Dynamics of Chaotic 3D Attractors: Part 1 Gallery of more than 100 three-dimensional attractors plotted by me in MATLAB based on General Algorithm of The Explicit Runge—Kutta Method .
A fair number of attractors I found on Jürgen Mayer's personal website , you can find references to primary sources there, so if some attractors lack references, those attractors were found there.
For attractors that have been found already by me, I will leave a reference to the primary source.
The plots are also available on Pinterest and Behance:
P.S. I give the title of attractors as the surnames of the authors of the paper where the attractor was found. For papers with a large number of authors, I take only the first 3 surnames.
$$
\begin{cases}
\dot{x} = \sigma(y - x), \\
\dot{y} = x(\rho - z) - y, \\
\dot{z} = xy - \beta z,
\end{cases}
$$
$$
\begin{bmatrix}
\sigma\\
\rho\\
\beta
\end{bmatrix} = \begin{bmatrix}
10 \\
28 \\
\dfrac{8}{3}
\end{bmatrix}.
$$
The Lorenz Mod 1 Attractor $$
\begin{cases}
\dot{x}=-\alpha x+y^2-z^2+\alpha\varsigma, \\
\dot{y}=x\left(y-\beta z\right)+\delta, \\
\dot{z}=-z+x\left(\beta y+z\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
0.1\\
4\\
14\\
0.08
\end{bmatrix}.
$$
The Lorenz Mod 2 Attractor $$
\begin{cases}
\dot{x}=-\alpha x+y^2-z^2+\alpha\varsigma, \\
\dot{y}=x\left(y-\beta z\right)+\delta, \\
\dot{z}=-z+x\left(\beta y+z\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
0.9\\
5\\
9.9\\
1
\end{bmatrix}.
$$
The Lotka—Volterra Attractor $$
\begin{cases}
\dot{x}=x-xy+\varsigma x^2-\alpha z x^2, \\
\dot{y}=-y+xy, \\
\dot{z}=-\beta z +\alpha z x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
2.9851\\
3\\
2
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = (z - \beta)x - \delta y, \\
\dot{y} = \delta x + (z - \beta)y, \\
\dot{z} = \varsigma + \alpha z - \dfrac{z^3}{3} - \left(x^2 + y^2\right)\left(1 + \varepsilon z\right) + \xi zx^3,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha \\
\beta \\
\varsigma \\
\delta \\
\varepsilon \\
\xi
\end{bmatrix}=
\begin{bmatrix}
0.95 \\
0.7 \\
0.6 \\
3.5 \\
0.25 \\
0.1
\end{bmatrix}.
$$
$$ \begin{cases}
\dot{x} =\left(x-\alpha y\right)\cos z-\beta y \sin z, \\
\dot{y} = \left(x+\gamma y\right)\sin z +\delta y\cos z, \\
\dot{z} = \varepsilon +\kappa z+\xi\arctan\left(\dfrac{1-\varsigma}{1-\omega}xy\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\gamma\\
\delta\\
\varepsilon\\
\kappa\\
\xi\\
\varsigma\\
\omega
\end{bmatrix}=
\begin{bmatrix}
1.013\\
-0.011\\
0.02\\
0.96\\
0\\
0.01\\
1\\
0.05\\
0.05
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = -\alpha x-4y-4z-y^2, \\
\dot{y} =-\alpha y-4z-4x-z^2, \\
\dot{z} = -\alpha z-4x-4y-x^2,
\end{cases}
$$
$$
\alpha=1.4.
$$
$$
\begin{cases}
\dot{x} =-\beta x+\sin y,\\
\dot{y} = -\beta y + \sin z, \\
\dot{z} = -\beta z + \sin x,
\end{cases}
$$
$$
\beta=0.19.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(x-y\right), \\
\dot{y} = -4\alpha y +xz+\varsigma x^3, \\
\dot{z} = -\delta\alpha z +xy+\beta z^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\delta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
1.8\\
-0.07\\
1.5\\
0.02
\end{bmatrix}.
