Introduction

We study three-dimensional systems of the form

$$ \begin{cases} \dot{x}=f(x,y,z)\\ \dot{y}=g(x,y,z)\\ \dot{z}=h(x,y,z) \end{cases} $$

where $f,g,h:\mathbb{R}^3\to\mathbb{R}$ are continuously differentiable functions. Given an initial condition $(x_0,y_0,z_0)$, there exists a unique solution $(x(t),y(t),z(t))$ passing through $(x_0,y_0,z_0)$ at time $t=0$. Like two-dimensional systems, three-dimensional systems can have fixed points that can be locally studied by linearization. They can also have limit cycles.Recall that in two-dimensional systems, trajectories are somewhat too much constrained and thus solutions have a simple fate. Typically, they are attracted in the long run either to a fixed point or a closed trajectory. We are going to see a completely new phenomenon occurring for higher dimensional systems, namely *deterministic chaos* which is the consequence of the possible existence of a so-called *strange attractor*. This is a set with fractal structure on which the motion is aperiodic and sensitive to tiny changes in the initial conditions.

If two-dimensional systems are very well understood from the mathematical point of view, we want to stress that this is far from being the case for higher dimensional systems! We are likely decades away from rigorously understanding all the fascinating phenomena that occur in the systems we will see below. So, needless to say that we do no more than scratching the surface of a vast, wide open, subject.

A last general word before moving on: we hope that we have convinced the reader, if required, about the usefulness of interactive digital experiments in exploring and visualizing two-dimensional systems. For three-dimensional systems and, in particular, for the discovery of strange attractors, such experiments were not only useful, they were crucial.

From meteorology to the Lorenz attractor

To study the possibly complicated behavior of three-dimensional systems, there is no better place to begin than with the famous model proposed by Lorenz in 1963. Before this model appeared, the only types of stable attractors known in differential equations were fixed points and closed trajectories. This model illustrates in particular the *sensitive dependence on intial conditions*, also known by the large public as the ‘butterfly effect’ (an expression coined by Lorenz himself).

The Lorenz system is given by the equations

$$ \begin{cases} \dot{x}= \sigma(y-x)\\ \dot{y}= rx-y-xz\\ \dot{z}= xy-bz \end{cases} $$

where $\sigma,r$ and $b$ are positive parameters. Lorenz first encountered chaotic phenomena for$$ \sigma=10, \, r=28,\, b=\frac{8}{3}\cdot $$

A small hint of the physics behind the model

Explaining how Lorenz got his equations would lead us away. We content ourselves with a few words. Lorenz was interested in setting up a simple model that would explain some of the unpredictable behavior of the weather.

Physical sensible models of atmospheric convection involve partial differential equations, and are extremely complicated to analyze. Lorenz sought a much simpler system. He considered a two-dimensional fluid cell that was heated from below and cooled from above. In Fourier modes, the fluid motion can be described by a system of differential equations involving infinitely many variables. Lorenz made a tremendous simplification by keeping only three variables!

Very roughly speaking, $x$ represents the rate of convective ‘overturning’, whereas $y$ and $z$ can be thought as the horizontal and vertical temperature, respectively. Notice that $x,y,z$ are thus not representing the position of a point in the ambient space, but instead an abstract three-dimensional phase space.

Regarding the three parameters, $\sigma$ is the Prandtl number (related to the fluid viscosity), $r$ is the Rayleigh number (related to the temperature difference between the top and bottom of the cell, and $b$ is a scaling factor (related to the aspect ratio of the rolls).

A few basic properties of Lorenz equations

A bit of ground work is needed to better understand the phase portraits that we shall see later on.

You can check that when $x=y=0$, we have $\dot{x}=\dot{y}=0$, so the $z$-axis is invariant. On this axis, we have simply $\dot{z}=-bz$, so all solutions tend to the origin on this axis.

