Lorenz system

The Lorenz system is a system of ordinary differential equations first studied by 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. In popular media the 'butterfly effect' stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.A solution in the Lorenz attractor plotted at high resolution in the x-z plane.A solution in the Lorenz attractor rendered as an SVG.An animation showing trajectories of multiple solutions in a Lorenz system.A solution in the Lorenz attractor rendered as a metal wire to show direction and 3D structure.An animation showing the divergence of nearby solutions to the Lorenz system.A visualization of the Lorenz attractor near an intermittent cycle. The Lorenz system is a system of ordinary differential equations first studied by 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. In popular media the 'butterfly effect' stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly. In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection. The model is a system of three ordinary differential equations now known as the Lorenz equations: 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 {displaystyle x} is proportional to the rate of convection, y {displaystyle y} to the horizontal temperature variation, and z {displaystyle z} to the vertical temperature variation. The constants σ {displaystyle sigma } , ρ {displaystyle ho } , and β {displaystyle eta } are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself. The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis. From a technical standpoint, the Lorenz system is nonlinear, non-periodic, three-dimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study. One normally assumes that the parameters σ {displaystyle sigma } , ρ {displaystyle ho } , and β {displaystyle eta } are positive. Lorenz used the values σ = 10 {displaystyle sigma =10} , β = 8 / 3 {displaystyle eta =8/3} and ρ = 28 {displaystyle ho =28} . The system exhibits chaotic behavior for these (and nearby) values. If ρ < 1 {displaystyle ho <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 {displaystyle ho <1} . A pitchfork bifurcation occurs at ρ = 1 {displaystyle ho =1} , and for ρ > 1 {displaystyle ho >1} two additional critical points appear at: ( β ( ρ − 1 ) , β ( ρ − 1 ) , ρ − 1 ) {displaystyle left({sqrt {eta ( ho -1)}},{sqrt {eta ( ho -1)}}, ho -1 ight)} and ( − β ( ρ − 1 ) , − β ( ρ − 1 ) , ρ − 1 ) . {displaystyle left(-{sqrt {eta ( ho -1)}},-{sqrt {eta ( ho -1)}}, ho -1 ight).} These correspond to steady convection. This pair of equilibrium points is stable only if which can hold only for positive ρ {displaystyle ho } if σ > β + 1 {displaystyle sigma >eta +1} . At the critical value, both equilibrium points lose stability through a subcritical Hopf bifurcation.