Replace the parabola with a corner

Stable cycles a hair from Hénon’s chaos described an uncomfortable fact about the most famous picture of a strange attractor. Hénon’s map at $a = 1.4$, $b = 0.3$ looks chaotic, and it has a positive Lyapunov exponent on any computation. But stable periodic windows crowd so closely around that parameter that a window thinner than any scan could contain it, and in that case the true long-run behaviour of almost every orbit would be a long stable cycle. Nobody can rule that out. Benedicks and Carleson’s theorem of 1991 gives chaos for a set of parameters of positive measure, and it does not reach any specific one.

In 1978 René Lozi tried something simple. He took Hénon’s formula $x' = 1 - a x^2 + y$, $y' = b x$, and replaced the square by an absolute value:

$$x' = 1 - a|x| + y, \qquad y' = b x.$$

The picture is nearly the same. The attractor still has a long folded shape, with layers within layers, except that all of its curves have become straight, with sharp corners at the folds. Two years later Michał Misiurewicz proved that this one is a strange attractor. He did it for an open set of parameters, with an argument that can be summarised on a page. A tiny change in a formula, of exactly the kind that seems to make no difference, removes the entire difficulty, and understanding why says a great deal about what was hard in the first place.

What a corner removes #

A one-dimensional comparison gives the idea. The logistic map, $x \mapsto 1 - a x^2$, has a critical point at nought, where its derivative vanishes. An orbit passing near that point is contracted by a large factor. And that is what makes periodic windows possible: a stable cycle has to pass near the critical point, which pays for all the expansion that occurs elsewhere along the orbit. The tent map, $x \mapsto 1 - a|x|$, has the same shape with a corner replacing the top of the parabola. Its slope is $\pm a$ everywhere, so with $a > 1$ every step expands, and no stable cycle can exist at all. In one precise case, the logistic map and a tent map are conjugate, and generically the logistic family has windows and the tent family has none. A corner has no critical point. That is the whole trick, and Lozi’s map is the same trick in two dimensions.

Is that a cheat? A corner is less artificial than it looks. Maps that bend sharply arise wherever a physical system switches between two modes: a converter in a power supply, a pair of impacts, a thermostat. For such systems the return map really is made of straight pieces with a crease between them, and the sudden changes they undergo as a parameter moves, known as border-collision bifurcations, have no equivalent in the smooth theory. So a system with a corner is not a smooth system with an approximation error. In its own domain it is the correct model, and it has its own kind of order, in which the road through period doubling can simply be absent, with a stable cycle jumping straight to chaos.

A cone that every step stretches #

In two dimensions, a single number will not do, because the Jacobian has one expanding direction and one contracting one. So the question becomes how to track directions. The Jacobian of Lozi’s map at $(x, y)$ is one of only two matrices:

$$\begin{pmatrix} -a & 1 \\ b & 0 \end{pmatrix} \ \text{for } x > 0, \qquad \begin{pmatrix} a & 1 \\ b & 0 \end{pmatrix} \ \text{for } x < 0.$$

Take a tangent vector $(u, v)$ with $|v| \le m|u|$, a direction within a wedge around the horizontal. Applying either matrix gives $u' = \mp a u + v$ and $v' = b u$. So $|u'| \ge (a - m)|u|$, and $|v'| = b|u| \le \tfrac{b}{a - m}|u'|$.

Two inequalities follow. If $b/(a - m) \le m$, the image lies in the same wedge. And if $a - m > 1$, the horizontal component has grown. With $a = 1.7$, $b = 0.3$ and $m = 0.3$, the first requires $0.3/1.4 = 0.214 \le 0.3$, and the second $1.4 > 1$. Both hold.

That is a complete argument. Every tangent vector in that cone stays in it and grows by at least a factor of 1.4 at each step, and it does not matter where the point is or how the orbit has been moving. Over 100,000 steps of one orbit, the measured stretch never went below 1.5, and the direction never left the cone. The stretch takes values clustering near 1.5 and 1.9, which reflects $a$ plus or minus the small slope $v/u$.

The consequence is immediate. A stable periodic orbit would need the product of its Jacobians to shrink every direction, and that cannot happen if one cone always grows. So an orbit that does not land exactly on the crease cannot be stable. The argument involves no search and no scan, and it holds for all parameters satisfying the inequalities. There is nothing like it for Hénon’s map, because there the Jacobian’s top-left entry is $-2ax$, which passes through nought at $x = 0$. Near that line, the expansion is lost and a cone can be flipped out of its wedge. The parabola has a place where the argument fails, and all the depth of the Benedicks–Carleson analysis goes into controlling how often an orbit comes close to that place. Corners have no such place to control.

