|
Lorenz 96 Model
The Lorenz 96 model is a dynamical system formulated by Edward Lorenz in 1996. It is defined as follows. For i=1,...,N: :\frac = (x_-x_)x_ - x_i + F where it is assumed that x_=x_,x_0=x_N and x_=x_1 and N \ge 4. Here x_i is the state of the system and F is a forcing constant. F=8 is a common value known to cause chaotic behavior. It is commonly used as a model problem in data assimilation. Python simulation from mpl_toolkits.mplot3d import Axes3D from scipy.integrate import odeint import matplotlib.pyplot as plt import numpy as np # These are our constants N = 5 # Number of variables F = 8 # Forcing def L96(x, t): """Lorenz 96 model with constant forcing""" # Setting up vector d = np.zeros(N) # Loops over indices (with operations and Python underflow indexing handling edge cases) for i in range(N): d[i] = (x[(i + 1) % N] - x[i - 2]) * x[i - 1] - x[i] + F return d x0 = F * np.ones(N) # Initial state (equilibrium) x0[0] += 0.01 # Add smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, a dynamical system has a State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