$$
The Hindmarsh—Rose Attractor $$
\begin{cases}
\dot{x} = -\alpha x^3 +\beta x^2+y -z+\iota, \\
\dot{y} =-\delta x^2-y+\varsigma, \\
\dot{z} = \rho\left(\xi\left(x-\chi\right)-z\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\iota\\
\rho\\
\xi\\
\chi
\end{bmatrix}=
\begin{bmatrix}
1\\
3\\
1\\
5\\
3.25\\
0.006\\
4\\
-1.6
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} =-\kappa x+\alpha y -yz, \\
\dot{y} = x, \\
\dot{z} = -z+y^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\kappa
\end{bmatrix}=
\begin{bmatrix}
6.7\\
2
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} =y, \\
\dot{y} = z, \\
\dot{z} = -\alpha x -\beta y -z+\varsigma x^3,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
-5.5\\
3.5\\
-1
\end{bmatrix}.
$$
The 3-Cells CNN Attractor $$
\begin{cases}
\dot{x} = -x+\alpha f(x)-\delta f(y)- \delta f(z), \\
\dot{y} = -y-\delta f(x)+\beta f(y)-\varsigma f(z), \\
\dot{z} = -z -\delta f(x)+\varsigma f(y) + f(z),
\end{cases}
$$
$$
f\left(\omega\right)=\dfrac{1}{2}\left(\left|\omega+1\right|-\left|\omega-1\right|\right),
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
1.24\\
1.1\\
4.4\\
3.21
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} =y-\rho x+\sigma yz,\\
\dot{y} = \xi y-xz+z, \\
\dot{z} = \varsigma xy-\varepsilon z,
\end{cases}
$$
$$
\begin{bmatrix}
\rho\\
\sigma\\
\xi\\
\varsigma\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
3\\
2.7\\
1.7\\
2\\
9
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} =-y-z,\\
\dot{y} = x+\alpha y, \\
\dot{z} = \beta+z\left(x-\varsigma\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=\begin{bmatrix}
0.1\\
0.1\\
14
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \left(\dfrac{1}{\beta}-\alpha\right)x+z+xy, \\
\dot{y} = -\beta y-x^2, \\
\dot{z} = -x -\varsigma z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=\begin{bmatrix}
0.001\\
0.2\\
1.1
\end{bmatrix}.
$$
The Chen—Celikovsky Attractor $$
\begin{cases}
\dot{x}=\alpha\left(y-x\right), \\
\dot{y}=-xz+\varsigma y, \\
\dot{z}= xy-\beta z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=\begin{bmatrix}
36\\
3\\
20
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = -y^2-z^2-\alpha x+\alpha\varsigma, \\
\dot{y} =xy -\beta xz-y+\delta, \\
\dot{z} =\beta xy+xz-z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
0.2\\
4\\
8\\
1
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(x-y\right)-yz, \\
\dot{y} = -\beta y+xz, \\
\dot{z} =-\varsigma z+\delta x+xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
0.977\\
10\\
4\\
0.1
\end{bmatrix}.
$$
The Wimol—Banlue Attractor $$
\begin{cases}
\dot{x} =y-x, \\
\dot{y} = -z\tanh x, \\
\dot{z} = -\alpha+xy+|y|,
\end{cases}
$$
$$
\alpha = 2.
$$
$$
\begin{cases}
\dot{x} = z (\lambda x - \mu y )+ (2-z) \left[ \alpha x \left( 1-\dfrac{x^2+y^2}{\rho^2} \right) -\beta y \right], \\
\dot{y} = z ( \mu x +\lambda y) + (2-z) \left[ \alpha y \left( 1- \dfrac{x^2+y^2}{\rho^2} \right)+\beta x \right], \\
\dot{z}= \dfrac{1}{\varepsilon} \left[z ( (2-z) \left( \varphi (z-2)^2+\psi \right) - \delta x)\left(z+\xi \left( x^2+y^2 \right)-\eta \right)-\varepsilon \varsigma(z-1) \right],
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\xi \\
\eta\\
\rho \\
\varepsilon\\
\lambda\\
\mu\\
\varphi\\
\psi
\end{bmatrix}=
\begin{bmatrix}
2.8\\
5\\
1\\
0.1\\
0.05\\
3.312\\
10\\
0.1\\
-2\\
1.155\\
3\\
0.8
\end{bmatrix}.
$$
The Shimizu—Morioka Attractor $$
\begin{cases}
\dot{x}=y, \\
\dot{y}=\left(1-z\right)x-\alpha y, \\
\dot{z}=x^2-\beta z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.75\\
0.45
\end{bmatrix}.
