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These figures — made using ρ = 28, σ = 10 and β = 8 / 3 — show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10 −5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one ...
The Lorenz attractor is an iconic example of a strange attractor in chaos theory.This three-dimensional fractal structure, resembling a butterfly or figure eight, reflects the long-term behavior of solutions to the Lorenz system, a set of three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer (of ...
Chen chaotic attractor [9] continuous: real: 3: 3: Not topologically conjugate to the Lorenz attractor. Chen-Celikovsky system [10] continuous: real: 3 "Generalized Lorenz canonical form of chaotic systems" Chen-LU system [11] continuous: real: 3: 3: Interpolates between Lorenz-like and Chen-like behavior. Chen-Lee system: continuous: real: 3 ...
The Lorenz attractor is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
A plot of Lorenz's strange attractor for values ρ = 28, σ = 10, β = 8/3. An attractor is called strange if it has a fractal structure, that is if it has non-integer Hausdorff dimension. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.
The Lorenz attractor arises in the study of the Lorenz oscillator, a dynamical system.. In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve.
The critical attractor. An attractor is a term used to refer to a region that has the property of attracting surrounding orbits, and is the orbit that is eventually drawn into and continues. The attractive fixed points and periodic points mentioned above are also members of the attractor family.
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.. In a dynamical system such as ¨ + (;) + =, which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case = is the symmetric pitchfork bifurcation.