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A double pendulum consists of two pendulums attached end to end.. In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. [1]
Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions. Chaos theory (or chaology [1]) is an interdisciplinary area of scientific study and branch of mathematics.
As a pedagogic tool, the Malkus waterwheel became a paradigmatic realization of a chaotic system, and is widely used in the teaching of chaos theory. [3] In addition to its pedagogic use, the Malkus waterwheel has been actively studied by researchers in dynamical systems and chaos. [4] [5] [6] [7]
A plot of Lorenz' strange attractor for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternative initial conditions on the attractor, the iterated points will become arbitrarily spread out from each other.
In 2017, Shijun Liao and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called the clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass.
chaos at = period-2 oscillation at γ = 0.65 {\displaystyle \gamma =0.65} Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below.
Since the system is invariant under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards: [1] r ( 0 ) = 0 {\displaystyle r(0)=0} The region close to the pivot is singular, since r {\displaystyle r} is close to zero and the equations of motion require dividing by r {\displaystyle r} .
A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems (especially partial differential equations).They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.