$$
The Nose—Hoover Attractor $$
\begin{cases}
\dot{x}=y, \\
\dot{y}=-x+yz, \\
\dot{z}=\alpha-y^2,
\end{cases}
$$
$$
\alpha=1.5.
$$
$$
\begin{cases}
\dot{x} =\alpha x +\varsigma yz,\\
\dot{y} = \beta x +\delta y -xz, \\
\dot{z} = \varepsilon z +\xi xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\delta\\
\varepsilon\\
\xi\\
\varsigma
\end{bmatrix}=\begin{bmatrix}
0.2\\
-0.01\\
-0.4\\
-1\\
-1\\
1
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=\alpha\left(y-x\right)+yz^2, \\
\dot{y}=\beta\left(x+y\right)-xz^2, \\
\dot{z}=-\varsigma z+\varepsilon y +xyz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
50\\
10\\
13\\
6
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=- \dfrac{\alpha\beta}{\alpha+\beta}x -yz+\varsigma, \\
\dot{y}=\alpha y +xz, \\
\dot{z}=\beta z+xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
-10\\
-4\\
18.1
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=-\alpha\left(x+y\right), \\
\dot{y}=-y-\alpha xz, \\
\dot{z}=\alpha xy +\beta,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
10\\
4.272
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \alpha x+\beta y +yz, \\
\dot{y} =\varsigma y-xz+\delta yz, \\
\dot{z} =\varepsilon z-xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
2.97\\
0.15\\
-3\\
1\\
-8.78
\end{bmatrix}.
$$
The Genesio—Tesi Attractor $$
\begin{cases}
\dot{x} = y, \\
\dot{y} = z, \\
\dot{z} = -\varsigma x-\beta y-\alpha z+x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
0.44\\
1.1\\
1
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right), \\
\dot{y} =\beta x-\varsigma xz, \\
\dot{z} = \exp{(xy)}-\delta z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
10\\
40\\
2\\
2.5
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = -x+y+yz, \\
\dot{y} =-x-y+\alpha xz, \\
\dot{z} = z-\beta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.4\\
0.3
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=\alpha\left(y-x-\left(\varsigma x + \dfrac{1}{2}\left(\delta-\varsigma\right)\left(\left|x+1\right|-\left|x-1\right|\right)\right)\right), \\
\dot{y}=x-y+z, \\
\dot{z}=-\beta y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
\dfrac{78}{5}\\
\dfrac{1279}{50}\\
-\dfrac{5}{7}\\
-\dfrac{8}{7}
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(y-x^3-\varsigma x\right), \\
\dot{y} = x-y+z, \\
\dot{z} = -\beta y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
10\\
16\\
-0.143
\end{bmatrix}.
$$
The Modified Chua Attractor $$
\begin{cases}
\dot{x} =\alpha\left(y+\delta\sin{\left(\dfrac{\pi x}{2\varsigma}+\varepsilon\right)}\right), \\
\dot{y} = x-y+z, \\
\dot{z} = -\beta y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
10.82\\
14.286\\
1.3\\
0.11\\
0
\end{bmatrix}.
$$
The Muthuswamy—Chua Attractor $$
\begin{cases}
\dot{x}=y, \\
\dot{y}=-\dfrac{x}{3}+\dfrac{y}{2}-\dfrac{yz^2}{2}, \\
\dot{z}=y-\alpha z-yz,
\end{cases}
$$
$$
\alpha=0.6.
$$
The Moore—Spiegel Attractor $$
\begin{cases}
\dot{x}=y, \\
\dot{y}=z, \\
\dot{z}=-z-\left(\beta-\alpha+\alpha x^2\right)y-\beta x,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
100\\
26
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=y, \\
\dot{y}=z, \\
\dot{z}=\alpha x + \beta y + \varsigma z + \delta x^3,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
0.8\\
-1.1\\
-0.45\\
-1
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=y, \\
\dot{y}=-x+yz, \\
\dot{z}=1-y^2
\end{cases}
$$
$$
\begin{cases}
\dot{x}=yz, \\
\dot{y}=x-y, \\
\dot{z}=1-xy
\end{cases}
$$
$$
\begin{cases}
\dot{x}=yz, \\
\dot{y}=x-y, \\
\dot{z}=1-x^2
\end{cases}
$$
$$
\begin{cases}
\dot{x}=-y, \\
\dot{y}=x+z, \\
\dot{z}=xz + \alpha y^2,
\end{cases}
$$
$$
\alpha=3.