There is a symmetry in the vector field: under the transformation $(x,y,z)\mapsto (-x,-y,z)$, the equations remain the same. Hence, if $(x(t),y(t),z(t))$ is a solution, so is $(-x(t),-y(t),z(t))$. In other words, all solutions are either symmetric by reflection trough the $z$-axis, or have a symmetric companion.

The origin $(0,0,0)$ is obviously a fixed point. After some easy algebra, one finds that there is a pair of symmetric fixed points, provided that $r>1$:

$$ P_{\pm}=\big(\pm \sqrt{b(r-1)}\, ,\pm \sqrt{b(r-1)}\, , r-1\big). $$

They represent left- or right-turning convection rolls. We observe that when $r=1$, they merge with the origin. So, it is very likely that we have a*pitchfork bifurcation*at $r=1$.

When $r<1$, one can show that all solutions are attracted to the origin. In other words, the fixed point $(0,0,0)$ is globally asymptotically stable. This can be proven by using Lyapunov's theorem.

We construct a Lyapunov function on all of $\mathbb{R}^3$. Let

$$ L(x,y,z)=x^2+\sigma y^2 + \sigma z^2. $$

Then we get after some easy algebra$$ \dot{L}(x,y,z)=-2\sigma\big( x^2+y^2-(1+r)xy\big)-2\sigma b z^2. $$

We therefore have $\dot{L}(x,y,z)<0$ away from $(0,0,0)$ provided that$$ \phi(x,y)=x^2+y^2-(1+r)xy>0 $$

for $(x,y)\neq (0,0)$. This is obviously true on the $y$-axis. Consider an arbitrary straight line $y=mx$ in the $xy$-plane that is not the $y$-axis.We have

$$ \phi(x,mx)=x^2(m^2-(1+r)m+1). $$

One easily checks that the quadratic term $x^2(m^2-(1+r)m+1)$ is positive for all $m$ if $r<1$. Hence we get the wanted conclusion by Lyapunov’s theorem.Linearizing about the fixed point $(0,0,0)$ is immediate: just omit the $xy$ and $xz$ nonlinearities in Lorenz equations. We get the linear system

$$ \begin{cases} \dot{x}= \sigma(y-x)\\ \dot{y}= rx-y\\ \dot{z}= -bz \end{cases} $$

in which the equation for $z$ is decoupled from the other two equations, and gives that $z(t)\to 0$ exponentially fast. The two other variables are governed by$$ \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} -\sigma & \sigma \\ r & -1 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}. $$

The trace of the matrix is $-\sigma-1<0$ and its determinant is $\sigma(1-r)$. If $r<1$, the origin is thus a sink (and it is the only fixed point). If $r>1$, the origin is a saddle. So far, we only encountered saddles in dimension two. \texttt*[Hyperlien \`a mettre]*They have a stable and an unstable manifold. Locally, this means that there is one incoming and one outgoing directions. For a saddle in dimension 3, the number of incoming directions plus the number of outgoing directions is equal to $3$. Here, we have one outgoing and two incoming directions.

Linearizing about the two other fixed points, that exist only if $r>1$, one can check that they are sinks provided that

$$ r < r^*=\frac{\sigma(\sigma+b+3)}{\sigma-b-1} $$

with the extra condition that $\sigma>b+1$. We omit the details. It turns out that a*Hopf bifurcation*occurs at $r^*$, but this is really hard to prove. The bifurcation is subcritical. We will observe it below.

Instead of leaving the reader out on his or her own, we are going to guide him/her by proposing several digital experiments, before proposing the full version.

Sensitive dependence on initial conditions and chaos

We use the parameters $\sigma=10, r=28, b=8/3$ that led to Lorenz’s discovery. Hence we have the three fixed points that are present.

You can observe that solutions that start out very differently seem to have the same fate, if we forget the ‘transient behavior’. They both eventually wind around the symmetric pair of fixed points, alternating at times which point they encircle. This forms a complicated set, the so-called *Lorenz attractor*, on which solutions stay for ever.

