45
votes
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How do computers "solve" the three-body-problem?
Numerical analysis is used to calculate approximations to things: the value of a function at a certain point, where a root of an equation is, or the solutions to a set of differential equations. It is ...
- 33.4k
42
votes
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Are double pendulums eventually periodic?
Short answer: No. General trajectories of double pendulum are not periodic.
You need to distinguish between two aspects: the trajectory in the spatial coordinate system and the trajectory in phase ...
- 4,239
31
votes
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Is there a formalization of the butterfly effect?
... if a physical system is deterministic, then it does not make sense to talk about perturbations of it ...
The future (and past) behaviour of a deterministic system is fixed once we know its state ...
- 60.3k
29
votes
Why does chaos preclude exact solutions?
You are right about this wording being sloppy. A chaotic system is a deterministic system whose solutions at late times are exponentially sensitive to early times. Nothing about this definition ...
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The Electron at the End of the Universe
The calculations are done in
Schwartz (2019), Lecture 3: Equilibrium (https://scholar.harvard.edu/files/schwartz/files/3-equilibrium.pdf)
Here's a summary of the key equations, using the notation ...
- 55k
22
votes
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Are all aperiodic systems chaotic?
No.
A system might be, for instance, stochastic, random - which is certainly not an example of deterministic chaos, but is aperiodic.
You can also have quasiperiodic behavior, where the system comes ...
- 12.7k
18
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Are double pendulums eventually periodic?
From wikipedia, the phase space of a dynamical system is defined as
... a space in which all possible states of a system are represented, with each possible state corresponding to one unique point ...
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17
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Why are we sure that integrals of motion don't exist in a chaotic system?
I think one must distinguish between chaotic "Hamiltonian" systems and chaotic dissipative systems. In the latter, the phase space volume is not conserved, so it is much more difficult to find "...
- 1,440
17
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A rule for when phase-space orbits may cross
The general idea is that trajectories that cross on phase space will have identical initial conditions from that point and will therefore satisfy identical initial-value problems, and from that you ...
- 135k
15
votes
What is a quantum scar?
Quantum scars were first conceived by Heller in 1984. They essentially describe the quantum phenonenon that certain classically chaotic systems leave behind traces, or scars, of their clasical ...
- 1,219
14
votes
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How do we know chaotic systems are actually chaotic and not periodic?
Well, on the one hand, yes, chaos is a mathematical abstraction so, for instance, there will never be an experimentally or numerically measured Lyapunov exponent, only finite Lyapunov exponents (FLEs) ...
- 12.7k
14
votes
Is there a formalization of the butterfly effect?
Usually butterfly effect is mathematized as "sensitivity to initial conditions". This is a bit different to what you are saying. Namely you consider two close initial conditions $x_1(0),x_2(...
- 2,484
13
votes
Are there necessary and sufficient conditions for ergodicity?
integrability, chaos and non-linearity are intimately related to ergodicity. Is there any set of conditions relating these concepts with ergodicity?
Yes.
In order to exhibit chaos, a system must be ...
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13
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What is the chain of cause(s) and effect(s) which does a butterfly's wing-flapping cause a hurricane 1000's of kilometers further away?
It's a mistake to attribute the hurricane to the butterfly. Rather, hurricanes are going to happen at various different times and in various different places, but the precise time and location are ...
- 61.8k
13
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How can one distinguish between a random process and a chaotic process?
To build on user304539's answer: what you want to know is whether starting in the same initial state $\mathbf{x}_0$ again will move you to the same next state $\mathbf{x}_1=\mathbf{f}(\mathbf{x}_0)$ (...
- 33.4k
12
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Why are we sure that integrals of motion don't exist in a chaotic system?
I am not an expert on these issues, but if a further integral existed the orbit would be confined in a codimesion-1 embedded submanifold (for almost all the possible values of that function due to ...
- 78k
11
votes
Accepted
Chaos and general relativity
Yes, we definitely have chaos in GR.
One of the earlier references on the topic is Barrow's Chaotic behaviour in general relativity (e-print).
A particularly relevant result is that Relativistic ...
- 12.7k
11
votes
Accepted
Are there any known models with limit cycles in their RG flow?
Such systems are quite possible, modelled copiously, the focus of a cottage industry, and have numerous applications. Beyond the Bulycheva & Gorsky review arXiv:1402.2431 that @Buzz links above, ...
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11
votes
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Dynamicity inside a stationary water drop
Based on the similar movement of inhomogeneities on soap bubbles, I'd say the flakes' movement is caused by air currents around the drop and perhaps thermal convection.
Source: https://youtu.be/...
- 12.7k
11
votes
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Are chaotic systems the same as dissipative systems in inverse time?
Yes, you are missing something. Looking at the change of phase-space volume ($∇·f$), you get three categories – if you have a constant sign of $∇·f$ (more on the alternative at the end):
dissipative (...
- 6,379
11
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Is there a formalization of the butterfly effect?
To your first comment, it makes sense to talk about a perturbation of a system. You are correct that if the system is deterministic, it can lead to only one outcome, but we can consider additional ...
- 51.7k
10
votes
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Is it possible for a system to be chaotic but not ergodic? If so, how?
A trivial example of a non-ergodic, chaotic system is a 2D conservative system that is not fully chaotic, i.e., with a mix of regular and chaotic regions in its phase space: each individual chaotic ...
- 12.7k
9
votes
What do physicists mean by an "integrable system"?
I am a bit late to the party, but I have had similar questions to yours in the past. I will summarise below what I know, which has been able to "quell" my dissatisfactions about integrable ...
- 25.2k
9
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Is it possible to quantify how chaotic a system is?
There are a number of ways of quantifying chaos. For instance:
Lyapunov exponents - Sandberg's answer covers the intensity of chaos in a chaotic system as measured by its Lyapunov exponents, which is ...
- 12.7k
9
votes
How are jerk equations connected to chaos theory?
A remark before. For chaos you need to have at least a 3D phase space.
A jerk equation consider a system of the form
$$\frac{\partial^3}{\partial t^3}x= f(x,\dot{x}, \ddot{x})$$
You can set ...
- 381
9
votes
Dynamicity inside a stationary water drop
Those are flakes of lampblack. Their wettability in water is unequal around their periphery. This allows surface tension forces to apply a netforce in some random direction on the flake and force it ...
- 97.8k
9
votes
Do all dynamical systems have attractors?
Conservative systems are an entire class of dynamical systems that doesn’t have attractors. You might argue that no real system (except perhaps the entire universe) is truly conservative, but then ...
- 6,379
9
votes
Why does chaos preclude exact solutions?
I think Connor is correct. For what it's worth, there are three different words people are using here.
"Analytical" means it can be represented by a convergent power series. "Closed-...
- 105
8
votes
Accepted
Self-study book for dynamical systems theory?
In no specific order:
Alligood K.T., Sauer T.D., Yorke J.A, Chaos. An Introduction to Dynamical Systems
That's a personal favorite of mine at the undergraduate level. It's clearly written and they ...
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