$$
$$
\begin{cases}
\dot{x}=yz, \\
\dot{y}=x^2-y, \\
\dot{z}=1-\alpha x,
\end{cases}
$$
$$
\alpha=4.
$$
$$
\begin{cases}
\dot{x}=y+z, \\
\dot{y}=-x+\alpha y, \\
\dot{z}=x^2-z,
\end{cases}
$$
$$
\alpha=\dfrac{1}{2}.
$$
$$
\begin{cases}
\dot{x}=\alpha x + z, \\
\dot{y}=xz-y, \\
\dot{z}=-x+y,
\end{cases}
$$
$$
\alpha=\dfrac{2}{5}.
$$
$$
\begin{cases}
\dot{x}=-y+z^2, \\
\dot{y}=x+\alpha y, \\
\dot{z}=x-z,
\end{cases}
$$
$$
\alpha=\dfrac{1}{2}.
$$
$$
\begin{cases}
\dot{x}=\alpha y, \\
\dot{y}=x+z, \\
\dot{z}=x+y^2-z,
\end{cases}
$$
$$
\alpha=-\dfrac{1}{5}.
$$
$$
\begin{cases}
\dot{x}=\alpha z, \\
\dot{y}=-\alpha y +z, \\
\dot{z}= -x+y+y^2,
\end{cases}
$$
$$
\alpha=2.
$$
$$
\begin{cases}
\dot{x}=xy-z, \\
\dot{y}=x-y, \\
\dot{z}=x+\alpha z,
\end{cases}
$$
$$
\alpha=\dfrac{3}{10}.
$$
$$
\begin{cases}
\dot{x}=y+\alpha z, \\
\dot{y}=\beta x^2 - y, \\
\dot{z}=1-x,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
3.9 \\
0.9
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=-z, \\
\dot{y}=-x^2-y, \\
\dot{z}=\alpha + \alpha x+y,
\end{cases}
$$
$$
\alpha=\dfrac{17}{10}.
$$
$$
\begin{cases}
\dot{x}=-\alpha y, \\
\dot{y}=x+z^2, \\
\dot{z}=1+y-\alpha z,
\end{cases}
$$
$$
\alpha=2.
$$
$$
\begin{cases}
\dot{x}=y, \\
\dot{y}=x-z, \\
\dot{z}=x+xz+\alpha y,
\end{cases}
$$
$$
\alpha=\dfrac{27}{10}.
$$
$$
\begin{cases}
\dot{x}=\alpha y + z, \\
\dot{y}=-x+y^2, \\
\dot{z}=x+y,
\end{cases}
$$
$$
\alpha=\dfrac{27}{10}.
$$
$$
\begin{cases}
\dot{x}=-z, \\
\dot{y}=x - y, \\
\dot{z}=\alpha x +y^2+\beta z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
3.4 \\
0.5
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=\alpha -y, \\
\dot{y}=\beta +z, \\
\dot{z}=xy-z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.9 \\
0.4
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=-x+\alpha y, \\
\dot{y}=x +z^2, \\
\dot{z}=1+x,
\end{cases}
$$
$$
\alpha=4.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right)+\varsigma xz, \\
\dot{y} = \varepsilon y-xz, \\
\dot{z} = \beta z+xy-\delta x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix} =
\begin{bmatrix}
40\\
0.833\\
0.5\\
0.65\\
20
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right)+\delta xz, \\
\dot{y} = \varsigma x-xz+\xi y, \\
\dot{z} = \beta z+xy-\varepsilon x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\xi
\end{bmatrix} =
\begin{bmatrix}
40\\
1.833\\
55\\
0.16\\
20\\
0.65
\end{bmatrix}.
$$
$$
\begin{cases}
\dot{x}=-\beta x + zy, \\
\dot{y}=-\beta y + \left(z-\alpha\right)x, \\
\dot{z}=1-xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
5 \\
2
\end{bmatrix}.
$$
The Newton—Leipnik Attractor $$
\begin{cases}
\dot{x}=-\alpha x+y+10yz, \\
\dot{y}=-x-0.4y+5xz, \\
\dot{z}=\beta z-5xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.4 \\
0.175
\end{bmatrix}.
$$
The Four—Wing 1 Attractor $$
\begin{cases}
\dot{x} = \alpha x -\beta yz, \\
\dot{y} = -\varsigma y +xz, \\
\dot{z} = \varepsilon x -\delta z +xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
4\\
6\\
10\\
5\\
1
\end{bmatrix}.
