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1
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1answer
103 views

Symbolic dynamics of a multidimensional system

Let $x_t = F(x_{t-1})$ be a discrete-time dynamical system in the chaotic regime. Starting from an initial condition $x_0$, we can generate a time series $(x_t)$ where $t =1,2,...,T$ indicates the ...
0
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0answers
12 views

Stability of a complex valued system using Jacobian

I want to find the conditions for stability of a system by constructing the Jacobian matrix and finding what values of my parameters will give me negative eigenvalues. My system is of the form $ ...
2
votes
2answers
27 views

When is an attractor meaningful?

I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of ...
0
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0answers
19 views

How is wave equation equivalent to the Turing reaction-diffusion equation with 3 species?

I was reading the paper, "Critical waves and the length problem of biology" by Robert B. Laughlin. He mentions a point like this in it : "This wave equation is equivalent to Turing Reaction-diffusion ...
-2
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1answer
98 views

What is the meaning of this Feynman's statement? [closed]

Richard Feynman has a strange statement in first lecture of his known book "Feynman Lectures on Physics. He says If a piece of steel or a piece of salt, consisting of atoms one next to the other, ...
0
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1answer
90 views

Is the double pendulum an example of a strange attractor?

Imagine a pendulum to which is attached another one (not necessarily the same length). Does this pendulum, when you let it go, which can be done in many ways but let's keep the total potential energy ...
1
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1answer
90 views

Randomness v. complexity

There are a few other topics I found that explore this idea from a different perspective: Is randomness deterministic? Can randomness exist? Is the universe fundamentally deterministic? My ...
0
votes
1answer
39 views

There isnt really an isolated system so why sometimes we consider a system to be isolated?

In real life, there's nothing to be called an isolated system then why at some applications do we assign isolated systems?
5
votes
1answer
71 views

Pink noise in low-dimensional systems

Pink noise (1/f) is often cited as a signature of complex or critical systems. Is it possible for a low-dimensional time-independent first-order system to generate pink noise? Intuitively it seems ...
0
votes
1answer
45 views

Why trajectories approach to origin tangent to the slower direction?

I am reading non-linear dynamics from Strogartz. Suppose, I have two solutions of a non linear system: $x(t) = x_0e^{at}$ and $y(t) = y_0e^{-t}$, where $a\in \mathbb{R}$. Now it is clear that,for ...
1
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0answers
17 views

Pre-, post- and analytical phases of physics data analysis? [closed]

I am writing a technical (physics) report about a GEM (Gas Electron Multiplier) system which has several weaknesses in several parts of the system: pre-analytic: wrong post-operative tool has been ...
4
votes
1answer
122 views

Analytic proof that Lyapunov exponents in Hamiltonian systems pairwise sum to zero

I have read that in Hamiltonian systems, Lyapunov exponents come in pairs $(\lambda_i, \lambda_{2N-i+1})$ such that their sum is equal to zero. Is there a way of proving this analytically? EDIT: ...
2
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1answer
103 views

Physical interpretation of the canonical scalar product in linear dynamics

Consider a unforced, undamped, linear mechanical system with a finite number of degrees of freedom. Its (second order) dynamical equations can be gathered in a matrix equation $$M\ddot X + K X=0$$ ...
5
votes
1answer
143 views

Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any ...
0
votes
1answer
161 views

What is the relationship between quantum physics and chaos theory?

I am not a physicist, I am looking for a non-technical explanation. Articles such as this one seem to hint at the fact that "macro reality" regulated by classical mechanics is somehow a pattern ...
0
votes
1answer
48 views

In reaction-diffusion processes what is the difference between oscillatory media and excitable media?

What is the basic differences between oscillatory media and excitable media? I know that both comes under reaction-diffusion processes. Where do Turing patterns come in the picture? Can some one give ...
0
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0answers
44 views

Heat Retention of Wine

This may seem odd and lengthy but I am going to attempt to consolidate my thoughts as much as possible. I'm sorry in advance for the length of my post and if this is confusing. I am currently ...
6
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4answers
115 views

Energy conservation without action principle?

The normal tagline for energy conservation is that it's a conserved quantity associated to time-translation invariance. I understand how this works for theories coming from a Lagrangian, and that this ...
3
votes
1answer
103 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
1
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0answers
24 views

Change of variables to apply Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
0
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1answer
78 views

Time it takes for a mass in a linked pendulum to flip?

