Aizawa
modified values
file: aizawa10.sca

From Lorenz to...

Attractors from a system of ordinary differential equations

3D attractors are point clouds generate from sequences of numbers pn{xn,yn,zn} ⇒ pnR3, nN, where n0→∞ denotes the step of the iteration process starting from a initial p0{x0,y0,z0} point.
This particular attractors class is defined from a ordinary differential equations: \begin{align} \text{ assigned:   } & & \dot{x} = \frac{\delta x}{\delta t}, & & \dot{y} = \frac{\delta y}{\delta t}, & & \dot{z} = \frac{\delta z}{\delta t} \end{align} They are derivatives of components of point p respect to t, where δt is the infinitesimal increment, the differential term, of equation.
In the cloud each next point is function of the previous one:

\[ \eqalign { \dot{x} = \xi (x, y, z) & \\ \dot{y} = \phi(x, y, z) & \\ \dot{z} = \psi(x, y, z) & } \qquad \Rightarrow \qquad \eqalign { x_{i+1} = x_i + \delta t \centerdot \xi(x_i, y_i, z_i) & \\ y_{i+1} = y_i + \delta t \centerdot \phi(x_i, y_i, z_i) & \\ z_{i+1} = z_i + \delta t \centerdot \psi(x_i, y_i, z_i) & } \qquad \Bigg\{ \eqalign { & x, y, z, t \in R \\ & [0, i, n_{\rightarrow\infty}[ \text{   } \Rightarrow i,n \in N \\ } \]


In the computational code:


 

In the ATTRACTORS window of glChAoS.P:


Colors are indicative of point speed: distance between pi and pi+1
 
You can to start wglChAoS.P with a specific attractor directly from  explore  button.
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\begin{align} \dot{x} & = (z - k_1)x- k_3 \centerdot y \\ \dot{y} & = k_3 \centerdot x+(z - k_1)y \\ \dot{z} & = k_2 + k_0 \centerdot z-\frac{z^3}{3}-(x^2+y^2)(1+ k_4 \centerdot z)+ k_5 \centerdot zx^3 \\ \end{align}

 

explore
    pNew.x =p.x + dt*((p.z-k[1])*p.x - k[3]*p.y);
    pNew.y =p.y + dt*((p.z-k[1])*p.y + k[3]*p.x);
    const float xQ =p.x*p.x;
    pNew.z =p.z + dt*(k[2] + k[0]*p.z - (p.z*p.z*p.z)/3.f - (xQ + p.y*p.y) * (1.f + k[4]*p.z) + k[5]*p.z*xQ*p.x);

\begin{align} \dot{x} & = y \\ \dot{y} & = z \\ \dot{z} & = - k_0 \centerdot x - k_1 \centerdot y - z + k_2 \centerdot x^3 \\ \end{align}

 

explore
    pNew.x = p.x + dt*p.y;
    pNew.y = p.y + dt*p.z;
    pNew.z = p.z + dt*(-k[0]*p.x - k[1]*p.y - p.z + k[2]*p.x*p.x*p.x);

\begin{align} \dot{x} & = x(4-y) + k_0 \centerdot z \\ \dot{y} & = -y(1-x^2) \\ \dot{z} & = - x (1.5 - k_1 \centerdot z) -0.005z \\ \end{align}

 

explore
    pNew.x = p.x + dt*( p.x*(k[0] - p.y) + k[4]*p.z);
    pNew.y = p.y + dt*(-p.y*(k[1] - p.x*p.x));
    pNew.z = p.z + dt*(-p.x*(k[2] - k[3]*p.z) - k[5]*p.z);

\begin{align} \dot{x} & = -k_0 \centerdot (x+y) \\ \dot{y} & = -y - k_0 \centerdot xz\\ \dot{z} & = k_0 \centerdot xy + k_1 \\ \end{align}

 

explore
    pNew.x = p.x + dt*(-k[0] * (p.x + p.y));
    pNew.y = p.y + dt*(-p.y -k[0]*p.x*p.z);
    pNew.z = p.z + dt*(k[0]*p.x*p.y + k[1]);

\begin{align} \dot{x} & = k_0 \centerdot (y-x) \\ \dot{y} & = -xz + k_2 \centerdot y \\ \dot{z} & = xy - k_1 \centerdot z\\ \end{align}

