Logistic Map, Euler & Runge-Kutta Method andLotka-Volterra Equations

S. Y. Ha and J. Park

Department of Mathematical SciencesSeoul National University

Sep 23, 2013

Contents

1 Logistic Map

2 Euler and Runge-Kutta MethodEuler MethodRunge-Kutta MethodEuler vs. Runge-KuttaR-K Method for System ODEs

3 Lotka-Volterra Equations

Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Logistic Map

Definition

Let f : [0, 1]→ [0, 1] such that f(x) = rx(1− x) where r ∈ [0, 4].Then f is called a logistic map.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Logistic Map f(x) = rx(1− x)

Intersections are fixed points.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Two iterations of logistic map f2(x)

Intersections are fixed points and period-2 points.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Three iterations of logistic map f3(x)

Intersections are fixed points and period-3 points.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Comparison of five iterations of logistic map f5(x) with repect to r

r=2.5

r=3.5

r=3

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Bifurcation of positions of fixed points and period-2 points

The figure indicates positions of x satisfying x = f2(x) with respectto 0 ≤ r ≤ 4.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Doubling iterations

The figure indicates positions of x satisfying x = f8(x) with respectto 2.8 ≤ r ≤ 3.6.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Let x0 = 0.5 and xn = fn(x0). Plot positions of xn whilen ∈ [500, 1500].

The figure indicates positions that xn wander with respect to3.5 ≤ r ≤ 4.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Euler Method

Euler and Runge-Kutta Method

Let given ODE be x = f(x, t) and ∆t = h.

dx

dt≈ xn+1 − xn

h

Explicit Euler Method

xn+1 − xn

h= f(xn, tn)

⇒ xn+1 = xn + hf(xn, tn), tn+1 = tn + h

h must be small.

Implicit Euler Method

xn+1 − xn

h= f(xn+1, tn+1)

⇒ xn+1 = g(xn, tn), tn+1 = tn + h

It is difficult to find out g.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Runge-Kutta Method

4th Order Runge-Kutta Method

xn+1 − xn

h=

1

6(k1 + 2k2 + 2k3 + k4)

⇒ xn+1 = xn +1

6h(k1 + 2k2 + 2k3 + k4)

weighted average of k1, k2, k3, k4 where,

k1 = f(xn, tn)

k2 = f(xn +1

2hk1, tn +

1

2h)

k3 = f(xn +1

2hk2, tn +

1

2h)

k4 = f(xn + hk3, tn + h)

tn+1 = tn + h

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Euler vs. Runge-Kutta

Solve x = x(1− x).Exact solution is

x(t) =x0e

t

1 + x0(et − 1)

Trajectory of x(t) with x0 = 0.05

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Euler vs. Runge-Kutta

Let x0 = 0.05, h = 0.01.

Trajectories of x(t) with Euler and Runge-Kutta methods

Euler method Runge-Kutta method

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Euler vs. Runge-Kutta

Difference from exact solution

Error of Euler method is less than 2.5× 10−3

Error of Runge-Kutta method is less than 2.5× 10−11

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Euler vs. Runge-Kutta

Using 4th order Runge-Kutta method needs 4 times morecalculations than using Euler method.

Let h1 = 0.01 for Runge-Kutta method and h2 = 0.0025 for Eulermethod in order to adjust the number of calculations.

Error of Euler method is less than 6× 10−4

Error of Runge-Kutta method is less than 2.5× 10−11

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

R-K Method for System ODEs

Runge-Kutta Method for System ODEs

A system ODE

x = f(x, y, t)

y = g(x, y, t)

Runge-Kutta method for the system

xn+1 = xn +1

6h(k1 + 2k2 + 2k3 + k4)

yn+1 = yn +1

6h(l1 + 2l2 + 2l3 + l4)

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

R-K Method for System ODEs

where

k1 = f(xn, yn, tn),

l1 = g(xn, yn, tn),

k2 = f(xn +1

2hk1, yn +

1

2hl1, tn +

1

2h),

l2 = g(xn +1

2hk1, yn +

1

2hl1, tn +

1

2h),

k3 = f(xn +1

2hk2, yn +

1

2hl2, tn +

1

2h),

l3 = g(xn +1

2hk2, yn +

1

2hl2, tn +

1

2h),

k4 = f(xn + hk3, yn + hl3, tn + h),

l4 = g(xn + hk3, yn + hl3, tn + h),

tn+1 = tn + h.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

R-K Method for System ODEs

Higher order ODE can be transformed into a system ODE

x + ax + b = 0

Let x1 := x, x2 := x then

x1 = x2

x2 = −ax1 − b

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Lotka-Volterra Equations

Lotka-Volterra equation

x = x(a− by)

y = y(−c + dx)

with positive a, b, c, d.

We use 4th order Runge-Kutta method for calculations.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Set a = 4, b = 2, c = 3, d = 1 and h = 0.01.

We plot 10000 iterations starting at square shaped points.We plot ’o’ shape at every 10 iterations.In this system, the phase follow the orbits counterclockwise.If the phase is far from the point (3, 2), the phase moves fast.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Lotka-Volterra equation with small perturbations.

x = x(a− ε1x− by)

y = y(−c + dx− ε2y)

Set a = 4, b = 2, c = 3, d = 1, ε1 = 0.0001, ε2 = 0.0001 .

In this system, the phase moves to (3, 2) slowly.

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Lotka-Volterra equation with intraspecific competition.

x = x(a− ex− by)

y = y(−c + dx− fy)

Set a = 6, b = 3, c = 4, d = 1, e = 2, f = 1 .

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Lotka-Volterra equation with intraspecific competition.

x = x(a− ex− by)

y = y(−c + dx− fy)

Set a = 18, b = 2, c = 1, d = 1, e = 3, f = 1 .

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Logistic Map Euler and Runge-Kutta Method Lotka-Volterra Equations

Thank You

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