The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. The term "butterfly effect" in popular media may stem from the real-world implications of the Lorenz attractor, namely that tiny changes in initial conditions evolve to completely different trajectories. This underscores that chaotic systems can be completely deterministic and yet still be inherently impractical or even impossible to predict over longer periods of time. For example, even the small flap of a butterfly's wings could set the earth's atmosphere on a vastly different trajectory, in which for example a hurricane occurs where it otherwise would have not (see Saddle points). The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly.

In 1963, Edward Lorenz, with the help of Ellen Fetter who was responsible for the numerical simulations and figures,^[1] and Margaret Hamilton who helped in the initial, numerical computations leading up to the findings of the Lorenz model,^[2] developed a simplified mathematical model for atmospheric convection.^[1] The model is a system of three ordinary differential equations now known as the Lorenz equations:

d�d�=�(�−�),d�d�=�(�−�)−�,d�d�=��−��.

The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: x is proportional to the rate of convection, y to the horizontal temperature variation, and z to the vertical temperature variation.^[3] The constants σ, ρ, and β are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself.^[3]

The Lorenz equations can arise in simplified models for lasers,^[4] dynamos,^[5] thermosyphons,^[6] brushless DC motors,^[7] electric circuits,^[8] chemical reactions^[9] and forward osmosis.^[10] The Lorenz equations are also the governing equations in Fourier space for the Malkus waterwheel.^[11][12] The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.

From a technical standpoint, the Lorenz system is nonlinear, aperiodic, three-dimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.^[3]

One normally assumes that the parameters σ, ρ, and β are positive. Lorenz used the values σ = 10, β = ⁠8/3⁠ and ρ = 28. The system exhibits chaotic behavior for these (and nearby) values.^[13]

If ρ < 1 then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin, which is a global attractor, when ρ < 1.^[14]

A pitchfork bifurcation occurs at ρ = 1, and for ρ > 1 two additional critical points appear at(�(�−1),�(�−1),�−1)and(−�(�−1),−�(�−1),�−1).These correspond to steady convection. This pair of equilibrium points is stable only if

�<��+�+3�−�−1,

which can hold only for positive ρ if σ > β + 1. At the critical value, both equilibrium points lose stability through a subcritical Hopf bifurcation.^[15]

When ρ = 28, σ = 10, and β = ⁠8/3⁠, the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set – the Lorenz attractor – a strange attractor, a fractal, and a self-excited attractor with respect to all three equilibria. Its Hausdorff dimension is estimated from above by the Lyapunov dimension (Kaplan-Yorke dimension) as 2.06±0.01,^[16] and the correlation dimension is estimated to be 2.05±0.01.^[17] The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:^[18][16][19]

3−2(�+�+1)�+1+(�−1)2+4��

The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model.^[20] Proving that this is indeed the case is the fourteenth problem on the list of Smale's problems. This problem was the first one to be resolved, by Warwick Tucker in 2002.^[21]

For other values of ρ, the system displays knotted periodic orbits. For example, with ρ = 99.96 it becomes a T(3,2) torus knot.

Example solutions of the Lorenz system for different values of ρ
ρ = 14, σ = 10, β = ⁠8/3⁠ (Enlarge)	ρ = 13, σ = 10, β = ⁠8/3⁠ (Enlarge)
ρ = 15, σ = 10, β = ⁠8/3⁠ (Enlarge)	ρ = 28, σ = 10, β = ⁠8/3⁠ (Enlarge)
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ > 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.

Sensitive dependence on the initial condition
Time t = 1 (Enlarge)	Time t = 2 (Enlarge)	Time t = 3 (Enlarge)
These figures — made using ρ = 28, σ = 10 and β = ⁠8/3⁠ — show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10−5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.

Divergence of nearby trajectories.
Evolution of three initially nearby trajectories of the Lorenz system. In this animation the equation is numerically integrated using a Runge-Kutta routine — made using starting from three initial conditions (0.9,0,0) (green), (1.0,0,0) (blue) and (1.1,0,0) (red). Produced with WxMaxima.
The parameters are: �=28, �=10 and �=8/3 . Siginificant divergence is seen at around �=24.0, beyond which the trajectories become uncorrelated. The full-sized graphic can be accessed here.

A recreation of Lorenz's results created on Mathematica. Points above the red line correspond to the system switching lobes.

In Figure 4 of his paper,^[1] Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a Poincaré plot, which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the tent map. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.

Over the past several years, a series of papers regarding high-dimensional Lorenz models have yielded a generalized Lorenz model,^[22] which can be simplified into the classical Lorenz model for three state variables or the following five-dimensional Lorenz model for five state variables:^[23]d�d�=�(�−�),d�d�=�(�−�)−�,d�d�=��−��1−��,d�1d�=��−2��1−�0�1,d�1d�=2��1−4��1.

A choice of the parameter �0=193 has been applied to be consistent with the choice of the other parameters. See details in.^[22][23]