// LvaPclm.cpo
include <fstream.h>
*include <math.h>
// x-coordinate of Coupled Logistic Map (CLMX) double CLMX(double x, double y, double r, double e)
return r*x*(1.0-x) e*(y-x);
// y-coordinate of Coupled Logistic Map (CLMY) double CLMY(double x, double y, double r, double e)
return r*v*(1.0-y) ± e*(x-y);
// Declare variational equations vl, v2
double vi (double x, double z1, double z2, double r, double e) . return (r-2.0*r*x-e)kz1 e*z2
double v2(double y, double zl, double z2, double r, double e) return e*zl (r-2.0*r*y-e)*z2,:
// The function wl represents an expression for w(t+1). Also w = w(t) double wl(double x, double y, double w, double r, double e)
return (r*r* (1 . 0-2 . 0*z.) * (1 . 0-2 . 0*y) -2. 0*e* . 0-x-y) )*w;
102
int main()
// 10 iterations to remove transients int t;
xinit = CLMX(x,y,r,e); vin±t = CLIA-17 (x,y,r,e);
zlinit = v1(x,z1,z2,r,e); z2_init = v2(y,z1,z2,r,e);
winit = wl(x,y,w,r,e);
double lambdal = log(fabs(zl_init)+ fabs(z2 init))/((double) T);
double lambda2 = log(fabs(w init))/((double) T);
data « r « • " « lambdal « " " « laMbda2 « "\n";
// cim2con.cpp
// Coupled logistic map is controlled by OGY // method to stabilize period two point // The parameter r is varied
*include <iostream.h>
*include <math.h>
*include <iomanip.h>
double CoupledLogisticMao(double x, double y, double r, double e) return r*x*(1 - x) e*(v - x);
double Secondlteratel (double x, double y, double r, double e).
double Fl = CouPledLogisticMap(x, y, r, e);
double F2 = CodpledLogisticHap(Y, x, r, e);
return CoupledLogistitMap(F1, F2, r, e);
double Seconditerate2(double x, double v, double r, double e) double Fl = CoubledLogisticMap(x, y, r, e);
double F2 = CoupledLogisticMap(y, x, r, e);
return CouoledLogisticMap(F2, Fl, r, e);
void ContrOlVector(double& Ki, double& K2, double a, double b, double r, double e, double all, double al2) double alpha = r*(1 - 2*a) -
double beta = r*(1 - 2*b) - e;
double gamma = alpha*beta e*e;
double hab = beta*a*(1 - a) b*(1 - b)*(1 e);
double hba = aloha*b*(1 - b) a*(1 - a)*(1 e);
double k = 2*e* (beta*hba*hba - alpha*hab*hab);
double ael = a ll -
K2 = (-aa2*hab T aal*(-gamma*hab T 2*beta*e*hba))/k;
104
void DifferenceVector(double& diffl, double& diff2, double x, double v, 'double a, double b)
diffl = x - a;
diff2 = v - b;
double DeltaParameter(double x, doUble y, double a, double b, double r, double e, double all, double al2)
5.
double KT.), KT2;
ControlVector(KT1, KT2, e, b, r, e, all, a1 7 );
double diffl, diff2;
DiZZ,LenLeVector(diffl, diff2, x, y, a, b);
return KTi*diffl KT2*diff2;
void Sphere(double& R, double x, double - y, double xf, double vf) (
double diffl, diff2;
DifferenceVector(diffl, diff2, x, y, xf, vi R = so-rt(diffl*diffl diff2*diff2);
void MagnitudeOfK(double& AbsK, double a, double b, double r, double e, double all, double a12)
f
double KT1, KT2;
ControlVector(KT1, KT2, a, b, r, e, all, al2);
AbsK = sart(KT1*KT1 +. KT2*KT2);
void main ( )
coast double pi = 3.141592654;
double theta_degrees;
cout << endl << "Enter Theta Value (Degrees) " ; cin >> theta degrees;
for(int k = 0; k < arid - 1; k++) double xk = k;
double rho =(xk 1)*drho;
double alphal = rho*sin(theta);
double alpha2 = rho*cos(theta) - 1 ; cout << endl << "alPh=, 1 = " << allobP1
<< " alpha2 = " << alpha2;
MagnitudeOfK(AbsK, a, b, r, epsilon,alphal,alpha2);
long nn = 0;
do {
xl = Seconditeratel(x, y, r, epsilon);
yl = Secondlterate2(x, y, r, epsilon);
x = x i ; y= yl;
n iterations++;
Sphere(Radius, x, y, x fix, yfix);
MagritudeOfK(AbsK, a, b, r,_epsilon,alphal,alpha2);
while (Radius > delta r/. 7AbsK);
_ —
long transient = niterations;
double Kl, K2;
COnti- olVactor(K1, K2, a, b, r, epsilon, alpha?, alpha2);
double diffl, diff2;
DifferenceVector(diffl, diff2, x, y, a, b);
double accuracy = 0.000000001;
niterations = 0;
double dr;
106
do
niterations++;
dr = - DeltaParameter(x,y,.a,b,r,epsilon,alphal,alpha2);
r = rnominal dr;
xl = Seconditeratel(x, y, r, epsilon);
vi = Seconditerate2(x, y, r, epsilon);
x = xl;
y = yl;
while (fabs(dr) > accuracy);
cout << endl << "x = << setw(10) << x << "
<< " Transients = " << transient
<< " Time = " << niterations ; 1
cout << endl;
y = " « setw(l0) << y
107
// logtarl.cDp
// Apoint in the attractor of the logistic map is targetted using the // method of Shinbrot et. al.
