NUMERICAL SOLUTION OF THE LORENZ SYSTEM

Recall the definition of the Lorenz system of coupled ordinary differential equations:

dx

dt= −σx+ σy, (1)

dy

dt= −y + rx− xz, (2)

dz

dt= −sz + xy. (3)

Here, the parameters σ, s, and r are given. We use the values of σ = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and sketchedbelow. The null equilibrium is stable for r < 1. For 1 < r < rc = σ(3 + σ + s)/(σ − 1 − s) there are two non-zeroequilibrium positions given by x = y = ±

√s(r − 1), z = r − 1. These two equilibria destabilize at r = rc where a

subcritical Hopf bifurcation occurs. There are no steady solutions for r > rc. For the parameter values chosen here,the subcritical Hopf bifurcation occurs at rc ≈ 24.73.

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NUMERICAL SOLUTION OF THE LORENZ SYSTEM

We describe numerical solutions of the Lorenz system of coupled ordinary di↵erential equations:

dx

dt= ��x + �y, (1)

dy

dt= �y + rx � xz, (2)

dz

dt= �sz + xy. (3)

Here, the parameters �, s, and r are given. We use the values of � = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and illustratedin figure 1:

These solutions were obtained using a fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsys-

tem.m provided on the course webpage).Orx1rc

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NUMERICAL SOLUTION OF THE LORENZ SYSTEM

We describe numerical solutions of the Lorenz system of coupled ordinary di↵erential equations:

dx

dt= ��x + �y, (1)

dy

dt= �y + rx � xz, (2)

dz

dt= �sz + xy. (3)

Here, the parameters �, s, and r are given. We use the values of � = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and illustratedin figure 1:

These solutions were obtained using a fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsys-

tem.m provided on the course webpage).Orx1rc

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NUMERICAL SOLUTION OF THE LORENZ SYSTEM

We describe numerical solutions of the Lorenz system of coupled ordinary di↵erential equations:

dx

dt= ��x + �y, (1)

dy

dt= �y + rx � xz, (2)

dz

dt= �sz + xy. (3)

Here, the parameters �, s, and r are given. We use the values of � = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and illustratedin figure 1:

These solutions were obtained using a fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsys-

tem.m provided on the course webpage).Orx1rc

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0 1 2 3 4 5 6 7 8 9 10ï2.5

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NUMERICAL SOLUTION OF THE LORENZ SYSTEM

We describe numerical solutions of the Lorenz system of coupled ordinary di↵erential equations:

dx

dt= ��x + �y, (1)

dy

dt= �y + rx � xz, (2)

dz

dt= �sz + xy. (3)

Here, the parameters �, s, and r are given. We use the values of � = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and illustratedin figure 1:

These solutions were obtained using a fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsys-

tem.m provided on the course webpage).Orx1rc

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x

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0 1 2 3 4 5 6 7 8 9 10ï2.5

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x

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NUMERICAL SOLUTION OF THE LORENZ SYSTEM

We describe numerical solutions of the Lorenz system of coupled ordinary di↵erential equations:

dx

dt= ��x + �y, (1)

dy

dt= �y + rx � xz, (2)

dz

dt= �sz + xy. (3)

Here, the parameters �, s, and r are given. We use the values of � = 10 and s = 8/3 originally used by Lorenz (1963),and we treat r as the bifurcation parameter. We recall the bifurcation diagram discussed in the lecture and illustratedin figure 1:

These solutions were obtained using a fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsys-

tem.m provided on the course webpage).Orx1rc

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Here we present numerical solutions of the system (1)–(3) obtained using a simple fourth-order Runge-Kutta time-marching scheme (Matlab code lorenzsystem.m provided on the course webpage).

Case I: r < 1

We first consider the regime where r < 1 in figure 1. We find that all solutions converge to the null equilibriumregardless of the initial condition. No oscillations are observed in the solutions.

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Figure 1: Numerical solution for r = 0.5 and initial condition (x(0), y(0), z(0)) = (−1,−3, 3). The solution quickly converges to the nullequilibrium.

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Case II: 1 < r < rc

The second regime corresponding to 1 < r < rc is illustrated in figure 2. In this case, the null solution is unstable,and the solution converges to either one of the two non-zero equilibria, depending on the initial condition. Typicallyoscillations are observed corresponding to a spiraling motion towards the fixed point.

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Figure 2: Numerical solution for r = 10 and initial condition (x(0), y(0), z(0)) = (1,−2,−2). The solution loops around one of the twoequilibrium solutions and converges to the second one after some spiraling motions.

Case III: r > rc

The third and most interesting regime occurs when r > rc. In this case, there is no steady solution of the equations.The numerical solutions are found to remain bounded and to oscillate between two nodes in space, which are neveractually reached: this is called a “strange attractor”. The solution circles around one node before jumping to thenext, and these complex dynamics repeat ad infinitum. There is no characteristic frequency to the motions and to thejumps. The solution is chaotic, and there is an extreme sensitivity to the choice of the initial condition.

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Figure 3: Numerical solution for r = 28 and initial condition (x(0), y(0), z(0)) = (1, 3,−2). The solution never reaches equilibrium andundergoes complex time dynamics centered around two points in space. The solution is chaotic in time and corresponds to a “strangeattractor”.

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