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The Beauty of Chaos
The Lorenz system is a set of three coupled, nonlinear ordinary differential equations, first described by Edward Lorenz in 1963. [2] It famously demonstrates chaotic behavior, where small changes in initial conditions lead to vastly different outcomes, a phenomenon known as the "butterfly effect." [3]
Equations
- dx/dt = σ(y - x)
- dy/dt = x(ρ - z) - y
- dz/dt = xy - βz
The Rössler Attractor
The Rössler attractor is another classic chaotic system, discovered by Otto Rössler in 1976. It is simpler than the Lorenz system, with only one nonlinear term. Its structure is like a band that is twisted and folded back onto itself, creating a fascinating, continuous loop.
Equations
- dx/dt = -y - z
- dy/dt = x + a * y
- dz/dt = b + z * (x - c)
Thomas' Cyclically Symmetric Attractor
Discovered by René Thomas, this attractor is notable for its simple, cyclically symmetric equations that produce a complex, looping structure. Unlike many other attractors, it is not bounded by a "box" and its paths can extend infinitely, though they always circle back to the central structure.
Equations
- dx/dt = sin(y) - b * x
- dy/dt = sin(z) - b * y
- dz/dt = sin(x) - b * z
The Chen Attractor
Guanrong Chen and Tetsushi Ueta discovered this attractor in 1999. While it is structurally similar to the Lorenz system, it exhibits a different and arguably more complex chaotic behavior, creating a distinct and tangled visual shape.
Equations
- dx/dt = a(y - x)
- dy/dt = (c - a)x - xz + cy
- dz/dt = xy - bz
References
- Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos.
- Lorenz, E. N. (1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences.
- Gleick, J. (1987). Chaos: Making a New Science.