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20

An orbit is stable because of conservation of angular momentum. Suppose we start with an object in an exactly circular orbit and slow it down slightly. That means it is moving at less than orbital velocity so it starts to fall inwards. However as its distance to the Sun decreases the tangential component of its velocity has to increase to conserve angular ...


20

Chaotic is not the same as random. A chaotic system is entirely deterministic, while a random system is entirely non-deterministic. Chaotic means that infinitesimally close initial conditions lead to arbitrarily large divergences as the system evolves. But it's impossible, practically speaking, to reproduce the same initial conditions twice. Given ...


12

Well, yes. In a purely mathematical world where you can specify initial conditions exactly, chaotic systems are fully deterministic. It's not like a quantum system with wavefunction collapse, whose evolution can never be specified exactly by the initial conditions. But in practice, we can never specify (or know) the initial conditions exactly. So there will ...


11

If we accept that the system (the Earth's atmosphere in this case) is chaotic and adopt the usual definitions of a chaotic system, e.g. one by Edward Lorenz Chaos: When the present determines the future, but the approximate present does not approximately determine the future. we immediately see the answer to your question. The small (impossible to ...


9

Classical mechanics is perfectly integrable for two bodies as a closed or isolated system. However, early on it was found that problems existed, where Newton found he could not find a solution for the motion of the planets in a complete form. He made his famous statement that God had to readjust the solar system now and them. Poincare solved the Sweden ...


9

I think the following from the wikipedia entry clears up well the terminology: Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is ...


8

The gravitational potential is what is known as a central force, which means that "how strong" the potential is only depends on how far away you are, and not on what angle you are relative to it. Having said that, gravitational systems are often treated in terms of an effective potential (full explanation provided on the Wikipedia page) which look like this ...


7

If you take a well-behaved physical system and perturb it a little bit, then you expect the total behavior or your system to be changed only a little bit. You can quantify this by saying that if your initial perturbation is $\delta$, then the final perturbation can never exceed $\gamma \delta$ for some constant $\gamma$. In many cases, such perturbations ...


6

The basic idea is that statistical properties of complex physical systems fall into a small number of universal classes. A very known example of this phenomenon is the universal law implied by the central limit theorem where the sum of a large number of random variables belonging to a large class of distrubutions converges to the normal distribution. Please ...


6

Good question. You are correct in that without any restoring force, an object balanced precariously in an equilibrium position will be unstable. In physics, we use the scalar quantity of "potential" to find the equilibrium positions. These will be the maxima and minima in the potential field. The negative gradient of the potential gives the force. You've ...


6

Qualitative discussion (almost math free) The real key to understand orbits is the conservation of angular momentum. A two body orbit is nice this way insofar as it is a planar system and we get an easy expression for the angular momentum (we'll assume a satellite much, much less massive than the primary and not bother with the canonical transformation ...


5

Jack Wisdom at MIT has extensively studied the question of the stability of the solar system. He has a list of papers with links to freely-readable PDF files on his website: http://groups.csail.mit.edu/mac/users/wisdom/ A good starting point might be "Is the Solar System Stable? and Can We Use Chaos to Make Measurements?" (PDF) (in Chaos, proceedings ...


5

It is true that the double pendulum exhibits integrable behavior, when the initial angles are very small, however, in general, it is very difficult to characterize the chaotic behavior of the double pendulum in terms of the initial angles. There are other representations which provide a clearer picture of its chaotic behavior. The introductory section of ...


5

Strictly speaking, there is no quantum chaos. Time evolution is unitary, which implies that small changes in state are not magnified in size. Thus the sensitive dependence on initial conditions, the prerequisite for chaos in classical mechanics, is absent in quantum mechanics. Even stronger, in discrete quantum systems with a finite-dimensional Hilbert ...


5

I'm following Steven H. Strogatz's explanation in his book Nonlinear Dynamics and Chaos throughout this answer. Great book, worth a read. See also the detailed explanation on the wolfram website. Let's use the logistic map $x_{n+1} = rx_n(1-x_n)$ as an example. Its bifurcation diagram looks like this (courtesy of wikipedia): The easiest window to ...