Two scans side by side #

The difference is easiest to see by following the largest Lyapunov exponent through a range of parameters.

For Hénon’s map with $b = 0.3$, between $a = 1.05$ and $a = 1.42$, the curve has windows everywhere. Of 301 parameter values, 49 have an exponent at or below nought, and that is a sampling of something still finer. The lowest value reached is $-0.485$. This is the picture on which all the uncertainty rests: any parameter might be in a window too narrow to find. For the Lozi map, across its range from $a = 1.64$ to $a = 1.84$, there is not one value at or below nought, and the curve looks like a ruled line, rising from about 0.463 to about 0.57.

The smoothness of that second curve is the unexpected part. It is not just that the exponent stays positive. It appears to be a continuous function of the parameter, and at the scale of this scan nearly linear. For Lozi maps, Yutaka Ishii and David Sands established the monotonicity of entropy in a certain parameter range near the tent maps. A scan such as this can show only that no window was found. But unlike the Hénon scan, there is a proof behind it to confirm that nothing has been missed. This is what is called robust chaos, a term introduced by Soumitro Banerjee, James Yorke and Celso Grebogi in 1998. They pointed out that it is what a good engineer wants from a chaotic system, because a slight drift of a component does not shut it down, and that piecewise-straight maps of this kind are natural models of switching electronic circuits, where corners in a return map are real.

Every periodic orbit, by linear algebra #

A map made of linear pieces has one more advantage: its periodic points can all be found exactly. A point of period dividing $n$ has some pattern of signs for $x$ along its orbit. Fix a pattern of length $n$. Then the recurrence $x_{i+1} = 1 - a s_i x_i + b x_{i-1}$ is a system of $n$ linear equations in $n$ unknowns, and it has one solution. If that solution has exactly the signs assumed, it is a real orbit. Otherwise it is discarded. So finding them all is a matter of solving $2^n$ small linear systems.

For $a = 1.7$, $b = 0.3$, the counts are 4 at $n = 2$, 8 at $n = 4$, 48 at $n = 8$, 464 at $n = 12$ and 3,680 at $n = 16$. Their growth per step at the end is about 0.502, which estimates the topological entropy, in the same way that counting folds does for a one-dimensional map. It stays near the Lyapunov exponent at this value. Pesin’s formula ties that exponent to the entropy of the natural measure, and the topological entropy can only be larger, so the two being close is what one would hope to see. Each of these orbits had its multiplier calculated, and all of them are saddles, as the cone requires: the smallest unstable multiplier over all orbits is 2.25.

For Hénon’s map, nothing of this sort is available. Each periodic orbit there is the solution of a set of polynomial equations, with up to $2^n$ solutions per pattern-free system and with no simple rule for which are real. Computer searches for all periodic orbits of Hénon’s map up to period 30, by Davidchack and Lai and by others, rely on stabilising transformations and have no guarantee of being complete, though they are believed to be. Here the list is exact, and the distinction matters. This is the reason why the theory of kneading was able to handle a one-dimensional map so well: there too the counting becomes symbolic.

Where the proof applies #

The cone argument was only half of Misiurewicz’s work. He also needed a region that orbits cannot leave, and a way to rule out orbits running through the crease repeatedly. His theorem has conditions:

$$b > 0, \quad a > b + 1, \quad 2a + b < 4, \quad b < \frac{a^2 - 1}{2a + 1}, \quad a\sqrt 2 > b + 2.$$

These conditions cut out a triangle-like region with its peak near $b = 0.46$. Within that region the attractor has been proved to be transitive, and to have a positive entropy. In fact it has a natural invariant measure of the Sinai, Ruelle and Bowen kind, which Michał Rychlik and then Pierre Collet and Yves Levy constructed in the early 1980s, and which Lai-Sang Young treated in general in 1985. So the statement that “almost every orbit behaves chaotically and has this exact distribution” is a theorem here, with no exceptions and no caveats about windows. A sample of 57 random parameters from inside the region all gave a positive exponent, the lowest being 0.377.

Interestingly, Lozi’s own original parameters, $a = 1.7$, $b = 0.5$, are just outside this region, because $(a^2 - 1)/(2a + 1) = 0.43$. So the theorem does not cover the pair he drew. The figures here use $b = 0.3$, which does lie inside the region, and has the added benefit of matching Hénon’s. Whether a similar proof reaches $b = 0.5$ has been studied, and much of the parameter plane outside the region does have similar behaviour by computation. Outside, trouble can occur. Orbits might escape, and an attracting cycle may touch the crease, where the matrix argument does not apply. What happens at the boundary of the region is more complicated, and it has been studied only in part.