$$
The Four—Wing 2 Attractor $$
\begin{cases}
\dot{x} = \alpha x+\beta y+\varsigma yz \\
\dot{y} = \delta y - xz \\
\dot{z}= \varepsilon z +\xi x y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\xi
\end{bmatrix}=
\begin{bmatrix}
-14\\
5\\
1\\
16\\
-43\\
1
\end{bmatrix}.
$$
The Four—Wing 3 Attractor $$
\begin{cases}
\dot{x}=x+y+yz, \\
\dot{y}=yz-xz, \\
\dot{z}=1-\alpha xy -z
\end{cases}
$$
$$
\alpha = 1.
$$
$$
\begin{cases}
\dot{x}=\alpha\left(y-x\right), \\
\dot{y}=\beta x - xz, \\
\dot{z}=xy+\varsigma z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
10 \\
16\\
-1
\end{bmatrix}.
$$
The Elhadj—Sprott Attractor $$
\begin{cases}
\dot{x}=\alpha\left(y-x\right), \\
\dot{y}=-\alpha x -\beta yz, \\
\dot{z}=-\varsigma+y^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
40 \\
33\\
10
\end{bmatrix}.
$$
The Sprott—Jafari Attractor Reference: Jafari, S., Sprott, J. C., & Nazarimehr, F. (2015). Recent new examples of hidden attractors. The European Physical Journal Special Topics, 224(8), 1469–1476.
$$
\begin{cases}
\dot{x} = y, \\
\dot{y} = -x+yz, \\
\dot{z}= z+\alpha x^2-y^2-\beta,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
8.888\\
4
\end{bmatrix}.
$$
The Sprott Strange Multifractal Attractor Reference: Sprott, J. (2020). Do We Need More Chaos Examples?. Chaos Theory and Applications, 2(2), 49-51.
$$
\begin{cases}
\dot{x}=y, \\
\dot{y}=-x-\text{sgn}(z) y, \\
\dot{z}=y^2-\exp\left(-x^2\right)
\end{cases}
$$
Reference: Liu, C. (2009). A novel chaotic attractor. Chaos, Solitons & Fractals, 39(3), 1037–1045.
$$
\begin{cases}
\dot{x} = \alpha\left(y-x+yz\right), \\
\dot{y} = \beta y - \varepsilon xz, \\
\dot{z}= \varsigma y-\delta z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
1\\
2.5\\
1\\
4\\
1
\end{bmatrix}.
$$
The Sundarapandian—Pehlivan Attractor Reference: Sundarapandian, V., & Pehlivan, I. (2012). Analysis, control, synchronization, and circuit design of a novel chaotic system. Mathematical and Computer Modelling, 55(7-8), 1904–1915.
$$
\begin{cases}
\dot{x} = \alpha y -x, \\
\dot{y} = -\beta x - z, \\
\dot{z}= \varsigma z + xy^2-x,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
1\\
0.46\\
0.46
\end{bmatrix}.
$$
The Sundarapandian Attractor Reference: Sundarapandian, V. (2013). Analysis and anti - synchronization of a novel chaotic system via active and adaptive controllers. Journal of Engineering Science and Technology Review, 6(4), 45–52.
$$
\begin{cases}
\dot{x} = \alpha\left(y -x\right)+yz, \\
\dot{y} = \beta x +\varsigma y -xz, \\
\dot{z}= -\delta z +x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
21.5\\
20.6\\
11\\
6.4
\end{bmatrix}.
$$
Reference: Pehlivan, I. (2011). Four-scroll stellate new chaotic system. Optoelectronics and Advanced Materials - Rapid Communications - OAM-RC - INOE 2000.
$$
\begin{cases}
\dot{x} = -\alpha x + y + yz, \\
\dot{y} = x-\alpha y +\beta xz, \\
\dot{z}= \varsigma z - \beta x y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
4\\
0.5\\
0.6
\end{bmatrix}.
$$
The Vaidyanathan Hyperbolic Sinusoidal Attractor Reference: Vaidyanathan, S. (2013). Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperboli c Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters. Journal of Engineering Science and Technology Review, 6(4), 53–65.