4

In general, no. It is possible to recognise a (system of) differential equation(s) as being nonlinear purely by inspection, but there are plenty of non-chaotic nonlinear systems. Chaos is a stronger (and, unfortunately, not well-defined) condition. Also, many (most?) systems, including the famed Lorenz attractor, only exhibit chaos under certain conditions. ...


4

I'm guessing that when you talk about randomness you're thinking about the collapse of the wavefunction and that the the result of the collapse is apparently random. If so, most us currently believe that the randomness is only apparent and is the result of decoherence. Decoherence describes the interaction of a quantum system with the environment around it. ...


4

Let's start out using notation similar to the example you linked to: $$ \ddot{y}+b\dot{y}+\sin(y)=a\cos(ct)+d $$ As in the example, we'll write this in autonomous form: $$ \mathbf{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}y\\\dot{y}\\ct\end{pmatrix} $$ the differential equation can now be written: $$ ...


4

Chaos isn't easy to define precisely, but I'll use the definition from Nonlinear Dynamics and Chaos by S.H. Strogatz to show the features everyone agrees on: Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions. Aperiodic long-term behaviour means there are no fixed points, closed ...


4

The dimension should be 3 or larger. If the dimension is smaller then 3 the existence and uniqueness theorem for differential equations will tell you that functions can't intersect (since you want them to be continuous and differentiable). In 1 dimension this means you can only have movement in one direction In 2 dimensions this means that your value ...


4

Indeed, in Kundu & Cohen, Fluid mechanics, the derivation of Lorenz system is rather sketchy. So let us turn to the source. Original paper by E.N. Lorenz is: Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2), 130-141. doi, Free pdf. (12000+ citations!) The text is quite accessible, so for all the ...


3

A stepping mortor or magnetic field lines in a Tokamak when two magnetic islands overlap are simple examples of chaotic behaviour (really different from turbulent) that can be easily controlled. http://www.epj.org/_pdf/HP_EPJB_slowly_rocking.pdf http://www-student.elec.qmul.ac.uk/people/josh/documents/ReissAlinSandlerRobert-ICIT2002.pdf Lyshevski S., ...


3

I am assuming that the idea of fractal cosmology expresses that the universe has some sort of fractional pattern to it, which is the nature of fractal construct, and implies homogeneity. No, this is basically wrong. Homogeneity is mutually exclusive with fractal. This isn't my area either, but I will try to describe the state of our understanding ...


3

Yes you can estimate the dimension by box counting. It is not quite hand waving but the idea has an advantage to be intuitive. 1) You consider the logistic map attractor like an analogy to a Cantor set whose dimension you can compute by box counting. 2) You remember that when the chaotic bands double, their sizes scale like $1/a$ and $1/a^2$ where $a$ is ...


3

First, you need a deterministic dynamical system. By deterministic it means that the state of the system is univocally determined at each time, ie. at each time you have one and only one possible state. In the counterpart are stochastic systems where, instead, the state of the system is determined by a distribution of possible states and is this distribution ...


3

According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are: Deterministic system (only one future for each state) Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature) A positive maximum Lyapunov exponent. 3) is pretty much the quantitative standard in journals of chaos, assuming you meet the ...


3

Perhaps a better question to ask is: why is a single pendulum non-chaotic? Almost all real systems are chaotic at least to some extent; the fact that we can write out the solution for a single pendulum for all points in time is really quite peculiar, and only true because it is a highly simplified system. The reason these non-chaotic systems are so prevalent ...


2

I think there is some debate about this, at least according to Steven Strogatz. I paraphrase a part of his lecture that I'm watching: Even systems with perfectly known, perfectly deterministic laws, where all the positions of all the particles, and all the forces, are known, can still be unpredictable. It is such systems that are called chaotic. This is ...



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