It drives me nuts when I struggle to get my head around something just to, when I eventually do, discover that it is not really as complicated as the documentation makes it out to be. Simple examples are what I need. Here is an example of how to implement the Lorenz System. Not that I really know much about the Lorenz System but it is always a simple system to use when you want to try out a new way of solving differential equations. More information about the
Lorenz System can be found in Wikipedia.
The first thing to do is to create a class containing the equations. The Lorenz system consists of the following three equations:
Doing this using the Apache Commons Math library means that we have to create a class that implements the FirstOrderDifferentialEquations interface. This interface requires two methods to be implemented, which are computeDerivatives and getDimension:
01 import org.apache.commons.math3.exception.DimensionMismatchException;
02 import org.apache.commons.math3.exception.MaxCountExceededException;
03 import org.apache.commons.math3.ode.FirstOrderDifferentialEquations;
04
05 public class Lorenz implements FirstOrderDifferentialEquations {
06
07 double sigma = 10.0;
08 double beta = 8 / 3;
09 double rho = 28.0;
10 double X, Y, Z;
11
12 @Override
13 public void computeDerivatives(double t, double[] y, double[] yDot)
14 throws MaxCountExceededException,
DimensionMismatchException {
15 X = y[0];
16 Y = y[1];
17 Z = y[2];
18 yDot[0] = sigma * (Y - X);
19 yDot[1] = X * (rho - Z) - Y;
20 yDot[2] = (X * Y) - (beta * Z);
21 }
22
23 @Override
24 public int getDimension() {
25 // TODO Auto-generated method stub
26 return 3;
27 }
28 }
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The next step is to write the code that will call the integrator. There are various integrators. For this application we'll use the ClassicalRungeKuttaIntegrator. You will also need a step handler to capture each step of the integration:
01 import org.apache.commons.math3.ode.FirstOrderDifferentialEquations;
02 import org.apache.commons.math3.ode.nonstiff.ClassicalRungeKuttaIntegrator;
03 import org.apache.commons.math3.ode.sampling.StepHandler;
04
05 public class LorenzMain {
06
07 public static void main(String[] args) {
08
09 ClassicalRungeKuttaIntegrator rk4 =
new ClassicalRungeKuttaIntegrator(0.01);
10 FirstOrderDifferentialEquations ode = new Lorenz();
11 double[] y = new double[] { 10.0, -2.0, 50 }; // initial state
12 StepHandler stepHandler = new
WriteToFileStepHandler("lorenz.csv");
13 rk4.addStepHandler(stepHandler);
14 rk4.integrate(ode, // equations
15 0.0, // start time
16 y, // initial conditions
17 100.0, // end time
18 y); // result
19 for (int i = 0; i < y.length; i++) {
20 System.out.println(y[i]);
21 }
22 }
23 }
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The step handler has to implement the StepHandler class. For the step handler below I created a constructor that takes a filename so that the results can be written to a csv file. Obviously you can do whatever you want with this data:
01 import java.io.File;
02 import java.io.PrintWriter;
03 import java.util.ArrayList;
04
05 import org.apache.commons.math3.exception.MaxCountExceededException;
06 import org.apache.commons.math3.ode.sampling.StepHandler;
07 import org.apache.commons.math3.ode.sampling.StepInterpolator;
08
09
10 public class WriteToFileStepHandler implements StepHandler {
11 ArrayList<String> steps = new ArrayList<String>();
12 String filename;
13
14 public WriteToFileStepHandler(String filename) {
15 this.filename = filename;
16 }
17 @Override
18 public void handleStep(StepInterpolator interpolator, boolean isLast)
19 throws MaxCountExceededException {
20 double t = interpolator.getCurrentTime();
21 double[] y = interpolator.getInterpolatedState();
22 String onestep = "" + t;
23 for (int i = 0; i < y.length; i++) {
24 onestep += ", " + y[i];
25 }
26 steps.add(onestep);
27 if (isLast) {
28 try {
29 PrintWriter writer =
30 new PrintWriter(new File(filename),
"UTF-8");
31 for (String step : steps) {
32 writer.println(step);
33 }
34 writer.close();
35 } catch (Exception e) {
36 }
37 ;
38 }
39 }
40
41 @Override
42 public void init(double t0, double[] y0, double t) {
43 // TODO Auto-generated method stub
44
45 }
46
47 }
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Last but not the least I read the file with R and created the graph:
