SIMULATION OF A SERVO SYSTEM USING VAN DER POL NON-LINEAR EQUATION

A servo system can be  described by the following second order differential equation:

                 (d²y/dt²) = P (1-y²)*(dy/dt) – Qy 

            Where P and Q are positive constants,  y is the  position coordinate which is a function of the time t.

This differential equation is the well-known Van der Pol non-linear equation.  To simulate this system , this second order differential equation is to be written as a set of two simultaneous equations of first order as follows:

By using the variable y1 in place of y, we get

                  dy1/dt = y2

                  dy2/dt = P(1- y1²)* y2 - Qy1

Assuming constants P to be 0.1 and Q =1.0 and the initial conditions to be y1(0)=1.0 and y2(0)=0, our equation becomes:

                                   

                  dy1/dt = y2

                  dy2/dt = 0.1(1- y1²)*y2- y1

The system has been simulated for 20 seconds with a step size of 0.001 seconds.  That is, 20000 computations have been done. The output is printed once for every  100 integration steps and the same has been plotted on graph.
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#include < stdio.h >
#include < conio.h >
#include < math.h >
#include < stdlib.h >

void main()
 {
     clrscr();
     float t=0,y1=1,y2=0,h=0.001,u11,u12,u21,u22,u31,u32,u41,u42;
     long i;
     for(i=1;i<=20000;i++)
       {
        u11=h*y2;
        u12=h*(0.1*(1-y1*y1)*y2-y1);
        u21=h*(y2+0.5*u12);
        u22=h*(0.1*(1-(y1+0.5*u11)*(y1+0.5*u11))*(y2+0.5*u12)-(y1+0.5*u11));
        u31=h*(y2+0.5*u22);
  u32=h*(0.1*(1-(y1+0.5*u21)*(y1+0.5*u21))*(y2+0.5*u22)-(y1+0.5*u21));
        u41=h*(y2+u32);
u42=h*(0.1*(1-(y1+u31)*(y1+u31))*(y2+u32)-(y1+u31));
        y1=y1+(u11+2*u21+2*u31+u41)/6;
        y2=y2+(u12+2*u22+2*u32+u42)/6;
        t=t+h;
if(i%100==0)
        printf("%f     %f     %f \n", t,y1,y2)
     }
   getch();
}