The previous experiment can be misleading because it can leave the impression that if you start with two very close initial conditions, the resulting solutions travel very close to each other, before they get on the attractor but also once they are on it. The following experiment shows that this is false! This time you can see how to initially close points evolve on the attractor.

You can observe that the two solutions move quite far apart during their journey around the attractor. Moreover, you can see that the trajectories are nearly identical for a certain time period, but then they differ significantly as one solution winds around one of the symmetric fixed points, while the other solution winds around the other one. No matter how close two solutions start, they always move apart in this manner when they are on the attractor. This is *sensitive dependence on initial conditions*, one of the main features of a chaotic system.

Moreover, we observe that solutions pass from one ‘lobe’ of the attractor to the other in an apparently unpredictable manner, leading to an irregular oscillation that never repeats: we have an *aperiodic* motion. This is called *deterministic chaos* because the equations are deterministic but the solutions can behave in a seemingly random way. Recall that ‘deterministic’ means, given the present state, the future (and the past) are completely determined. Mathematically, this means that, given an initial condition $(x_0,y_0,z_0)$, there is a unique solution $(x(t),y(t),z(t))$ passing through $(x_0,y_0,z_0)$ at time $t=0$. But, to predict the future evolution, we need to know *exactly* the initial condition, which is impossible in practice. In a chaotic system, this leads to unpredictability. To illustrate this, consider a tiny blob if initial conditions around $(x_0,y_0,z_0)$. We observe that, rapidly, this blob smears out over the entire attractor! This means that a tiny error on the initial condition is amplified quickly in such a way that we only know that we are on the attractor, but we don’t know precisely where.

More on the (strange) Lorenz attractor

**All solutions are bounded**. We want to prove the observed fact that solutions that start far from the origin do at least move closer to it. In other words, we want to show that all trajectories are confined in a bounded set. We already now that this and more is true when $r<1$ (all solutions converge to the origin). To prove this fact for any $r$, define the function

$$ V(x,y,z)=rx^2+\sigma y^2+\sigma(z-2r)^2. $$

Note that the surfaces of constant $V$ are concentric ellipsoids centered at $(0,0,2r)$. We claim that there exists $v^*$ such that any solution that starts outside the ellipsoid $V(x,y,z)=v^*$ eventually enters and then remains trapped therein for all future time.

We compute the derivative of $V$ along solutions of the systems:

$$ \begin{aligned} \dot{V} &=-2\sigma\big(rx^2 +y^2+b(z^2-2rz)\big)\\ &=-2\sigma\big(rx^2 +y^2+b(z-r)^2-br^2 \big). \end{aligned} $$

The equation$$ rx^2 +y^2+b(z-r)^2=\rho $$

also defines an ellipsoid when $\rho>0$. When $\rho>br^2$ we have $\dot{V}<0$. Hence we can choose $v^*$ large enough so that the ellipsoid $V=v^*$ strictly contains the ellipsoid$$ rx^2 +y^2+b(z-r)^2=br^2 $$

in its interior. Then $\dot{V}<0$ for all $v\geq v^*$.Consequently, all solutions starting far from the origin are attracted to a set that lies inside the ellipsoid $V(x,y,z)=v^*$.

**The Lorenz attractor has zero volume**.The next fact we want to establish is that the set to which all solutions are attracted have *zero volume* ! Again this is obvious when $r<1$ since, in this case, all solutions converge to the origin which has zero volume. For the values for which one observes the Lorenz attractor, that is, when $\sigma=10, r=28,b=8/3$, this is not obvious at all. To prove this fact, we have to answer a basic general question: how do volumes evolve? More precisely, take any three-dimensional system

