 Lorenz system

The Lorenz system is the three-variable autonomous differential equationIt arose as a low-dimensional model of thermal convection. Edward Norton Lorenz used its numerical solutions in 1963 to study nonperiodic motion and sensitivity to the initial condition, with consequences for long-range weather forecasting.^[ref]

The equilibrium points follow directly from the equations. For $\sigma >0$ and $\beta >0$, the first gives $y=x$ and the third gives $z=x^2/\beta$. The second then becomes $x(\rho -1-x^2/\beta )=0$. Thus the origin always exists, while for $\rho >1$ there are two additional equilibriawith matching signs in the first two coordinates. The symmetry $(x,y,z)\mapsto (-x,-y,z)$ exchanges them.

The Equilibrium bifurcations of the Lorenz system determine how their stability changes. The separate instantaneous Lorenz error growth at a zero-y crossing calculation concerns a perturbation at one instant; it should not be confused with the long-term Lyapunov exponent.

The Lorenz system is a set of three nonlinear ordinary differential equations originally studied by mathematician and meteorologist Edward Lorenz in 1963. It is famous for its chaotic solutions, which exhibit sensitive dependence on initial conditions—an essential feature of chaotic systems, often referred to as the "butterfly effect." The Lorenz system is defined by the following equations: 1. \(\frac{dx}{dt} = \sigma (y - x)\) 2.