Lecture 6 made the motion of an integrable system free in action–angle variables, on tori labelled by the actions. Real systems are at best close to integrable. So the question is

Two theorems answer the first half, one destructive and one constructive; the section map answers the second. The worked example is Hénon–Heiles (§7.6), set beside the Toda ring that Lecture 8 and Lecture 9 will solve.

Imported from Lecture 6: near a torus of an integrable system there are action–angle coordinates $(\varphi,I)$, $\varphi\in T^{n}$, $I$ in an open set $D\subset\R^{n}$, in which $H=H_0(I)$ and the motion is $\dot\varphi=\freq(I)=\partial H_0/\partial I$, $\dot I=0$ (Theorem 6.5, Corollary 6.4). $H_0$ is non-degenerate if $\det(\partial^{2}H_0/\partial I^{2})\ne0$, and then, for $n\ge2$, the resonant tori of Definition 6.9 are dense and of measure zero (Figure 6.7).

Poincaré called the following the fundamental problem of dynamics: a nearly integrable system $$H(\varphi,I)=H_0(I)+\varepsilon H_1(\varphi,I)+O(\varepsilon^{2}), \qquad H_1=\sum_{k\in\Z^{n}}h_k(I)\,e^{\ii k\cdot\varphi},$$(7.1) with $\varepsilon$ small and $H_1$ expanded in a Fourier series in the angles. The planets are the original case: $H_0$ is the motion of each planet round the Sun, and $\varepsilon H_1$ their mutual attraction.

What separates the two cases is which Fourier modes act on which resonant tori.

For general $n$, $I\in\mathfrak P$ asks for $n-1$ linearly independent such $k$. In Example 7.1, $k\cdot\freq(I)=k\cdot I=0$ is a line through the origin of the $I$-plane. With $V=\cos q_1+\cos q_2$ only $k=\pm(1,0),\pm(0,1)$ have $h_k\ne0$, and $\mathfrak P$ is the two axes. With $V=P_rP_r$ every rational line through $0$ is in $\mathfrak P$: it is dense.

Here independent is Definition 5.1 read order by order in $\varepsilon$ (Kozlov, ch. IV §1). So the second choice of $V$ in Example 7.1 has no second integral of this kind, and “non-integrable” is a theorem about it, not a report that nobody has found one.

The engine of the proof is visible at first order, and we prove that part in the next section.

The obvious strategy is to remove $\varepsilon H_1$ by a canonical transformation close to the identity, as Hamilton–Jacobi removed all of $H$ in Lecture 6. Take a type-2 generating function (Proposition 4.9) with old positions $\varphi$ and new momenta $I'$, $$S(\varphi,I')=I'\cdot\varphi+\varepsilon S_1(\varphi,I'), \qquad I=\frac{\partial S}{\partial\varphi}=I'+\varepsilon\frac{\partial S_1}{\partial\varphi}, \qquad \varphi'=\frac{\partial S}{\partial I'}=\varphi+\varepsilon\frac{\partial S_1}{\partial I'} .$$ Substitute into (7.1) and expand $H_0(I'+\varepsilon\,\partial_\varphi S_1)$: $$H=H_0(I')+\varepsilon\Big(\freq(I')\cdot\frac{\partial S_1}{\partial\varphi} +H_1(\varphi,I')\Big)+O(\varepsilon^{2}).$$(7.2) The new Hamiltonian is integrable to first order if the bracket does not depend on the angles. Its angle average is $h_0$, since $\partial_\varphi S_1$ averages to zero; so we need the homological equation $$\freq(I')\cdot\frac{\partial S_1}{\partial\varphi}=h_0(I')-H_1(\varphi,I').$$(7.3)

If $H_0$ is non-degenerate and $H_1$ has finitely many Fourier modes, such open sets exist: everything off the finitely many active resonance hypersurfaces $k\cdot\freq(I')=0$. If the Poincaré set is dense, as for the second rotor of Example 7.1, no open set qualifies.