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right)+yz, \\
\dot{y} = \beta x - \varsigma xz, \\
\dot{z}=-\delta z + \sinh\left(xy\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
10\\
92\\
2\\
10
\end{bmatrix}.
$$
The Vaidyanathan Hyperbolic Cosinusoidal Attractor Reference: Vaidyanathan, S. (2013). Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperboli c Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters. Journal of Engineering Science and Technology Review, 6(4), 53–65.
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right)+yz, \\
\dot{y} = \beta x - \varsigma xz, \\
\dot{z}=-\delta z + \cosh\left(xy\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix}=
\begin{bmatrix}
10\\
98\\
2\\
10
\end{bmatrix}.
$$
The Neamah—Shukur Attractor Reference: Neamah, A. A., & Shukur, A. A. (2023). A novel conservative chaotic system involved in hyperbolic functions and its application to design an efficient colour image encryption scheme. Symmetry, 15(8), 1511.
$$
\begin{cases}
\dot{x} = y, \\
\dot{y} =-x-yz, \\
\dot{z}= \cosh y-1-\alpha\cos x^2-\beta\cos y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
1\\
0.3
\end{bmatrix}.
$$
Reference: Li, X., & Ou, Q. (2010). Dynamical properties and simulation of a new Lorenz-like chaotic system. Nonlinear Dynamics, 65(3), 255–270.
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right), \\
\dot{y} = \varsigma y-xz, \\
\dot{z}=-\beta z + \delta x^2 +\varepsilon xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix}=
\begin{bmatrix}
10\\
3\\
6\\
1\\
0
\end{bmatrix}.
$$
The Sprott—Li Chaotic Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = 1+yz, \\
\dot{y} = -xz, \\
\dot{z}= y^2+\alpha yz,
\end{cases}
$$
$$
\alpha=2.
$$
The Sprott—Li SL$_1$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -x+\alpha y^2 - xy, \\
\dot{y} = xz, \\
\dot{z}= z^2 -\beta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
2\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_2$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -\alpha x + xy, \\
\dot{y} = z^2 + xz, \\
\dot{z}= y^2 -\beta yz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
2\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_3$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = x + \alpha y^2-z^2, \\
\dot{y} = x^2-\beta y^2, \\
\dot{z}= xz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
2.4\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_4$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -x + \beta y^2 + xz, \\
\dot{y} = xz, \\
\dot{z}= -\alpha xy + yz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.1\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_5$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -x+\alpha z^2, \\
\dot{y} = z^2 - \beta xz, \\
\dot{z}= xy - yz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
1\\
2
\end{bmatrix}.
$$
The Sprott—Li SL$_6$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = y - z^2, \\
\dot{y} = -\alpha xz, \\
\dot{z}= x^2 - yz,
\end{cases}
$$
$$
\alpha=0.9.
$$
The Sprott—Li SL$_7$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -y-yz, \\
\dot{y} = x^2+\alpha xz, \\
\dot{z}= z^2 + \beta yz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
14\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_8$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = y - y^2, \\
\dot{y} = \alpha z^2 + xy, \\
\dot{z}= -x^2 - \beta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.3\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_9$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = y, \\
\dot{y} = \alpha y^2 - xz, \\
\dot{z}= x^2 +xy-\beta xz,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.4\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_{10}$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = y + \alpha xz, \\
\dot{y} = xy - xz, \\
\dot{z}= x^2 +\beta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.2\\
3
\end{bmatrix}.
$$
The Sprott—Li SL$_{11}$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = y + y^2 - \alpha yz, \\
\dot{y} = -z^2+\beta yz, \\
\dot{z}= xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.9\\
1
\end{bmatrix}.
$$
The Sprott—Li SL$_{12}$ Attractor Reference: Li, C., & Sprott, J. C. (2014). Chaotic flows with a single nonquadratic term. Physics Letters A, 378(3), 178–183.
$$
\begin{cases}
\dot{x} = -y+x^2-y^2, \\
\dot{y} = -xz, \\
\dot{z}= \alpha x + \beta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
0.3\\
1
\end{bmatrix}.
$$
Reference: Zhang, J., & Liao, X. (2017). Synchronization and chaos in coupled memristor-based FitzHugh-Nagumo circuits with memristor synapse. AEU - International Journal of Electronics and Communications, 75, 82–90.
