IDC621: Non-Linear Dynamics

The Lorenz System

Nikhil KumarMS08035

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Abstract

In this paper we study the Lorenz system, finding solutions to thecoupled differential equations describing the system. We also study a

number of bifurcation diagrams, depicting the ρ (one of theparameter involved in the model) dependence of the attractor.

1 Introduction.

The Lorenz system is a system of ordinary differential equations (theLorenz equations) first studied by Edward Lorenz. It is notable for havingchaotic solutions for certain parameter values and initial conditions. Inparticular, the Lorenz attractor is a set of chaotic solutions of the Lorenzsystem which, when plotted, resemble a butterfly or figure eight.

2 History.

The Lorenz system of differential equations arose from the work ofmeteorologist/mathematician Edward N. Lorenz, who was studyingthermal variation in an air cell underneath a thunder. As he wascomputing numerical solutions to the system of three differential equationsthat he came up with, he noticed that initial conditions with smalldifferences eventually produced vastly different solutions. What heobserved was sensitivity to initial conditions, a characteristic of chaos. Hisobservations led him to further study of the system, and since that time,about 1963, the Lorenz system has become one of the most widely systemsof ODEs because of its wide range of behaviors.

3 The Lorenz system.

The system of differential equations Lorenz use was :

x = σ(y − x) (1)

y = x(ρ− z)− y (2)

z = xy − βz (3)

where σ, ρ and β are positive parameters which denote the physicalcharacteristic of air flow. The variable x corresponds to the amplitude ofconvective current of air cell, y to the temperature difference betweenrising and falling currents, and to z to the deviation of the temperaturefrom the normal temperature in the cell.Even though a definition of chaos has not been agreed upon by

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mathematicians, two properties that are generally agreed to characterize itare:

• sensitivity to initial conditions and

• the presence of limit cycles which repeatedly double their period as ρis varied in one direction until the orbits begin to wander chaotically.

We will explore these dynamics and other behaviors of the Lorenz system.

4 Analysis of the system.

4.1 Steady States and Stability.

Solving equations (1)-(3) at equilibrium, i.e.:

σ(y∗ − x∗) = 0x∗(ρ− z∗)− y∗ = 0x∗y∗ − βz∗ = 0,

which yields that the steady states are:

(x∗, y∗, z∗) = (0, 0, 0) & (±√β(ρ− 1),±

√β(ρ− 1), ρ− 1).

This pair is stable only if ρ = σ σ+β+3σ−β−1 , which can hold only for positive ρ if

σ > β + 1. For ρ < 1, all solutions are attracted to the origin. At ρ = 1,the two equilibrium points appear with a period doubling bifurcation.Further, they are stable until some ρ∗, the figure below shows the unstablemanifold of the origin for σ = 10, ρ = 10, β = 8/3 which end up as part ofthe stable manifold of the two equilibrium points.

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4.2 Jacobian and solutions.

The system of equations (1)-(3) can be represented as X = F (X) has theJacobian matrix,

DF (x, y, z) =

−σ σ 0ρ− z −1 −xy x −β

. At the steady state (0,0,0), the

Jacobian DF(0,0,0) is

−σ σ 0ρ −1 00 0 −β

, which is a block diagonal. The

eigenvalues are −b, −1−s±√

(1−s)2+4rs

2 . For r < 1, where√(1− s)2 + 4sr < (1 + s), all three eigenvalues are negative. For r > 1, we

have one positive eigenvalue and one negative eigenvalue, of which thepositive eigenvalue belongs to unstable manifold, which is in turn part ofthe Lorenz attractor.

At the two other steady state, the eigenvalues are the roots of a polynomialof degree 3. For σ > b+ 1 and 1 < ρ < ρ∗ = (σ(σ + b+ 3)/(σ − b− 1), alleigenvalues have a negative real part and the two fixed points are stable.At ρ = ρ∗, a Hopf bifurcation happens, the two stable fixed points collide,each with an unstable cycle and become unstable. For σ = 10, b = 8/3 wehave ρ∗ = 470/19 = 24.7. For large r parameters, the attractor can besingle periodic orbit. Some periodic solutions are knots.