 

explore
    pNew.x = p.x + dt*(k[0]*(p.y-p.x));
    pNew.y = p.y + dt*(-p.x*p.z + k[2]*p.y);
    pNew.z = p.z + dt*( p.x*p.y - k[1]*p.z);

\begin{align} \dot{x} & = k_0 \centerdot x - yz \\ \dot{y} & = k_1 \centerdot y + xz \\ \dot{z} & = k_2 \centerdot z + x \frac{y}{3} \\ \end{align}

 

explore
    pNew.x = p.x + dt*(k[0] * p.x - p.y * p.z);
    pNew.y = p.y + dt*(k[1] * p.y + p.x * p.z);
    pNew.z = p.z + dt*(k[2] * p.z + p.x * p.y/3.f);

\begin{align} \dot{x} & = y \\ \dot{y} & = z \\ \dot{z} & = k_1 \centerdot x + k_2 \centerdot y + k_3 \centerdot z + k_4 \centerdot x^3 \\ \end{align}

 

explore
    pNew.x = p.x + dt*p.y;
    pNew.y = p.y + dt*p.z; 
    pNew.z = p.z + dt*(k[0]*p.x + k[1]*p.y + k[2]*p.z + k[3]*p.x*p.x*p.x);

\begin{align} \dot{x} & = y - k_0 \centerdot x + k_1 \centerdot yz\\ \dot{y} & = k_2 \centerdot y - xz + z \\ \dot{z} & = k_3 \centerdot xy - k_4 \centerdot z \\ \end{align}

 

explore
    pNew.x = p.x + dt*(p.y - k[0]*p.x + k[1]*p.y*p.z);
    pNew.y = p.y + dt*(k[2]*p.y - p.x*p.z + p.z); 
    pNew.z = p.z + dt*(k[3]*p.x*p.y - k[4]*p.z);

\begin{align} \dot{x} & = k_0 \centerdot x - yz \\ \dot{y} & = k_1 \centerdot y + xz \\ \dot{z} & = k_2 \centerdot z + x \frac{y}{3} \\ \end{align}

 

explore
    pNew.x = p.x + dt*(k[0]*(p.y-p.x) + k[2]*p.x*p.z);
    pNew.y = p.y + dt*(k[4]*p.x + k[5]*p.y - p.x*p.z); 
    pNew.z = p.z + dt*(k[1]*p.z + p.x*p.y - k[3]*p.x*p.x);

\begin{align} \dot{x} & = k_0 \centerdot x- k_1 \centerdot yz \\ \dot{y} & = - k_2 \centerdot y+xz \\ \dot{z} & = - k_4 \centerdot x - k_3 \centerdot z + xy \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(k[0]*p.x - k[1]*p.y*p.z);
    pNew.y =p.y +  dt *(p.x*p.z - k[2]*p.y);
    pNew.z =p.z +  dt *(k[4]*p.x - k[3]*p.z +p.x*p.y);

\begin{align} \dot{x} & = k_0 \centerdot x+ k_1 \centerdot y+ k_2 \centerdot yz \\ \dot{y} & = k_3 \centerdot y-xz \\ \dot{z} & = k_4 \centerdot z + k_5 \centerdot xy \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(k[0]*p.x + k[1]*p.y + k[2]*p.y*p.z);
    pNew.y =p.y +  dt *(k[3]*p.y -p.x*p.z);
    pNew.z =p.z +  dt *(k[4]*p.z + k[5]*p.x*p.y);

\begin{align} \dot{x} & = x+y+yz \\ \dot{y} & = yz-xz \\ \dot{z} & = -z- k_0 xy +1 \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(k[0]*p.x + k[1]*p.y + k[2]*p.y*p.z);
    pNew.y =p.y +  dt *(k[3]*p.y*p.z - k[4]*p.x*p.z);
    pNew.z =p.z +  dt *(1.f - k[5]*p.z - k[6]*p.x*p.y);

\begin{align} \dot{x} & = y \\ \dot{y} & = z \\ \dot{z} & = x^2-k_2 \centerdot x- k_1 \centerdot y - k_0 \centerdot z \\ \end{align}