*include <iostream.h>
*include <math.h>
*include <iomanip.h>
double logmap(double x, double p) {
return D*x*(1-x);
void main()
double epsilon;
cout << endl << "Ent-=r Required Accuracy << endl;
cin >> epsilon;
double target, x;
cout << endl << "Enter Target Value" << end1;•
cin >> target;
cout << endl << "Enter Initial Value of x " << endl;
cin >>.x;
while (delta x epsilon) int hit interval = 0;
Waile(hit_interval == 0) x_0 = logmap(x0,o_11);
x minus = logmap(x minus, p_h - deltap);
x_ plus = logmap(x_51us, ph deltap);
deltax = fabsctarget - x_0);
n_c--Ols++;
i -F(delta_x < epsilon) break;
if (((target < x0). & (target > x minus)) ((target < x_minus) & (target > x_0) . )) hitinterval = 1;
if(((taraet < x9) & (target > xplus)) 11 - ((target < xplus) & (target > x_0))) hit interval = 2;
if(hitinterval == 1) delta_p /= 2;
"o_h -= delta p;
deltax = fabs(tar.et - x_0);
)
<< "n_iterations = " << n_itel-ations << endl cout <<
<<
endl endl;
<< "Total Number of Function Calls : " << n_calls
109
7/ polvgon.cpp
// Program tests whether a given point lies inside or // outside a polygon
*include <iostream.h>
double min (double x, double y) if(x<v) return x;
else return y;
double max(double x, double v) if(x>v) return x;
else return v;
int inpolvgon(double* xc, double* yc,. int n, double xtarcet, double Ytarget) int counter = 0;
void main()
cout << "point outside"; colt « endl;
x = 3.0; v = 2.5;
int result2 = inpolygon(xc,vc,n,x,y);
if(result2 == 0)
cout << "point inside";
else
cout << "point outside"; cout << endl;
x = 0.9; y = 1.5;
int result3 = inpolygon(xc,yc,n,x,v);
if(result3 == 0)
cout « "point inside";
else
cout << "point outside"; cout « endl;
x = 1.0; v = 2.0;
// tarbis2.cmp
*include <iostream.h>
*include <math.h>
double CLM(double x, double y, double r, double e) return r*x*(1-x) e*(y - x);
double SCLMX(double x, double y, double r, double e) double Fl = C .1.14(x,y,r,e);
double F2 = CLM(y,x,r,e);
return CLM(F1,F2,r,e);
double SCLMY(double x, double Y, double r, double e) double Fl = CLM(x,y,r,e);
double F2 = CL1;1(y,x,r,e);
return CLM(F2,Fl,r,e);
double min(double x, double v) if(x<y) return ;
else return y;
}
double max (double x, double y) if(x>y) return x;
else return v;
}
// "inside-outside test algorithm"
int inoolygon(double* xc, double* vc, int n,
double x_target, double y_target)
for(i=1; i<=n; i++)
{xinters = (vtarget-vt1)*(xt2-xt1)/(vt2 - yt1)+xt1;
if((xt1==xt2) II (x_target<= xinters))
cout << "Enter Required Accuracy: ";
cin >> acc;
int n;
cout << "Enter Number of iterations: ";
cin >> n;
for(int iter=0; iter<n; iter++)
double Dl = (double)deltal/pow(2,iter);
double D2 = (double)delta2/now(2,iter);
113
// Define the nine grid-points of the auadrilaterAl jut n auarl = 9;
double* xq = new double(npaad);
double* va = new double[n_auad];
double r0 = r - Dl;
// Determine into which of the four sub-quads // :the taraet point falls
int nc = 4;
double* xc = new double[nc];
double* yc = new double[nc];
//First sub-quad: top left
//Redefine the points according to which sub-quad they fall into : // 1 -> (-,+), 2 -> (-,-), 3 -> (+,-), 4 -> (÷,÷)
if(resulti == 0) r = r - D1/2.0;
e = e D2/2.0;
cout << "Target in sub-quad 1" « endl;
cout << "r = " « r « endl;
cout << "e = " « e « endl;
if(result2 =1= 0) r = r - D1/2.0;
e = e - D2/2-0;
cout << "Target in sub-quad 2" « cout << "r = " << r << endl;
cout << "e = " << e « endl;
115
if(resUlt3 == 0)
« endU r D1/2.0;
= e - D2/2.0;
cout << "Target in sub-crud 3"
cout « "r = " << r << enr11;
cout << "iteration " << iter cout << "D 1 = " << Dl << "
et - xq[4]) < acc) && (fabs(vLtarget - yq[4) < acc)))
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125
Index
Coupled Logistic Map 43
Crisis 51
Feedback Gain Matrix 23
Feedback System 22
Fixed Point 23
Functional Determinant 46
G
General Structure Theorem 31
Gronwall Inequality 8
I
Implicit Function Theorem 26
Information Dimension 73
Input 2
Input-output Theory 3
Internal Model Principle
126
J
Jacobian 46
Joint Probability Distribution 28
L
Laplace Transform 30
Lipschitz Constant 6
Lipschitz Continuous 6
Local Existence and Uniqueness 5
Local Stable Manifold 55 .
Local Unstable Manifold 55
Lyapunov Dimension 75
Lyapunov Equation 22
Lyapunov Exponents 49
M
_Markov Property 28
Matrix Differential Equation 12
Matrix Functions 12'
Minimal Realisation 20
-Monic Polynomial 35
N
Natural Probability Measure 73
Null-controllable 24
Pole Placement Problem 37
R
Stabil isability Problem 23
Stabilisable 23
Stable 54
Stable Manifold.Theorem .55
State Feedback Controller 23
State Observer 39
System 1
System State 4
T.
Targetting 74
Taylor Expansion 73
Transfer Matrix 31
Transition Density Function 28 .
Transition Matrix 12
U
Unstable 54
V
Variational Equation 50
Weighting Pattern 19
Well Determined System 62
. 128