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I assume that you are talking about an undamped undriven double pendulum. In this case the motion exhibited by the double pendulum may be chaotic (depending on the intial conditions, lengths of the pendulum arms and masses), but it exhibits no attractors in the sense that trajectories converge to a certain invariant set (the attractor) in phase space. ...


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I found the (shamefully simple) answer myself: The simple double pendulum is a conservative system, hence due to Liouville’s theorem the phase-space volume given by a given ensemble of trajectories is constant over time. However, if the system exhibited transients and thus attractors, all trajectories starting within the basin of attraction of a given ...


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This is not unlike the question "can you theoretically balance a perfect pencil on its tip". The answer is always "no". No, because the initial conditions cannot be perfectly obtained; and the equations of motion are such that a small perturbation from the initial condition will grow. You cannot generate perfectly laminar flow on a molecular scale, ...


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The simplest answer to your question would be: Yes, put the flag out on a day when there is no wind. If you want to be able to distinguish which country it represents, tilt it so that its pole is horizontal. But I guess that this is not the situation that you have in mind. I imagine that you are asking if it is possible that wind will be strong enough to ...


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As somebody who works in the field of chaos theory (for whatever that’s worth), I confirm Dmckee’s assessment: There is no reasonable relation to any concepts from chaos theory. There is, however, an attempt in your quote to relate this to the phenomenon of criticality – which is not chaos theory, but like chaos theory is related to the field of complex ...


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$$\lambda= \lim_{n → \infty} \frac{1}{n} \sum\limits^{n-1}_{i=0} \log \left |f'(x_i) \right|$$ When $n\rightarrow \infty$ I think that the Lyapunov exponent, $\lambda$, should be zero because of the $1/n$ factor, so what am I not understanding? No, the number of summands in your sum is going towards $∞$ as fast as your $\frac{1}{n}$ factor is ...


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Weather models have significant positive feedback loops built in because without them the return to average is too fast for accurate prediction. This is due to the grossly inadequate spatial density of weather collection stations, and results in weather models being subject to the butterfly effect. However, the actual circumstances in which significant ...


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I think both answers can be found if you look at car designs in F1. Red Bull were famous for abusing 'blown defusers' which is blowing exhaust gas under the car, and a flexible front wing which bends depending mostly on air resistance. My basic understanding of designing cars in F1 is that there is a LOT of trial and error, which is one of the reasons why ...


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Could a butterfly flapping its wings cause a hurricane? Yes. However, only if the required conditions (which are very precise) existed elsewhere. There would be a critical state of the atmosphere after which a hurricane will form, and a small injection of energy from a butterfly flapping its wings could cause this threshold to be crossed. Is it likely that ...


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This question already has an answer (by me) on Earth Science: The butterfly is a colourful illustration of Chaos Theory, and the word butterfly came from the diagram of the state space (see below). (Apparently, my claim on the origin of the word butterfly may be historically inaccurate. Could be of interest for HSM SE) A system that is chaotic is ...


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Ok, I do not know the Sage algorithm but I am going to offer a conjecture of what is happening. You have to verify the conjecture by further numerical investigations. I assume that the Sage algorithm works optimally for bifurcations of a single equilibrium and can run into problems such as we see here when dealing with equilibria (AKA fixed points) ...


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Your question or confusion is mostly based on several misconceptions of the premises: Chaos theory is not a theory in the scientific sense like, e.g., the theories of relativity, evolution or quantum mechanics. It does not make predictions about the laws of nature. You do cannot make statements about reality like: “According to chaos theory, …”, or: “This ...


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Suppose I go outside, flap my hands in the air and then go about my business as usual. A few months later I see on the news that some town in the US has been devastated by a tornado. Is the counterfactual statement that says that had I not flapped my hands, the town would not have been hit by the tornado, a rigorously correct statement? Let's assume for ...


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Does the flap of a butterfly's wing in Brazil set off a tornado in Texas? This was the whimsical question Edward Lorenz posed in his 1972 address to the 139th meeting of the American Association for the Advancement of Science. Some mistakenly think the answer to that question is "yes." (Otherwise, why would he have posed the question?) In doing so, they ...


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The butterfly effect is a popularization of chaos theory. This diagram is part of the narrative of chaos theory for the hoi polloi. A plot of Lorenz attractor for values r = 28, σ = 10, b = 8/3 It does look like a butterfly after all :; ( tongue in cheek) . Let us set up the chaos backround first, from the Wiki article: Chaos theory is the ...


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Yes - but only as far as one is willing to believe the mathematical model fits reality. In the mathematics we can demonstrate the butterfly effect; the sensitivity of particular nonlinear dynamic system models to initial conditions. And we can contrive certain experiments of systems that seem to behave in the context of a butterfly effect. But even for ...


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The effect is real in the sense that the movement of the butterfly may have a huge effect on the weather in a far away place. There is however no way to control this. Don't imagine some mad scientist holding the world for ransom with a cage full of butterflies. It is better to think of it as an illustration of the theory of chaos. The idea is that chaotic ...


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First the statement is a poetical way to express how in chaotic systems, small changes can trigger drastically different results. This statement does not attempt to relate butterfly movement to large scale weather changes. I would say we have elements to say it is not, for example, waves are a chaotic behaviour of the sea surface, but we have never found ...


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Obviously the effect needs time to propagate (probably the speed of sound), but the effect is quite real. Imagine a incalculably large pool table where half of table has balls, if you shot the cue ball so it clipped one of the balls on the half with balls, the effect could propagate to the end of the table, but if you missed that half nothing would happen. ...


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The uncertainity principle does indicate indeterminism in quantum mechanics, in the sense that uncertainity in position measurement necessitates a corresponding uncertainity in momentum measurement. There is another aspect of indeterminacy in quantum mechanics in the context of measurements, which states that the exact outcome cannot be predicted. On the ...


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Other answers have pointed to the fact that the question in classical mechanics has only limited applicability in real life (CuriousOne's comment in particular points out that this is at best an academic question), but let's discuss it out of curiosity nevertheless. Most systems in classical mechanics are deterministic. Here is a simple heuristic: The ...


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You are making to the following mistakes: Your initial displacement $d_0$ is 5 % of the variation of the respective variable, while it should be orders of magnitude lower. The Lyapunov exponent is defined for the limit of infinitesimal displacements, i.e., $d_0→0$. Your observation time is far too short. You are looking at four oscillations of your ...



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