is hyperbolicity — that is, no eigenvalue having zero real part. Inside the fence the Hartman–Grobman theorem ties the linearisation to the nonlinear system locally, and the eigenvalues do decide. Outside it, two systems with the same linearisation part company into convergence and divergence in finite time. Paper 171 identified the fence around least squares as linearity, and Paper 177 identified the fence around arguments about continuity as compactness — here the fence is hyperbolicity.
Observation 7.4 The centre sits precisely on this fence. Its eigenvalues are purely imaginary, so their real parts are zero — which is why the linearisation has no answer, and why in Section 5 the sign of the perturbation decided. Sections 5 and 7 are two appearances of one fact.
8. The Separator: What You Perturb
Observation 8.1 line up the six senses of “stable” by what is perturbed.
| Name | What is perturbed | What is asked |
|---|---|---|
| Lyapunov stable | the initial state | does it stay near |
| Asymptotically stable | the initial state | does it come back |
| Exponentially stable | the initial state | does it come back at a fixed rate |
| Structurally stable | the vector field itself | is the phase portrait unchanged |
| Topologically stable (Paper 185) | the configuration, continuously | can it be removed |
| Energetically stable (Paper 185) | the scale | does it sit at a minimum |
Observation 8.2 The six are not a ladder of strength. The first three do form a ladder (exponential implies asymptotic, asymptotic implies Lyapunov), but the fourth stands outside that ladder — as Check 6.3 showed, neither direction of implication holds. And the last two are not words about a dynamical system at all, but about a configuration of a field.
Observation 8.3 So the sentence “this system is stable” by itself conveys nothing. It conveys something only when what was perturbed is attached to it.
9. The Census and the Close
What this paper checked:
- Staying near and coming back are different (Check 2.1): the centre holds its radius to one part in $10^{14}$ and never approaches.
- Coming back and the rate are different (Checks 3.2, 3.3): each halving costs four times the last, and the effective rate slides from $10^{-1}$ to $10^{-6}$.
- Attracting and staying near are different (Check 4.1): starting $0.001$ away, the orbit travels $3.141380$ away.
- Perturbing the state and perturbing the equation are different (Check 5.1): even at $10^{-6}$ the sign decides the fate.
- The six are mutually independent (Check 6.3): the saddle is structurally stable and Lyapunov unstable, and the centre is the reverse.
- The fence is hyperbolicity (Check 7.1): two systems with the same linearisation part into convergence and finite-time divergence.
- The separator is what you perturb (Observation 8.1).
References
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Reference
Yamagishi, Y. (2026). “Stable” Is Six Separate Words, and What Separates Them Is What You Perturb [Paper 195]. YOI: https://www.yamagishiyuuki.com/yoi/195
DOI: 10.5281/zenodo.22107170
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 195. This paper is a preprint and has not been peer reviewed.