$$ \dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x}) $$

where $\boldsymbol{f}(\boldsymbol{x})=(f(\boldsymbol{x}),g(\boldsymbol{x}),h(\boldsymbol{x}))$ and $\boldsymbol{x}=(x,y,z)$. Take an arbitrary region $D$ (with smooth boundary) in $\mathbb{R}^3$. Think of the points of $D$ as initial conditions for trajectories, and let them evolve. At time $t$, we get a set $D(t)$. Let $V(t)$ be the volume of $D(t)$. Then Liouville’s theorem asserts that$$ \dot{V}=\int_{D(t)} \text{div}(\boldsymbol{f})\, \text{d}x\, \text{d}y\, \text{d}z $$

where$$ \text{div}(\boldsymbol{f})= \frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}+\frac{\partial h}{\partial z} $$

is the divergence of the vector field $\boldsymbol{f}$.Let’s apply the previous formula for the Lorenz equations. This gives immediately

$$ \dot{V}=-(\sigma+1+b) V $$

since the divergence is the constant $-(\sigma+1+b)$. Solving this simple equation yields

$$ V(t)=e^{-(\sigma+1+b)t}V(0) $$

where $V(0)$ is the volume of $D$. The conclusion is that any volume in phase space must shrink exponentially fast to $0$. Hence, in particular, the Lorenz attractor has zero volume.As you know, a point, a curve or a surface has zero volume in $\mathbb{R}^3$. The Lorenz attractor looks like a pair of surfaces which merge into one in the lower portion. But this is difficult to believe in view of the uniqueness of solutions of the differential equation: trajectories cannot cross or merge! In fact, a more-in-depth study would reveal that the attractor is not a surface but a *fractal set*, whose Hausdorff dimension is numerically found to be about $2.06$ (nobody has so far been able to prove this mathematically). If the attractor was a surface, the Hausdorff dimension would coincide with the usual dimension and would be equal to $2$.

The staggering conclusion is the following: when $\sigma=10, r=28,b=8/3$, all solutions of the Lorenz system get confined to a bounded set of zero volume on which they manage to move forever, and their trajectories do not intersect themselves or do not intersect any of the other trajectories. Thus, the Lorenz attractor is really a strange and extremely complex object!

The full interactive digital experiment

You can play here with the Lorenz system by changing the three parameters in wide ranges.

Here is a little sample of what you can observe if you fix $\sigma=10$, $b=8/3$, and look at the $r$ dependence of solutions of the Lorenz system:

- For $0 < r < 1$, the origin is the only fixed point and all solutions are attracted to it, as we proved above. At $r=1$, a pitchfork bifurcation takes place: for $r$ just a bit above this threshold (say, equal to $1.5$), the origin looses its stability and two attractive fixed points appear.
- At the value $r=13.93$, two repulsive limit cycles appear, each encircling one of the two symmetric fixed points. Increasing $r$, but staying below the value $r\approx 24$, we observe that these cycles shrink to these fixed points.
- At the parameter $r^*=470/19\approx 24.74$ (see the formula for $r^*$ above), there is a subcritical Hopf bifurcation: the repulsive limit cycles merge with the attractive fixed point and render them repulsive.
- At the parameter $r=28$, one observes the Lorenz attractor we already saw before.
- Decreasing $r$ wihtin the interval $[99,101]$, one can observe a kind of bifurcation we haven’t seen yet: the so-called
*period doubling bifurcation*: there is an attracting limit cycle whose period doubles as we decrease $r$ inside a tiny interval. When we leave this interval by further decreasing $r$, the previous period doubles, and so forth. In fact there is an infinite series of period doubling bifurcations! - Take for instance $r=99.96$. We get a knotted closed trajectory. It is called a $T(3,2)$ torus knot, meaning that the knot thread goes around the torus 2 times and threads through the torus hole 3 times).

To conclude on the Lorenz system, let’s say that the behavior of the Lorenz system as the parameter $r$ increases is the subject of contemporary research in mathematics, and we are a long time away from understanding all of the phenomena that occur as the parameters change.