 

explore
    pNew.x = p.x + dt*p.y;
    pNew.y = p.y + dt*p.z;
    pNew.z = p.z + dt*(-k[2]*p.x - k[1]*p.y - k[0]*p.z + p.x*p.x);

\begin{align} \dot{x} & = -2 \pi \centerdot k_2 \centerdot y - k_0 \centerdot z(x+1) \\ \dot{y} & = 2 \pi \centerdot k_2 \centerdot x \\ \dot{z} & = k_0 \centerdot (x+1)^2 - k_1\\ \end{align}

 

explore
    const float omega = 2.f * M_PI * k[2];
    const float x1 = v.x+1.f;
    pNew.x = p.x + dt*(-omega*p.y - k[0]*p.z*x1);
    pNew.y = p.y + dt*omega*p.x;
    pNew.z = p.z + dt*(k[0]*x1*x1 - k[1]);

\begin{align} \dot{x} & = k_0 \centerdot k_3 -x^2-z^2- k_0 \centerdot x \\ \dot{y} & = xy - k_1 \centerdot xy - y + k_3 \\ \dot{z} & = k_1 \centerdot xy + xz - z \\ \end{align}

 

explore
    pNew.x = p.x + dt*(-p.y*p.y -p.z*p.z -k[0]*p.x + k[0]*k[2]);
    pNew.y = p.y + dt*(p.x*p.y - k[1]*p.x*p.z - p.y + k[3]);
    pNew.z = p.z + dt*(k[1]*p.x*p.y + p.x*p.z - p.z);

\begin{align} \dot{x} & = - k_0 \centerdot x -4y -4z -y^2 \\ \dot{y} & = - k_0 \centerdot y -4z -4x -z^2 \\ \dot{z} & = - k_0 \centerdot z -4x -4y -x^2 \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(-k[0]*p.x - 4.f*p.y - 4.f*p.z -p.y*p.y);
    pNew.y =p.y +  dt *(-k[1]*p.y - 4.f*p.z - 4.f*p.x -p.z*p.z);
    pNew.z =p.z +  dt *(-k[2]*p.z - 4.f*p.x - 4.f*p.y -p.x*p.x);

\begin{align} \dot{x} & = k_0 \centerdot y+ k_1 \centerdot x+ k_2 \centerdot yz \\ \dot{y} & = k_3 \centerdot y-z+ k_4 \centerdot xz \\ \dot{z} & = k_5 \centerdot z+ k_6 \centerdot xy \\ \end{align}

 

explore
    pNew.x = p.x + dt*(k[0]*p.y + k[1]*p.x + k[2]*p.y*p.z);
    pNew.y = p.y + dt*(k[3]*p.y - p.z + k[4]*p.x*p.z);
    pNew.z = p.z + dt*(k[5]*p.z + k[6]*p.x*p.y);

\begin{align} \dot{x} & = k_0 \centerdot (y-x) \\ \dot{y} & = k_1 \centerdot x - y - xz \\ \dot{z} & = - k_2 \centerdot z + xy \end{align}

 

explore
    pNew.x = p.x + dt*(k[0] * (p.y - p.x));
    pNew.y = p.y + dt*(p.x * (k[1] - p.z) - p.y);
    pNew.z = p.z + dt*(p.x * p.y - k[2] * p.z);

\begin{align} \dot{x} & = k_0 \centerdot (y-\xi(x)) \\ \dot{y} & = x - y + z \\ \dot{z} & = - k_1 \centerdot y \end{align}

\[ \xi(x) = m_5 x + \frac{1}{2} \sum_{i=1}^5 (m_{i-1}-m_i)(|x+c_i| - |x-c_i|) \]

explore
    float f(const float x) {
        const int M = 2, C = 7; float sum = 0;
        for(int i=1; i<=5; i++) sum+= (k[M+(i-1)]-k[M+i])*(fabs(x+k[C+i]) - fabs(x-k[C+i]));
        return k[M+5]*x + sum*.5f;  }
    pNew.x = p.x + dt*(k[0]*(p.y-f(p.x)));
    pNew.y = p.y + dt*(p.x - p.y + p.z); 
    pNew.z = p.z + dt*(-k[1]*p.y);