Every term is divided by $k\cdot\freq(I')$, the combination whose vanishing is resonance. On a resonant torus with $h_k\ne0$ — a point of the Poincaré set — the $k$-th mode of (7.3) reads $0=-h_k$, and there is no solution at all. Near one the coefficient is enormous. And for $n\ge2$ the resonant tori are dense, so every torus has resonant ones arbitrarily close: these are the small denominators (Figure 7.1). They are not a defect of the method. Near resonance, a small perturbation is not a small change.

The same obstruction kills integrals, which is Poincaré’s point.

Poincaré’s theorem destroys the integrals, but not necessarily the tori. A torus is an invariant set, and it may survive even where no integral does; Proposition 7.4 cannot build it, because of the resonances. The tori to hope for are those furthest from every resonance.

It is quantitative non-resonance: every term of (7.4) is then at most $\abs{h_k}\abs k^{\tau}/\gamma$, which an analytic $H_1$, with $h_k$ decaying exponentially, easily beats. And almost every vector is Diophantine.

To see what becomes of the tori we need one tool: the section map.

Take $n=2$ and fix the energy, $H=E$, a three-dimensional level set. Choose a surface $\Sigma$ in it that the flow crosses transversally — for instance $q_1=0$ with $\dot q_1>0$ — with coordinates $(q_2,p_2)$, the remaining momentum $p_1$ being fixed by $H=E$. If the orbit from $x\in\Sigma$ comes back to $\Sigma$, call the first return $P(x)$: the Poincaré map (Figure 7.2).

The proof is Stokes on the tube of orbits joining $\gamma$ to $P\circ\gamma$, on which $\symp$ vanishes; it is in the reading.

Two readings follow. An invariant torus meets $\Sigma$ in a closed curve, and $P$ maps the curve to itself. And for the unperturbed system, with $\Sigma=\{\varphi_2=0\}$, coordinates $(\varphi_1,I_1)$ and $\freq_2>0$ (reverse $\varphi_2$ if necessary), the return time is $2\pi/\freq_2$, in which $\varphi_1$ advances by $2\pi\freq_1/\freq_2$: $$P_0(\varphi_1,I_1)=\big(\varphi_1+2\pi\rho(I_1),\,I_1\big), \qquad \rho=\freq_1/\freq_2\ \text{on the level }H_0=E .$$(7.5) Each circle $I_1=c$ is turned rigidly by $2\pi\rho$. If $\rho$ changes with $I_1$ — which is what isoenergetic non-degeneracy says — $P_0$ is a twist map: circles at different heights turn by different angles.

Hénon and Heiles wanted to know whether a star in an axisymmetric galaxy has a third integral besides the energy and the angular momentum. The question reduces to two degrees of freedom, and they took as the simplest model $$H=\tfrac12\big(p_x^{2}+p_y^{2}\big)+\tfrac12\big(x^{2}+y^{2}\big)+x^{2}y-\tfrac13y^{3}.$$(7.6) The potential has a minimum at $0$ and three saddles, at $(0,1)$ and $(\pm\sqrt3/2,-\tfrac12)$, all at height $1/6$; below $E=1/6$ the motion near the origin is confined to a curved triangle.⁴

Its potential is the isotropic oscillator plus a cubic term. Rescaled to a fixed energy, the cubic term’s strength is proportional to $\sqrt E$ (a reading has the computation): the higher the energy, the stronger its effect.

Take the section $\Sigma=\{x=0,\ p_x>0\}$ at fixed $E$, with coordinates $(y,p_y)$, and plot the crossings of a dozen orbits (the tutorial at the end of this section has the code). By Proposition 7.9 the picture is a portrait of an area-preserving map, and three kinds of orbit look different:

• an orbit on an invariant torus fills a smooth closed curve;

• an orbit in an island chain fills several small curves, visited in turn;

• a chaotic orbit fills a region, erratically, and its points scatter.

