数理与编程 · Bilibili · 3:03
This video introduces the Chen-Lee attractor, proposed by Chinese scholars Guo-Liang Chen and Tian-Yan Li in the 1990s—a significant example of a chaotic system. A set of nonlinear differential equations describes its complex geometric structure in three-dimensional space. The video shows the trajectories of a single particle and 300 particles, illustrating how multiple particles better reveal the attractor's chaotic behavior and spatial structure. This video presents a three-dimensional visualization of the Chen–Lee attractor, a chaotic system in nonlinear dynamics. Its mathematical definition appears on screen: a system of three ordinary differential equations, with parameters a equals 5, b equals negative 10, and c equals negative zero point three eight. A dynamic glowing trajectory vividly demonstrates the attractor's complex, non-repeating, and bounded nature in three-dimensional space, highlighting the beauty of chaotic systems.
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At the beginning of the video, the screen displays the name of the Chen-Lee attractor and information about its proposers.
Next, the mathematical expression of the Chen-Lee attractor appears at the bottom of the screen: a system of nonlinear differential equations.
Then, the video shows the trajectory of a single particle in three-dimensional space, but points out that a single particle cannot fully reveal the attractor's complex spatial structure.
To more clearly illustrate the attractor's chaotic behavior and spatial structure, the video adds trajectories of 300 particles.
Finally, the video summarizes the importance of using multiple particles to demonstrate the Chen-Lee attractor's complex spatial structure.
The video shows the mathematical definition of the Chen–Lee attractor: a system of three coupled nonlinear ordinary differential equations, specifically dx over dt equals a times x minus y times z, dy over dt equals b times y plus x times z, dz over dt equals c times z plus x times y over three; where parameter a equals 5, b equals negative 10, and c equals negative zero point three eight. Meanwhile, a three-dimensional animation traces the system's trajectory, revealing its complex, butterfly-shaped chaotic structure.
The Chen-Lee attractor is an important example in chaotic systems, proposed by Chinese scholars Guo-Liang Chen and Tian-Yan Li in the 1990s; its mathematical expression is a system of nonlinear differential equations.
Although a single particle's trajectory follows the differential equations of the Chen-Lee attractor, it cannot fully display its complex three-dimensional spatial structure.
Through the trajectories of 300 particles, the chaotic behavior and complex three-dimensional spatial structure of the Chen-Lee attractor become clearer.
The Chen-Lee attractor is defined by three coupled nonlinear ordinary differential equations. The specific parameter values shown in this video are crucial for generating this particular chaotic structure.
This animation demonstrates how a trajectory evolves over time under the Chen-Lee equations: the path never repeats exactly, yet remains confined to a specific geometric region, thereby illustrating the concept of a strange attractor in nonlinear dynamics.
The Chen-Lee attractor is a chaotic attractor composed of three specific nonlinear ordinary differential equations. This video presents its exact mathematical expression and the parameter values used to simulate its three-dimensional structure.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
appears in the system of differential equations, such as = ax - yz
x
a coordinate variable in three-dimensional space
a real number
appears in the system of differential equations, such as = by + xz
y
a coordinate variable in three-dimensional space
a real number
appears in the system of differential equations, such as = cz + xy/3
z
a coordinate variable in three-dimensional space
a real number
parameter
a
a parameter of the system of differential equations
a real number
Parameter
b
Parameters of the differential equation system
Real number
Parameter
c
Parameters of the differential equation system
Real number
On-screen text displays the differential equations of the Chen-Lee attractor.
The system of ordinary differential equations defining the Chen-Lee attractor.
x, y, and z are state variables; t is time.
On-screen text shows parameter values (a equals 5, b equals negative 10, c equals negative zero point three eight).
, ,
Specific constant parameters used to generate the displayed chaotic trajectory.
Real numbers.
The Chen–Lee attractor, proposed by Chinese scholars Guoliang Chen and Tianyan Li in the 1990s
The equations of the Chen–Lee attractor are: the derivative of x with respect to time t equals a times x minus y times z; the derivative of y with respect to time t equals b times y plus x times z; the derivative of z with respect to time t equals c times z plus x times y divided by three; where a equals five, b equals negative ten, and c equals negative zero point three eight
The Chen–Lee attractor is an important example of a chaotic system proposed by Chinese scholars Guoliang Chen and Tianyan Li in the 1990s; its mathematical expression is a set of nonlinear differential equations.
a equals five
b equals negative ten
c equals negative zero point three eight
A single particle cannot reveal the attractor's complex spatial structure
Add 300 particles; their trajectories better reveal the attractor's chaotic behavior and spatial structure
By observing a single particle's trajectory, we find it cannot fully reveal the attractor's complex spatial structure. Therefore, we need to add multiple particles (for example, 300 particles) to better demonstrate the attractor's chaotic behavior and spatial structure.
The text 'Chen-Lee attractor =' followed by the system of ordinary differential equations and parameters remains visible throughout the video.
The Chen-Lee attractor is a chaotic attractor defined by a specific set of three nonlinear ordinary differential equations. This video presents its exact mathematical expression and the parameter values used to simulate its three-dimensional structure.
This visualization must use parameters a equals 5, b equals negative 10, and c equals negative zero point three eight.
Display a single particle's trajectory in three-dimensional space
A single particle
Particle position changes over time
The system of differential equations remains unchanged
Although a single particle's trajectory follows the Chen-Lee attractor's differential equations, it cannot fully reveal its complex three-dimensional spatial structure.
Display 300 particles' trajectories in three-dimensional space
300 particles
particle positions changing over time
the trajectory gradually forms a complex three-dimensional structure
the system of differential equations remains unchanged
Through the trajectories of 300 particles, the chaotic behavior and complex three-dimensional spatial structure of the Chen-Lee attractor become clearer.
A glowing, semi-transparent line traces a complex, non-repeating three-dimensional path against a black background, overall resembling a butterfly or double-loop shape.
glowing trajectory line
black background
This trajectory continuously evolves in three-dimensional space, spiraling around two main centers and repeatedly crossing between them, yet its path never exactly repeats.
As the trajectory gradually fills the entire structure, the attractor's overall bounded shape remains unchanged.
This illustrates the long-term behavior of a dynamical system starting from a specific initial condition, demonstrating sensitive dependence on initial conditions and the bounded chaotic nature characteristic of strange attractors.
A single particle cannot reveal the attractor’s complex spatial structure
believing that the trajectory of a single particle alone fully reveals the attractor’s complex spatial structure
In fact, the trajectory of a single particle shows only partial information; multiple particle trajectories are needed to more comprehensively reveal the attractor’s complex spatial structure.
A single particle cannot reveal the attractor’s complex spatial structure
Adding 300 particles, their trajectories better reveal the attractor’s chaotic behavior and spatial structure
By observing the trajectories of multiple particles, one can better understand the chaotic behavior and spatial structure of the Chen-Lee attractor.
The Chen-Lee attractor, proposed by Chinese scholars Guo-Liang Chen and Tian-Yan Li in the 1990s
A single particle cannot reveal the attractor's complex spatial structure.
Adding 300 particles, their trajectories better illustrate the attractor's chaotic behavior and spatial structure.
The explicit formula for the Chen–Lee attractor appears on screen.
Covered · This video thoroughly introduces the definition of the Chen-Lee attractor and the method of revealing its complex spatial structure using multiple particles.
Covered · This segment simultaneously displays the formula and animates the corresponding three-dimensional structure.
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