5 Attractors and Bifurcation.

one normally assumes σ, ρ, β > 0, but usually σ = 10, β = 8/3, and ρ isvaried. The system exhibits chaotic behavior for ρ = 28 but displaysknotted periodic orbits for other values of ρ. For example, with ρ = 99.96it becomes a T(3,2) torus knot(as shown in Plot-2). A Saddle-nodebifurcation occurs at β(ρ− 1) = 0. When σ 6= 0 and β(ρ− 1) > 0, theequations generate three critical points. The critical points at (0,0,0)correspond to no convection, and the critical points at(±

√β(ρ− 1),±

√β(ρ− 1), ρ− 1) correspond to steady convection. This

pair is stable only if ρ < σ σ+β+3σ−β−1 , which can hold only for positive ρ if

σ > β + 1. When ρ = 28, σ = 10, and β = 8/3, the Lorenz system haschaotic solutions (not all solutions are chaotic). The set of chaoticsolutions make up the Lorenz attractor, a strange attractor and a fractal ofHausdorff dimension between 2 and 3 (see Plot-1). Grassberger(1983) hasestimated the Hausdorff dimension to be 2.06± 0.01 and the correlationdimension to be 2.05± 0.01.1

1P.Grassberger and I. Procaccia(1983). ”Measuring the strangeness of strange attrac-tors.”Physica D 9(1-2):189-208.

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Plot-1. Chaotic solution of the Lorenz system , the Lorenz attractor.

Plot-2. T(3,2) torus knot obtained for ρ = 99.96.

Plot-3. At ρ = 24.74 = 470/19, the unstable cycles collide with the stableequilibrium points, and render them unstable. This is called subcritical

Hopf bifurcation.

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6 Results.

Although only a few plots have been added here, but there other plotswhich can be of interest, as:

• For 0 < ρ < 1, the origin is the only equilibrium point. At ρ = 1, apitchfork bifurcation takes place, the origin becomes unstable andtwo stable equilibrium points appear.

• Between ρ = 0.99524 and ρ = 100.795, one observes a series of perioddoubling bifurcations of stable periodic points (one has to start withthe larger value and decrease r). These bifurcations are analogue tothe Feigenbaum scenario.

The system has sensitive dependence on the initial condition anddivergence of orbits can be clearly observed by varying t, keeping ρ, σ, βfixed.

7 References.

1. Strogatz, Steven H.(1994). Nonlinear Systems and Chaos. Perseuspublishing.

2. P.Grassberger and I. Procaccia(1983). Measuring the strangeness ofstrange attractors. Physica D 9(1-2):189-208.

3. lorenz, E.N.(1963). Deterministic nonperiodic flow. J. Atmos. Sci.20 (2):130-141.

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Appendix 1. Simulating the Lorenz system.

Following is a code written in SAGE to simulate the Lorenz system.

import numpy as npfrom numpy import *from scipy.integrate import odeintimport scipydef lorenz_int(initial, t):

x = initial[0]y = initial[1]z = initial[2]sigma = 10rho = 28beta = 8.0/3x_dot = sigma * (y - x)y_dot = x * (rho -z) - yz_dot = x * y - beta* zreturn [x_dot, y_dot, z_dot]

#initial conditions.initial = [0, 1, 1.05]

t = scipy.arange(0, 200, 0.01)lorenz_sol = odeint(lorenz_int, initial, t)x = [i[0] for i in lorenz_sol]y = [i[1] for i in lorenz_sol]z = [i[2] for i in lorenz_sol]DataOut = column_stack((x,y,z))savetxt(’out.dat’, DataOut)

Change the value of variable ’rho’, to get the solutions of the Lorenzsystem for different values of ρ. Now plot the data file out.dat usingGNUPlot to get the bifurcation diagrams for the system.

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