An island chain is what a resonant circle of a perturbed twist map generically breaks into. Every point of a circle with rotation number $a/b$ is fixed by $P^{\,b}$ before the perturbation. After it, the Poincaré–Birkhoff theorem (a reading) guarantees at least two periodic orbits, and in the generic picture there are $2kb$ periodic points for some $k\ge1$, alternately elliptic, ringed by small closed curves, and hyperbolic, saddles of $P^{\,b}$. The two curves of points that approach a hyperbolic point, forwards and backwards in time, join into a separatrix in an integrable system; after a perturbation they generically cross, into a homoclinic tangle, which forces chaotic dynamics nearby; the reading computes one for a model twist map, the standard map. The Hénon–Heiles sections below show island chains and scatter that resemble this picture, but they are not a consequence of the theorem: the unperturbed Hénon–Heiles map has a constant rotation number and is no twist map.

Figure 7.3 is the result, and it is what Hénon and Heiles saw in 1964. At low energy the plotted orbits lie on apparent curves, as if each had a second integral. At higher energies the scatter spreads, and at $E=1/6$ most plotted orbits scatter. A finite-time test classifies launch points as regular or not (Figure 7.4): the fraction classified regular is $1$ to the resolution at small $E$ and falls near $E\approx0.1$.

Take $\Sigma=\{x=0,\ p_x>0\}$ at fixed $E$, with coordinates $(y,p_y)$; the allowed region is $\tfrac12p_y^{2}+\tfrac12y^{2}-\tfrac13y^{3}\le E$. Launch an orbit from a point of it, with $p_x=\sqrt{2E-p_y^{2}-y^{2}+\tfrac23y^{3}}$; integrate Hamilton’s equations accurately; record $(y,p_y)$ each time $x$ passes through $0$ upwards; plot; repeat for a dozen starting points.

The whole computation is a dozen lines of Python.

def rhs(t, z):                      # Hamilton's equations for (eq:6:HH)
    x, y, px, py = z
    return [px, py, -(x + 2*x*y), -(y + x*x - y*y)]
def crossing(t, z): return z[0]     # the section x = 0 ...
crossing.direction = 1              # ... crossed upwards, px > 0

def section(E, y, py, tmax=2000):
    px = sqrt(2*E - py**2 - y**2 + 2*y**3/3)
    s = solve_ivp(rhs, (0, tmax), [0, y, px, py], events=crossing,
                  rtol=1e-10, atol=1e-12)
    return s.y_events[0][:, 1], s.y_events[0][:, 3]   # (y, py)

Two things decide whether the picture can be trusted. The accuracy: $H$ must stay constant along the orbit, and checking it is part of the computation — the generator measures a drift below $10^{-8}$ over $t\approx2000$, and refuses to draw above $5\times10^{-8}$. And the event detection: the crossing must be located by root finding on $x(t)=0$, not by the nearest time step. scripts/figures/henon_heiles_sections.py is this code with a loop over launch points.

Three particles on a ring, each pushed by its neighbours through the exponential potential of Toda, with the centre of mass removed and the coordinates, time and energy rescaled (Toda, §2.9), have the Hamiltonian $$H_{\mathrm{Toda}}=\tfrac12\big(p_x^{2}+p_y^{2}\big) +\tfrac1{24}\Big(e^{2y+2\sqrt3x}+e^{2y-2\sqrt3x}+e^{-4y}\Big)-\tfrac18 .$$(7.7) This is the periodic Toda chain, a model that Lecture 9 will prove integrable; here we claim nothing else about it. Expand the exponentials: the constants cancel the $-\tfrac18$, the linear terms cancel ($2y+2y-4y=0$), the quadratic ones give $\tfrac12(x^{2}+y^{2})$, and the cubic ones give $$\tfrac1{144}\Big[(2y+2\sqrt3x)^{3}+(2y-2\sqrt3x)^{3}-64y^{3}\Big] =x^{2}y-\tfrac13y^{3}.$$ So $H_{\mathrm{Toda}}$ agrees with Hénon–Heiles (7.6) through cubic order, and differs from it at quartic order.⁵ Hénon–Heiles is the cubic truncation of the Toda chain.

Figure 7.5 puts the two side by side at $E=1/6$, the highest energy of Figure 7.3. Two Hamiltonians that agree through cubic order disagree about everything that matters. Integrability belongs to the whole Hamiltonian: the cubic Taylor polynomial cannot see it, and it need not survive truncation.