A comparison that reaches back #

Lozi’s map is not the only place that corners or special structure have rescued an analysis. The closest parallel is the Lorenz attractor. There too, a singularity in the return map replaces what would otherwise be a critical point. The cusp in Lorenz’s one-dimensional map has an infinite slope, and it is a strong source of expansion, not a loss of it. Because of that Warwick Tucker was able to prove in 2002 that Lorenz’s attractor exists at the classical parameters, using a cone and interval arithmetic. The essays on a flow that doubles like a parabola show what happens when the structure fails. A kind of rule emerges. A map is provable when its singular set helps the expansion, and unprovable, at a fixed parameter, when a smooth critical point works against it.

There is a family resemblance between the Lozi and Hénon maps that makes all this matter. Lozi’s map can be seen as the limiting shape of a family, in which the parabola gets narrower until it is a corner. A smooth map with a very sharp fold has nearly the same picture, and it has a critical point again, so nothing prevents it from having windows, though they should be very thin if the corner is sharp. So real systems that look like Lozi’s map, with a corner that is merely sharp, may have windows after all, and those windows may be far too thin to matter. The practical question is how thin a window has to be before it cannot be said to exist, which a proof about exact corners does not answer.

What these figures are not #

A picture of 7,000 points is a picture of a computed orbit in double precision, and it cannot establish anything about the true orbit in the long run. The theorem is what does that, and the figures only illustrate it. The cone inequalities were checked directly by arithmetic, and that is a proof for the pair of values. The Lyapunov scan is an estimate from 30,000 steps at each parameter, so it is accurate to perhaps a percent, and the sampling step is about $7 \times 10^{-4}$. For Lozi, the theorem removes the need for finer scanning within the region, but only there.

A rounding point also deserves a mention. A double-precision orbit cannot land exactly on the crease, except by a very special coincidence, and it is not obvious that its rounding does not change anything. In a hyperbolic system there is a good answer: shadowing says that a computed orbit stays near a true orbit for all time, which is a stronger guarantee than the Hénon picture has. Whether that strictly applies in this piecewise-linear setting, where the map is not smooth, needs its own care, and it was not addressed here.

What was computed and what was quoted #

The cone arithmetic is exact. The stretch measurement ran over 100,000 steps of one orbit. The Lyapunov exponents were computed at 301 parameter values, with 30,000 steps at each. Periodic points up to period 16 came from one linear solve per sign pattern, $2^{16}$ at the last, with signs checked and multipliers calculated from the exact matrix product, in floating point. The solutions have a rounding error much smaller than their distance from the crease, though no interval arithmetic was used. The random sample in the parameter region used 57 points and 6,000 steps per point.

The theorems of Misiurewicz, Rychlik, Collet and Levy, Young, Benedicks and Carleson, Tucker, and Ishii and Sands, are quoted. The statement on the entropy and Lyapunov exponent’s proximity was noticed in these runs at one parameter value, and it is mentioned rather than proved here. The remarks on smoothed corners and on robust chaos are quoted from the literature without any calculation here.

Still open: outside the box #

How much of the Lozi parameter plane has a strange attractor? Misiurewicz’s region is one explicit set, and the actual region where chaos occurs looks larger in computations, up to and including Lozi’s own parameters. Inside the larger region, the behaviour at the boundary, where the attractor ceases to exist or where a cycle touches the crease, has no complete description. Whether the entropy is monotonic everywhere it is defined, in any natural family, is a question that has partial answers from Ishii and Sands.

And there is the original question, which all this makes sharper: does Hénon’s map at $a = 1.4$, $b = 0.3$ have a strange attractor or a stable cycle? Lozi’s map shows what the difference is, in that the only obstacle is a critical line and the way that orbits return to it. It does not help to answer the question, because the obstacle is really there for Hénon’s map. Replacing the parabola by a corner proved something else, and it did not bring the original question closer.

A trade of one curve for two lines #

Replace the critical point of a parabola with the corner of an absolute value, and a single wedge of directions is stretched by every step, so that no periodic orbit can be stable. A problem that has resisted all attempts at the classical parameters becomes three lines of arithmetic, and the exponent that wandered through window after window becomes a ruled line.

The cost is that the map is no longer smooth. It is not what Hénon wrote down, and it is not really a model of anything in particular, except perhaps circuits. But it locates the difficulty exactly. Chaos in Hénon’s map does not look any different from chaos in Lozi’s, and the two pictures are close enough to be confused. All the difficulty lies in one line where a derivative vanishes, and in the question of how an orbit behaves near it.