 
\[ \text{   } m_j = k_s \rightarrow s = j+2, \text{   } \text{   } j \in [1,5]; \text{   } \text{   } \text{   } \text{   } c_j = k_t \rightarrow t = j+7, \text{   } \text{   } j \in [1,5]\]

\begin{align} \dot{x} & = y - k_0 \centerdot x + 10yz \\ \dot{y} & = 5xz - x -0.4y \\ \dot{z} & = k_1 z - 5xy \\ \end{align}

 

explore
    pNew.x = p.x + dt*(-k[0]*p.x + p.y + 10.0*p.y*p.z);
    pNew.y = p.y + dt*(-p.x - 0.4*p.y + 5.0*p.x*p.z);
    pNew.z = p.z + dt*(k[1]*p.z - 5.0*p.x*p.y);

\begin{align} \dot{x} & = y \\ \dot{y} & = - x + yz \\ \dot{z} & = k_0 - y^2 \\ \end{align}

 

explore
        pNew.x = p.x + dt*p.y;
        pNew.y = p.y + dt*(-p.x + p.y*p.z);
        pNew.z = p.z + dt*(k[0] - p.y*p.y);

\begin{align} \dot{x} & = k_0 \centerdot (y-x) + k_4 \centerdot yz\\ \dot{y} & = k_2 \centerdot x + k_3 \centerdot y - xz \\ \dot{z} & = xy - k_1 \centerdot z \\ \end{align}

 

explore
        pNew.x = p.x + dt*(k[0]*(p.y-p.x)+k[4]*p.y*p.z);
        pNew.y = p.y + dt*(k[2]*p.x+k[3]*p.y-p.x*p.z);
        pNew.z = p.z + dt*(p.x*p.y-k[1]*p.z);

\begin{align} \dot{x} & = - k_0 \centerdot x + k_0 \centerdot y \\ \dot{y} & = k_2 \centerdot x - y - xz \\ \dot{z} & = xy - k_1 \centerdot z \\ \end{align}

 

explore
    pNew.x = p.x + dt*(-k[0]*(p.x - p.y));
    pNew.y = p.y + dt*(k[2]*p.x - p.y - p.x*p.z);
    pNew.z = p.z + dt*(p.x*p.y - k[1]*p.z);

\begin{align} \dot{x} & = -y -z \\ \dot{y} & = x+ k_0 \centerdot y \\ \dot{z} & = k_1 +z(x - k_2) \\ \end{align}

 

explore
    pNew.x = p.x + dt *(-p.y - p.z);
    pNew.y = p.y + dt *(p.x + k[0]*p.y);
    pNew.z = p.z + dt *(k[1] + p.z*(p.x - k[2]));

\begin{align} \dot{x} & = - k_1 \centerdot x+ k_0 \centerdot y - yz \\ \dot{y} & = x \\ \dot{z} & = - z - y^2 \\ \end{align}

 

explore
    pNew.x = p.x + dt *(-k[1]*p.x + k[0]*p.y - p.y*p.z);
    pNew.y = p.y + dt *p.x;
    pNew.z = p.z + dt *(-p.z + p.y*p.y);

\begin{align} \dot{x} & = -x +y +yz \\ \dot{y} & = -x-y+ k_0 \centerdot xz \\ \dot{z} & = z - k_1 \centerdot xy \\ \end{align}

 

explore
    pNew.x = p.x + dt *(-p.x + p.y + p.y*p.z);
    pNew.y = p.y + dt *(-p.x - p.y + k[0]*p.x*p.z);
    pNew.z = p.z + dt *(p.z - k[1]*p.x*p.y);

\begin{align} \dot{x} & = y \\ \dot{y} & = x(1 - z) - k_0 \centerdot y \\ \dot{z} & = x^2 - k_1 \centerdot z \\ \end{align}