On a torus of an integrable system, in action–angle coordinates $\varphi(t)=\varphi(0)+\freq(I)t$, so a small change $\delta I$ gives $\delta\varphi(t)=\delta\varphi(0)+t\,(\partial\freq/\partial I)\,\delta I$: nearby orbits separate linearly, by shear, and every $\lambda$ is $0$. In a chaotic region they separate exponentially (Figure 7.6). Numerically one computes a finite-time exponent, as in Figure 7.4.

• Integrability is exceptional, in a stated sense. Poincaré’s theorem rules out a second integral that is a power series in $\varepsilon$, for a non-degenerate $H_0$ and a dense Poincaré set; and truncating the Toda ring at cubic order turns its regular section at $E=1/6$ into a mostly chaotic one (Figure 7.5). So the integrable systems of Lecture 8 and Lecture 9 have to be found exactly as they are.

• The tori are robust. KAM keeps most of them, so an integrable system organises the systems near it.

• A diagnostic: a section’s curves against its scatter, and Lyapunov exponents.

• Status. Poincaré’s theorem, KAM and the Poincaré–Birkhoff theorem (a reading) are cited. What is proved is the first-order step of Poincaré’s argument (Proposition 7.5): the leading term of any such integral is a function of $H_0$, which is not yet non-existence. Proposition 7.4 is proved; Lemma 7.7 is proved in a reading, and Proposition 7.9 in a reading modulo Stokes.

• What KAM does not say. No size for “sufficiently small”, no rate for “small”, and no integral. The Diophantine family it guarantees does not fill the resonance zones, and the theorem does not say what is there. The two theorems do not conflict: Poincaré’s rules out an integral of a specified form, and KAM supplies many invariant tori, not an integral.

• Hénon–Heiles is outside both theorems. It perturbs the isotropic oscillator, which is degenerate, so neither Theorem 7.3 nor Theorem 7.8, as stated, applies to it. Its sections, and the regular fraction of Figure 7.4, are measurements, not predictions.

• A positive Lyapunov exponent alone is not chaos: a hyperbolic periodic point has one. For a bounded orbit that is not periodic it is a numerical indicator, and the threshold in Figure 7.4 is a classifier.

• Genericity, and no more. The course puts no measure on the space of all Hamiltonians. It claims Poincaré’s theorem, the effect of truncation in Figure 7.5, and, in a reading, that the integrable members of the Hénon–Heiles family are isolated, for meromorphic integrals.

On a circle with $\rho=a/b$ rational (in lowest terms), every point is a fixed point of $P_0^{\,b}$: the torus is resonant, filled with periodic orbits (Proposition 6.10). On an irrational circle every orbit is dense. Perturb: $P_\varepsilon$ is still area-preserving (Proposition 7.9) and close to $P_0$. By Theorem 7.8 most of the irrational circles survive as invariant curves. A circle of fixed points is not robust — nothing forces a whole circle of solutions of $P^{\,b}x=x$ to persist — and what remains of it is fixed by a theorem of Poincaré.

(Arnold, App. 9: Poincaré announced it just before his death, and Birkhoff proved it.) Apply it between two surviving invariant curves whose rotation numbers $\rho_-<a/b<\rho_+$ straddle the resonance, to $P_\varepsilon^{\,b}$ followed by a rotation by $-2\pi a$: the inner boundary turns back, the outer forward. The result is periodic orbits of period $b$ that survive the break-up — for a sufficiently small perturbation of an isoenergetically non-degenerate system, at least two geometrically distinct ones (Arnold, App. 9, p. 422).

Two steps hide in “the inner boundary turns back”. Arnold’s hypothesis is pointwise: every point of the inner circle must move backwards. That holds because a circle homeomorphism with rotation number $\rho_-<a/b$ moves every point by less than $2\pi a$ in $b$ steps (if some point moved by exactly $2\pi a$ the rotation number would be $a/b$). And the two fixed points of the theorem might lie on one orbit; that there are two distinct periodic orbits coming from this resonant torus is Arnold’s corollary (p. 422): when an $n$-dimensional torus filled with closed trajectories of an isoenergetically non-degenerate system breaks up under a sufficiently small perturbation, at least $2^{n-1}$ closed trajectories form, counted with multiplicity, of which at least $n$ are geometrically distinct.

