I first encountered the Lorenz system in the late 1980s when chaos and fractals were popular. I remember extending the programming source included in Mitsuo Morimoto's "Differential Equations Using Computers to 3D and drawing the solution trajectory of the Lorenz system.

The Lorenz system became well known because of its sensitive dependence on initial conditions: even a very small difference in the starting values can lead to completely unexpected trajectories, despite the solutions remaining bounded. This phenomenon is widely known as the “butterfly effect.” The name “butterfly” comes from the shape of the attractor formed by the trajectories, which resembles a butterfly with its wings spread. In recent years, the term “butterfly effect” has been frequently used in NHK programs... :-(

The Lorenz system forms a single strange attractor that resembles a butterfly with its wings spread. For the parameter values shown below, even if the initial point (the starting point of the trajectory) is far from the attractor, the trajectory will gradually be drawn into the butterfly. This is a characteristic feature of the Lorenz system.

You can find plenty of hits if you search for the Lorenz system. For reference, I'll mention the following site.

1. Lorenz_system

2. Butterfly effect

The parameters often used are $a = 10.0, b = 28.0, c = 8.0 / 3.0$. This tool also uses these values by default. Initial values are set as $(0, 1, 2)$ and $(0, 1, 2.01)$.

While it's hard to believe this equation leads to chaos, it indeed does.

For the Thomas model, please refer to the following site.

1. Thomas' cyclically symmetric attractor

In the simulation tool, the parameter is set to $a = 0.21$, and the initial conditions are $(x, y, z) = (0, 1, 2)$ and $(0, -1, -2)$. Under these conditions, two attractors exist symmetrically with respect to the origin. When the value of $a$ is increased (e.g., to around 3), the trajectories converge to a single point. When $a$ is decreased (e.g., to around 0.19), the trajectories begin to mix while preserving symmetry about the origin, eventually forming a single attractor and exhibiting chaotic behavior.

This can be interpreted as follows: when $a$ is large, the damping terms $-ax, -ay, -az$ are strong, preventing trajectories from crossing the basin boundary, resulting in two separate attractors. As $a$ decreases, the oscillatory terms ($\sin$) become dominant, allowing trajectories to cross the boundary and move between regions, thereby forming a single attractor.

The trajectory resembles two Lorenz systems combined together. This formulation was based on the first reference site I found; it differs slightly from other published Four-Wing equations. However, when plotted, it clearly produces a Four-Wing attractor.

The following site was the first reference I found for the Four-Wing system.

1. STRANGE ATTRACTORS

The tool uses parameters $a = 0.2$, $b = 0.01$, $c= -0.4$, and initial values $(x, y, z) = (0, 1, 2)$ and $(0, 1, 2.01)$.

This system behaves similarly to the Thomas model, showing relatively regular motion while still exhibiting chaos. Although increasing the rendering speed makes it harder to see, it exhibits sensitivity to initial conditions.

For information about the Halvorsen system, please refer to the following site.

1. A Symmetric Chaotic Flow

In the simulation tool, the parameter is $a = 1.499$, and the initial conditions are $(x, y, z) = (0, 1, 2)$ and $(0, 1, 2.01)$.

This system was proposed by the German biologist Otto Rössler in a 1976 paper.

Although the Rössler system is very simple, with only one nonlinear term ($zx$ in the third equation), it still generates chaotic behavior. It was originally discovered during research on chaos and was later found to be useful for modeling chemical reaction equilibria.

For details on the Rössler system, please refer to the site below.

1. Rössler attractor

In the simulation tool, the parameters are $a = 0.2, b = 0.2, c = 5.7$, and the initial conditions are $(0, 1, 2)$ and $(0, 1, 2.01)$.

This equation was designed by Julien C. Sprott as part of his search for minimal chaotic systems, and several such systems exist. Despite its extremely compact form, it is capable of producing chaos and exhibits sensitivity to initial conditions.