 

explore
    pNew.x = p.x + dt * p.y;
    pNew.y = p.y + dt * (1.f-p.z)*p.x - k[0]*p.y);
    pNew.z = p.z + dt * p.x*p.x - k[1]*p.z);

\begin{align} \dot{x} & = k_0 \centerdot yz \\ \dot{y} & = k_1 \centerdot x - k_2 \centerdot y \\ \dot{z} & = 1 - k_3 \centerdot xy \\ \end{align}

 

explore
    pNew.x = p.x + dt * k[0]*p.y * p.z;
    pNew.y = p.y + dt *(k[1]*p.x - k[2]*p.y);
    pNew.z = p.z + dt *(1 - k[3]*p.x*p.y);

\begin{align} \dot{x} & = y + z \\ \dot{y} & = -x + k_0 \centerdot y \\ \dot{z} & = x^2 -z \\ \end{align}

 

explore
    pNew.x = p.x + dt *(p.y + p.z);
    pNew.y = p.y + dt *(-p.x + k[0]*p.y);
    pNew.z = p.z + dt *(p.x*p.x - p.z);

\begin{align} \dot{x} & = (x - k_0 \centerdot y)cos(z) - k_1 \centerdot ysin(z) \\ \dot{y} & = (x + k_2 \centerdot y)sin(z) - k_3 \centerdot ycos(z) \\ \dot{z} & = k_4 + k_5 \centerdot z + k6 \centerdot atan(y(1-k_7)/x(1-k_8)) \\ \end{align}

 

explore
    pNew.x =p.x +  dt *((p.x - k[0]*p.y)*cos(p.z) - k[1]*p.y*sin(p.z));
    pNew.y =p.y +  dt *((p.x + k[2]*p.y)*sin(p.z) + k[3]*p.y*cos(p.z));
    pNew.z =p.z +  dt *(k[4] + k[5]*p.z + k[6]*atan(((1-k[7])*p.y) / ((1-k[8])*p.x)));

\begin{align} \dot{x} & = - k_0 \centerdot x+sin(y) \\ \dot{y} & = - k_0 \centerdot y+sin(z) \\ \dot{z} & = - k_0 \centerdot z+sin(x) \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(-k[0]*p.x + sinf(p.y));
    pNew.y =p.y +  dt *(-k[1]*p.y + sinf(p.z));
    pNew.z =p.z +  dt *(-k[2]*p.z + sinf(p.x));

\begin{align} \dot{x} & = k_0 \centerdot (y-x) + k_3 \centerdot xz \\ \dot{y} & = k_1 \centerdot x - xz + k_5 \centerdot y \\ \dot{z} & = k_2 \centerdot z + xy -k_4 \centerdot x^2 \\ \end{align}

 

explore
    pNew.x =p.x +  dt*(k[0]*(p.y -p.x) + k[3]*p.x*p.z);
    pNew.y =p.y +  dt*(k[1]*p.x + k[5]*p.y -p.x*p.z);
    pNew.z =p.z +  dt*(k[2]*p.z +p.x*p.y - k[4]*p.x*p.x);


\[ \text{TSUCS1: } \text{   } \text{   } k_1 = 0.0 \]

\begin{align} \dot{x} & = k_0 \centerdot (y-x) \\ \dot{y} & = k_1 \centerdot x - k_2 \centerdot xz \\ \dot{z} & = e^{xy} - k_3 \centerdot z \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(k[0]*(p.y -p.x));
    pNew.y =p.y +  dt *(k[1]*p.x - k[2]*p.x*p.z);
    pNew.z =p.z +  dt *(exp(p.x*p.y) - k[3]*p.z);

\begin{align} \dot{x} & = k_0 \centerdot x + k_1 \centerdot y + yz \\ \dot{y} & = k_2 \centerdot y - xz + k_3 \centerdot yz \\ \dot{z} & = k_4 \centerdot z - xy \\ \end{align}

 

explore
    pNew.x =p.x +  dt *(k[0]*p.x + k[1]*p.y+ p.y*p.z);
    pNew.y =p.y +  dt *(k[2]*p.y - p.x*p.z - k[3]*p.y*p.z);
    pNew.z =p.z +  dt *(k[4]*p.z - p.x*p.y);