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The following interpretations are taken from Thorne [2014]. Chapter 17, entitled Miller's Planet, discusses the issue of the large waves on the water planet in the movie Interstellar. There Kip mentions that the waves are due to tidal bore waves with height of ~1.2 km. In the appendix entitled Some Technical Notes, Kip estimates the density of Miller's ...

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The phase potrait for this physical system describes the oscillation of the particle described above to be a Homoclinic orbit-that is, the particle oscillating between extreme ends of a double well. No, not even close. I assume that your misunderstanding originates from confusing phase space and geometrical space. Suppose, we describe the dynamical ...

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For a general answer concerning these types of systems, we can take the simple gravity pendulum as a representative example. The equation of motion is generally non-linear, but can be approximated to be linear at small angles (small amplitudes). See also the small angle approximation section in the linked wikipedia article. For progressive waves you will ...

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Many dynamical systems involve recursive sequences of simple functions, $f(f(f(...f(x))))$ which are iterated n times to lead to some discrete iteration sequence (a Picard sequence) with often tractable properties. It is possible however, to analytically continue the discrete iteration index n to a continuous non integer one, fractional, infinitesimal, or ...

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Will there be a symbolic sequence for each dimension or will a symbol be assigned to a point $(x,y)$? This depends what you eventually want to do with your symbol sequence, but for typical applications, such as determining the entropy or modelling, you want to assign one symbol to the point. The general reason behind this is that (for a proper ...

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Take a potential energy of the form $$U(x)= x(x-1)\:.$$ Next consider a smooth function $g: \mathbb R \to [0,1]$ such that vanishes in an open neighborhood of $0$ and takes the constant value $1$ form $x= 1/4$ on. Finally consider the dynamical system defined by $$\dot{x}=y$$ $$\dot{y}= -U'(x) - cg(x)y$$ for some $c>0$. With these choices, $(x,y)=(0,0)$ ...

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