$$
\begin{cases}
\dot{x} = y - x\left(\beta+0.5\left(\alpha-\beta\right)\left(\text{sgn}\left(z+1\right)-\text{sgn}\left(z-1\right)\right)\right)+\dfrac{\varepsilon}{\vartheta}\cos\left(\vartheta t\right), \\
\dot{y} = -\varsigma y-\varsigma x, \\
\dot{z}= \delta x,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\vartheta
\end{bmatrix}=
\begin{bmatrix}
-1.03\\
-0.5\\
0.98\\
1\\
0.15\\
0.75
\end{bmatrix}.
$$
The Kountchou—Louodop Attractor Reference: Kountchou, M., Louodop, P., Bowong, S., Fotsin, H., & Kurths, J. (2016). Optimal Synchronization of a Memristive Chaotic Circuit. International Journal of Bifurcation and Chaos, 26(06), 1650093.
$$
\begin{cases}
\dot{x} = \cos\left(\beta t\right)+\alpha y\left(1+z^2-z^4\right), \\
\dot{y} = x-y\cos\left(\beta t\right), \\
\dot{z}= -\varsigma y,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix}=
\begin{bmatrix}
2\\
2.5\\
6
\end{bmatrix}.
$$
The Sambas—Vaidyanathan—Zhang Attractor Reference: Sambas, A., Vaidyanathan, S., Zhang, S., Zeng, Y., Mohamed, M. A., & Mamat, M. (2019). A New Double-Wing Chaotic System with Coexisting Attractors and Line Equilibrium: Bifurcation Analysis and Electronic Circuit Simulation. IEEE Access, 1–1.
$$
\begin{cases}
\dot{x} =yz, \\
\dot{y} = x-y, \\
\dot{z}= \alpha\left|x\right|-\beta x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
5\\
2
\end{bmatrix}.
$$
The Kingni—Jafari—Simo Attractor Reference: Kingni, S. T., Jafari, S., Simo, H., & Woafo, P. (2014). Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. The European Physical Journal Plus, 129(5).
$$
\begin{cases}
\dot{x} = -z, \\
\dot{y} = -x-z, \\
\dot{z}= 3x-\alpha y + x^2-z^2-yz+\beta,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix}=
\begin{bmatrix}
1.3\\
1.01
\end{bmatrix}.
$$
The Nazarimehr—Sprott Attractor Reference: Nazarimehr, F., & Sprott, J. C. (2020). Investigating chaotic attractor of the simplest chaotic system with a line of equilibria. The European Physical Journal Special Topics, 229(6-7), 1289–1297.
$$
\begin{cases}
\dot{x} = \alpha y, \\
\dot{y} = xz, \\
\dot{z}= y-z-y^2,
\end{cases}
$$
$$
\alpha=289.
$$
The Sun—Tian—Fu Attractor Reference: Sun, M., Tian, L., & Fu, Y. (2007). An energy resources demand–supply system and its dynamical analysis. Chaos, Solitons & Fractals, 32(1), 168–180.
$$
\begin{cases}
\dot{x} = \alpha x\left(1 - \dfrac{x}{\varphi}\right) - \beta\left(y + z\right), \\
\dot{y} = -\varsigma y - \delta z + \xi x\left[\vartheta - \left(x - z\right)\right], \\
\dot{z} = \eta z\left(\zeta x - \varepsilon\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\xi \\
\eta\\
\zeta \\
\varepsilon\\
\varphi\\
\vartheta
\end{bmatrix} = \begin{bmatrix}
0.09\\
0.15\\
0.06\\
0.082\\
0.07\\
0.2\\
0.5\\
0.4\\
1.8\\
1
\end{bmatrix}.
$$
The Wang—Sun—Cang Signum Attractor Reference: Wang, Z., Sun, Y., & Cang, S. (2011). Acta Physica Polonica B, 42(2), 235. Herrera-Charles, R., Afolabi, O. M., Núñez-Pérez, J. C., & Ademola, V. A. (2024). Secure communication based on chaotic spherical 3D attractors. In Applications of Digital Image Processing XLVII (Vol. 13137, pp. 78-89).
$$
\begin{cases}
\dot{x} = \alpha x - \beta y + \varsigma z + 2\text{sgn}\left(\sin y\right), \\
\dot{y} = -\delta xz+\xi+ \eta x, \\
\dot{z}= \zeta xy + \varepsilon yz + \varphi z + \vartheta,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\xi \\
\eta\\
\zeta \\
\varepsilon\\
\varphi\\
\vartheta
\end{bmatrix} = \begin{bmatrix}
-4.1\\
1.2\\
13.45\\
1.6\\
0.161\\
0\\
2.76\\
0.6\\
13.13\\
3.5031
\end{bmatrix}.