For details on the Sprott B system, please refer to the site below.

1. Simple Chaotic Flow GIF Animations

2. 3D Chaotic Attractors

In the simulation tool, the parameters are $a = 0.4, b = 1.2, c = 1$, and the initial conditions are $(0.1, 1, 1)$ and $(0.11, 1, 1)$.

The trajectories move among three attractors in a complex manner. As in the Lorenz system, sensitivity to initial conditions is observed.

For details on the Rabinovich–Fabrikant system, please refer to the site below.

1. Rabinovich–Fabrikant equations

In the simulation tool, the parameters are $a = 0.14, b = 0.1$, and the initial conditions are $(-1, 0.9, 0.6)$ and $(-1, 0.9, 0.60001)$.

This system exhibits sensitivity to initial conditions and produces chaotic trajectories within attractors. Unlike the Lorenz system, which has a single attractor, the Chen–Lee system has multiple attractors, and the specific attractor approached depends on the initial conditions.

For details on the Chen-Lee system, please refer to the site below.

1. 3D Chaotic Attractors – The Sequelaen Collection

In the simulation tool, the parameters are $a = 5.0, b = -10.0, c = -0.38$, and the initial conditions are $(5, 10, 10)$ and $(-7, -5, -10)$.

This system was proposed by the Chinese mathematician Chen Guanrong and can be regarded as a nonlinear extension of the Lorenz system. Like the Lorenz system, it has a single attractor and exhibits sensitivity to initial conditions.

The Chen system has applications in secure communications, synchronization, and control.

For details on the Chen system, please refer to the site below.

1. Fully Integrated Chen Chaotic Oscillation System

In the simulation tool, the parameters are $a = 40, b = 3, c = 28$, and the initial conditions are $(-0.1, 0.5, -0.6)$ and $(-0.1, 0.5, -0.601)$.

The Chua circuit, which exhibits chaotic behavior, was invented in 1983 by the Chinese-American computer scientist Leon O. Chua . It can be constructed using capacitors, an inductor, resistors, and a nonlinear element (the Chua diode).

Because its dynamics can be accurately modeled by differential equations, the Chua system is one of the rare chaotic systems that can be physically realized and experimentally verified.

For details on the Chua system, please refer to the site below.

1. Multiscroll attractor

In the simulation tool, the parameters are $a = 15.41, b = 28, c = -0.714, d = -1.143$, and the initial conditions are $(-0.000001, 0, 0)$ and $(-0.0000011, 0, 0)$.

With these parameters, a double-scroll attractor appears and exhibits sensitivity to initial conditions. By adjusting the parameters, single-scroll and periodic attractors can also be obtained.

This equation is often cited as a typical example of a quasiperiodic attractor. A similar system is the Aizawa system; although some coefficients and nonlinear terms differ, they share very similar properties.

For details on the Langford system, please refer to the site below.

1. Numerical Studies of Torus Bifurcations

In the simulation tool, the parameters are $a = 0.7, b = 3.5, c = 0.6, d = 1, e = 0.25, f = 0$, and the initial conditions are $(1, 1, 1)$ and $(1, 1, 1.1)$.
In this case, the trajectory is quasiperiodic and evolves on a torus, forming a very beautiful attractor.

However, when the coefficient $c$ is increased to around 0.8, the torus begins to break down and the system transitions to chaos, exhibiting sensitivity to initial conditions.

This system was introduced by Dadras and Momeni in 2009 and generates multi-scroll attractors.

For details on the Dadras system, please refer to the site below.

1. A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors

In the simulation tool, the parameters are $a = 3, b = 2.7, c = 1.7, d = 2, e = 5.4$, and the initial conditions are $(1, 1, 0)$ and $(1, 1, 0.1)$.
In this case, a four-scroll attractor is obtained; however, increasing $e$ from 5.4 to around 9 changes it to a three-scroll attractor.