t	y1	y2		t	y1	y2		t	y1	y2		t	y1	y2
0.1	0.995	-0.09984		3.7	-0.94681	0.59661		7.3	0.64308	-1.06776		10.901	-0.10418	1.40049
0.2	0.98007	-0.19871		3.8	-0.88247	0.6892		7.4	0.53293	-1.13381		11.001	0.03686	1.41794
0.3	0.95532	-0.29572		3.9	-0.80916	0.77593		7.5	0.41664	-1.19033		11.101	0.17893	1.42117
0.4	0.92101	-0.39002		4	-0.7275	0.85618		7.6	0.2952	-1.23656		11.201	0.32059	1.40944
0.5	0.87744	-0.48085		4.1	-0.63817	0.92928		7.7	0.1697	-1.27169		11.301	0.4603	1.38219
0.6	0.82498	-0.56749		4.2	-0.5419	0.99461		7.8	0.04126	-1.29494		11.401	0.5965	1.33911
0.7	0.7641	-0.64926		4.3	-0.43953	1.05149		7.9	-0.08887	-1.30555		11.501	0.7276	1.2802
0.8	0.69531	-0.7255		4.4	-0.33191	1.09925		8	-0.2194	-1.30286		11.601	0.85203	1.20582
0.9	0.6192	-0.79561		4.5	-0.22	1.13721		8.1	-0.34898	-1.28632		11.701	0.96827	1.11672
1	0.53642	-0.85895		4.6	-0.10482	1.16467		8.2	-0.47619	-1.25557		11.801	1.07492	1.014
1.1	0.44766	-0.91492		4.7	0.01256	1.181		8.3	-0.59962	-1.2105		11.901	1.17067	0.89912
1.2	0.3537	-0.96291		4.8	0.13099	1.18559		8.4	-0.71782	-1.15126		12.001	1.25439	0.77382
1.3	0.25536	-1.00231		4.9	0.24927	1.17796		8.5	-0.82941	-1.0783		12.101	1.32515	0.64003
1.4	0.15354	-1.03253		5	0.36616	1.15774		8.6	-0.93305	-0.99242		12.201	1.38218	0.4998
1.5	0.04918	-1.053		5.1	0.48039	1.12475		8.7	-1.0275	-0.89467		12.301	1.42496	0.35517
1.6	-0.05672	-1.06321		5.2	0.59068	1.07901		8.8	-1.11163	-0.78641		12.401	1.45314	0.20813
1.7	-0.1631	-1.06272		5.3	0.69578	1.02079		8.9	-1.18448	-0.66918		12.501	1.46657	0.0605
1.8	-0.26889	-1.05119		5.4	0.79444	0.9506		9	-1.24523	-0.5447		12.601	1.46527	-0.08607
1.9	-0.37297	-1.0284		5.5	0.88552	0.86921		9.1	-1.29324	-0.41473		12.702	1.44944	-0.23014
2	-0.4742	-0.99432		5.6	0.96795	0.77762		9.2	-1.32805	-0.28106		12.802	1.41937	-0.37052
2.1	-0.57146	-0.94905		5.7	1.04075	0.67704		9.3	-1.34938	-0.14538		12.902	1.37549	-0.50619
2.2	-0.66365	-0.89295		5.8	1.1031	0.56882		9.4	-1.35712	-0.0093		13.002	1.31831	-0.63634
2.3	-0.7497	-0.82653		5.9	1.15431	0.45445		9.5	-1.35128	0.12574		13.102	1.24843	-0.76025
2.4	-0.82863	-0.75052		6	1.19383	0.33545		9.6	-1.33204	0.25848		13.202	1.16649	-0.87734
2.5	-0.89952	-0.66584		6.1	1.22129	0.21333		9.7	-1.2997	0.38781		13.302	1.07321	-0.98704
2.6	-0.96155	-0.57354		6.2	1.23645	0.08957		9.8	-1.25463	0.51278		13.402	0.96935	-1.08879
2.7	-1.01401	-0.4748		6.3	1.2392	-0.03446		9.9	-1.19731	0.6326		13.502	0.85574	-1.18201
2.8	-1.05633	-0.3709		6.4	1.22959	-0.1575		10	-1.1283	0.74654		13.602	0.73325	-1.26603
2.9	-1.08806	-0.26311		6.5	1.20777	-0.27841		10.1	-1.04822	0.85397		13.702	0.60286	-1.3401
3	-1.10887	-0.15274		6.6	1.17401	-0.39619		10.2	-0.95774	0.9543		13.802	0.46559	-1.40337
3.1	-1.11856	-0.04105		6.7	1.12867	-0.50994		10.301	-0.85762	1.0469		13.902	0.32258	-1.45487
3.2	-1.11707	0.07079		6.8	1.07218	-0.61888		10.401	-0.74864	1.13114		14.002	0.17504	-1.4936
3.3	-1.10444	0.18166		6.9	1.00507	-0.7223		10.501	-0.63169	1.20633		14.102	0.02432	-1.51848
3.4	-1.08081	0.29056		7	0.92793	-0.81953		10.601	-0.50771	1.27171		14.202	-0.12816	-1.52848
3.5	-1.04642	0.39656		7.1	0.84139	-0.90995		10.701	-0.3777	1.32646		14.302	-0.28085	-1.52262
3.6	-1.00162	0.49883		7.2	0.74619	-0.99291		10.801	-0.2428	1.3697		14.402	-0.43213	-1.50013
t	y1	y2		t	y1	y2
14.502	-0.5803	-1.46047		18.402	1.51587	0.68195
14.602	-0.72364	-1.40344		18.501	1.57595	0.51887
14.702	-0.86042	-1.32926		18.601	1.61952	0.3522
14.802	-0.98894	-1.23856		18.701	1.64635	0.18429
14.902	-1.10761	-1.13245		18.801	1.65641	0.01728
15.002	-1.21497	-1.01247		18.901	1.64989	-0.14705
15.102	-1.30971	-0.8805		19.001	1.62714	-0.3072
15.203	-1.39074	-0.73868		19.101	1.58863	-0.46201
15.303	-1.45719	-0.58933		19.201	1.53495	-0.61059
15.403	-1.50843	-0.43478		19.301	1.46675	-0.75224
15.503	-1.54405	-0.27729		19.401	1.38475	-0.88646
15.603	-1.56386	-0.11896		19.501	1.28972	-1.01279
15.703	-1.56788	0.03834		19.601	1.18247	-1.13082
15.803	-1.55628	0.19301		19.701	1.06384	-1.24011
15.903	-1.52941	0.34371		19.801	0.93475	-1.34013
16.003	-1.48771	0.48937		19.901	0.79615	-1.4302
16.103	-1.43173	0.62913		20.001	0.64907	-1.50953
16.203	-1.3621	0.7623
16.303	-1.27951	0.88833
16.403	-1.18469	1.00672
16.503	-1.07844	1.11698
16.603	-0.96158	1.21858
16.702	-0.83503	1.31091
16.802	-0.69973	1.39326
16.902	-0.55674	1.46478
17.002	-0.40717	1.52448
17.102	-0.25227	1.57126
17.202	-0.09339	1.60395
17.302	0.06801	1.62134
17.402	0.23033	1.62226
17.502	0.39188	1.60569
17.602	0.55086	1.57086
17.702	0.70542	1.51733
17.802	0.8537	1.4451
17.902	0.99384	1.35468
18.002	1.12406	1.24709
18.102	1.24274	1.12389
18.202	1.34839	0.98704
18.302	1.43977	0.83888