$$
The Akgul—Hussain—Pehlivan Attractor Reference: Akgul, A., Hussain, S., & Pehlivan, I. (2016). A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik - International Journal for Light and Electron Optics, 127(18), 7062–7071.
$$
\begin{cases}
\dot{x} = \alpha \left(x-y\right), \\
\dot{y} = -4\alpha y + xz+\varsigma x^3, \\
\dot{z}=\alpha\delta z + x^3y+\beta z^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\end{bmatrix}=
\begin{bmatrix}
1.8\\
-0.07\\
0.12\\
1.5
\end{bmatrix}.
$$
The Wang—Sun—Cang Hyperbolic Tangent Attractor Reference: Wang, Z., Sun, Y., & Cang, S. (2011). Acta Physica Polonica B, 42(2), 235. Herrera-Charles, R., Afolabi, O. M., Núñez-Pérez, J. C., & Ademola, V. A. (2024). Secure communication based on chaotic spherical 3D attractors. In Applications of Digital Image Processing XLVII (Vol. 13137, pp. 78-89).
$$
\begin{cases}
\dot{x} = \alpha x - \beta y + \varsigma z + 2\tanh\left(100\sin y\right), \\
\dot{y} = -\delta xz+\xi+ \eta x, \\
\dot{z}= \zeta xy + \varepsilon yz + \varphi z + \vartheta,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\xi \\
\eta\\
\zeta \\
\varepsilon\\
\varphi\\
\vartheta
\end{bmatrix} = \begin{bmatrix}
-4.1\\
1.2\\
13.45\\
1.6\\
0.161\\
0\\
2.76\\
0.6\\
13.13\\
3.5031
\end{bmatrix}.
$$
The Huang—Yang Attractor (Chemical Chaotic Reactor) Reference: Huang, Y., & Yang, X.-S. (2005). Chaoticity of some chemical attractors: a computer assisted proof. Journal of Mathematical Chemistry, 38(1), 107–117. Vaidyanathan, S. (2015). A Novel Chemical Chaotic Reactor System and its Adaptive Control. International Journal of ChemTech Research, 8(11), 654–668.
$$
\begin{cases}
\dot{x} = \alpha x-\delta x^2-xy-xz, \\
\dot{y} = xy-\varsigma y, \\
\dot{z}=\beta z-xz-\varepsilon z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix} = \begin{bmatrix}
30\\
16.5\\
10\\
0.5\\
0.5
\end{bmatrix}.
$$
The Wang—Hu—Tian Attractor Reference: Wang, Q., Hu, C., Tian, Z., Wu, X., Sang, H., & Cui, Z. (2023). A 3D memristor-based chaotic system with transition behaviors of coexisting attractors between equilibrium points. Results in Physics, 56, 107201.
$$
\begin{cases}
\dot{x} = \alpha y +\beta\left(0.1 + 0.5y^2\right)z, \\
\dot{y} = z, \\
\dot{z}=x-\varsigma y -z - 2.682\cdot10^{-4}\sinh\left(4.0485x\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
2\\
2.5\\
2
\end{bmatrix}.
$$
The Qiu—Xu—Jiang Attractor Reference: Qiu, H., Xu, X., Jiang, Z., Sun, K., & Cao, C. (2023). Dynamical behaviors, circuit design, and synchronization of a novel symmetric chaotic system with coexisting attractors. Scientific Reports, 13(1).
$$
\begin{cases}
\dot{x} = y-2xz, \\
\dot{y} = -x+0.5\left(1-x^2\right)y-0.5yz, \\
\dot{z}=0.1xy+\alpha x^2-0.8,
\end{cases}
$$
$$
\alpha = 0.21.
$$
The Vaidyanathan—Volos Attractor Reference: Vaidyanathan, S., & Volos, C. (2015). Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Archives of Control Sciences, 25(3), 333–353.
$$
\begin{cases}
\dot{x} = \alpha y + xz, \\
\dot{y} = -\beta x + yz, \\
\dot{z} = 1 - x^2 - y^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix} = \begin{bmatrix}
0.05\\
1
\end{bmatrix}.
$$