In the generic case there are $kb$ periodic points of each of two kinds, and they alternate round the old circle (Ott, §7.2.2): elliptic points, surrounded by small invariant curves of their own, and hyperbolic points, saddles of $P_\varepsilon^{\,b}$. The resonant torus has become an island chain (Figure 7.7).

The simplest area-preserving perturbation of a twist map is the standard map on the cylinder, $$p'=p+K\sin x,\qquad x'=x+p'\qquad(x\bmod2\pi),$$(7.8) the section map of a rotor kicked once per unit time by the potential $K\cos x$. Its Jacobian has determinant $1$, and at $K=0$ it is (7.5) with rotation $p$.⁶ Figure 7.8 shows the whole story: at $K=0.5$ the golden circle survives, deformed, while the resonant circle $\rho=1/3$ has broken into three islands and three hyperbolic points, exactly as Figure 7.7 predicts.

The hyperbolic points carry the chaos. A hyperbolic fixed point has a stable curve $W^{s}$ of points that approach it, and an unstable curve $W^{u}$ of points that approach it backwards in time. In an integrable system they join smoothly into a separatrix, as for the pendulum (§6.4). Perturbed, they generically cross. One crossing $x$ forces infinitely many, since every image $P^{m}x$ is again on both curves; the lobes between successive crossings have equal areas because the map preserves area, and as they crowd towards the fixed point they must stretch into long thin tongues. This homoclinic tangle (Figure 7.9), discovered by Poincaré, forces chaotic dynamics nearby (Ott, §§4.3, 7.2.2): a mechanism for the chaotic dynamics that Figure 7.8 shows, not a computation of that sea’s origin.

Two consequences of KAM and of the Diophantine condition. For $n=2$ it has a sharp consequence. On a three-dimensional energy level a two-dimensional torus separates, like a surface of revolution in space: an orbit starting between two surviving tori stays between them forever, and its actions stay near their initial values (Arnold, App. 8, p. 407, under the analogous isoenergetic non-degeneracy condition). For $n\ge3$ the tori separate nothing, and a slow drift through the gaps is possible — Arnold diffusion, which we only name.

Positive exponents without chaos. A positive exponent alone is not chaos: the hyperbolic fixed point of the standard map has one. For a bounded orbit that is not periodic, $\lambda>0$ is the standard numerical indicator of chaos, and that is how it is used here.

The energy as the perturbation parameter. The energy is the perturbation parameter. Rescaling $(x,y,p)\mapsto s\,(x,y,p)$ gives $$\frac{1}{s^{2}}H(sx,sy,sp) =\tfrac12\big(p_x^{2}+p_y^{2}+x^{2}+y^{2}\big)+s\big(x^{2}y-\tfrac13y^{3}\big),$$ so the dynamics at energy $E$ is that of the isotropic oscillator plus a cubic perturbation of size $\varepsilon=s=\sqrt E$ at energy $1$.

The Hénon–Heiles family. In the family $V_\lambda=\tfrac12(x^{2}+y^{2})+x^{2}y+\tfrac\lambda3y^{3}$, $\lambda\ne0$ (Hénon–Heiles is $\lambda=-1$), the known integrable members are $\lambda=1$, which separates in the coordinates $(y\pm x)/\sqrt2$, and $\lambda=6$, which has a second integral that is a polynomial of degree four, quadratic in the momenta.⁷ For every other $\lambda$ there is no second integral that is meromorphic in the complexified coordinates and momenta — in particular no rational or polynomial one (cited: Ito 1985, completed by Morales-Ruiz, Ramis and Simó 2007). Integrals that are only smooth, or defined on part of phase space, are not excluded. So the statement is: within this family, and for integrals of this class, the integrable members are isolated points of the parameter line.