I have created Mathematica code that simulates a double pendulum. So I've numerically solved for $\theta_{1}(t)$ and $\theta_{2}(t)$. I have also found the momentum from the Lagrangian as well. My ...
2
votes
1answer
281 views

Importance of periodic orbits

In the study of dynamical systems, one often talks about solutions that repeat themselves after a certain time, hence their name of "periodic orbits". Then one moves to the distinction of "stable" ...
4
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2answers
3k views

Understanding the Jacobian Matrix

Taking the example of a two dimensional system, desribred by the following ODE's: \begin{align} \frac{dx_1}{dt}&=f_1(x_1,x_2)\\ \frac{dx_2}{dt}&=f_2(x_1,x_2) \end{align} The Jacobian Matrix ...
1
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3answers
74 views

Complexity of a physical system

Are there any accepted definitions quantifying the complexity of: a) macroscopic, classical mechanical systems (e.g., a bicycle) b) microscopic systems (ensembles of atoms)? By the way, I'm not ...
3
votes
0answers
156 views

What is the future of complexity theory in black-hole physics and string theory? [closed]

I found the recent work by Hayden and Harlow and Susskind very fascinating. I have also heard talks by Scott Aaronson about this emerging connection. In particular this idea of understanding ...
15
votes
1answer
445 views

The natural metric of a phase space and the Lyapunov exponent

For me, it seems that there is no apparent metric on a phase space of a dynamical system. Of course one can naively define an Euclidean metric on it, but it seems that this metric has not much to do ...
1
vote
1answer
179 views

Hill's and Mathieu's equation [closed]

I am supposed to apply Hill's and Mathieu's equation to parametric pendulum. Can you tell me what is the difference between them? Why are they used? What do they describe?
2
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0answers
56 views

Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. ...
1
vote
2answers
104 views

Resources on Master Equations

Presently I am reading about "Introduction to dynamical process theory and simulation" which uses the notion of Master Equations to solve Markov process. I am very new to this. Can someone provide me ...
4
votes
0answers
49 views

Solutions of nonlinear systems invariant wrt. perturbations (looking for applications)

I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics. The problem in a nutshell: describe properties of solution sets of real ...
2
votes
0answers
47 views

What does unfolding of attractor mean?

What does unfolding of attractor mean? Effect of time scales on the unfolding of neural attractors paper talks about Takens embedding theorum. It says that the embedding dimension should be large ...
1
vote
0answers
62 views

Logistic map and attractors

Does the logistic map have a strange attractor for some "chaotic" values of the parameter?
1
vote
1answer
242 views

Does Lyapunov exponent equate to exponential inflation?

Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors ...
8
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2answers
4k views

What creates the chaotic motion on a double pendulum?

As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random? I'm just ...
4
votes
1answer
828 views

Calculating Lyapunov exponents from a multi-dimensional experimental time series

Wolf's paper Determining Lyapunov Exponents from a Time Series states that: Experimental data typically consist of discrete measurements of a single observable. The well-known technique of phase ...
3
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0answers
60 views

Reference for the Landau-Lifshitz system

I'm interested in understanding the dynamics of the discrete Landau-Lifshitz system. It's solutions to equations like $$\frac{\partial X_n}{\partial t} = X_n\times (X_{n-1}+X_{n+1})$$ where the $X_n$ ...
2
votes
2answers
734 views

Two suns, one moon, and one planet?

I have a question about how would seasons and the moon cycle be affected in a system where one planet orbits Sun #1, and Sun #1 orbits a second sun. Online I found this description: "Type II: ...
4
votes
1answer
696 views

About Poincare section for the double pendulum

I am reading Prof. Louis N. Hand's Analytical Mechanics. In the chapter about chaos, it introduces the concepts of Poincare section based on the example of double pendulum. Also, it plot the section ...
1
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0answers
25 views

Self-contained book about complex systems and nonlinear dynamics [duplicate]

I am a student at the 2nd year of a B.Sc. in Biotechnology. I started reading some papers about complex systems and nonlinear dynamics applied to economy, biology of course, climate model etc... I ...
5
votes
1answer
91 views

Stability theory [closed]

I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any ...
3
votes
2answers
259 views

Linearized equations

What is $V_{\alpha\beta}$? And what is a symmetric, positive definite potential energy matrix? And why is there a linearized equation like this?
2
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2answers
184 views

Mapping between numbers and symbolic representations

I am not a physicist but applying symbolic dynamics for information coding in signal processing. Is there any mapping between symbols and numbers?
0
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0answers
33 views

Is there a thermodynamic law or theorem that expresses how systems “break” or “change” when enough energy is added?

I have a simple question about thermodynamic laws, and I am hoping you can help me. Let's say that I have a sphere container with some pressurized gas in it. I can slowly increase the pressure over ...
1
vote
2answers
497 views

What are the *necessary* conditions to deterministic chaos?

What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos? We collected yet: A positive ...
3
votes
2answers
979 views

What are the principles of deterministic chaos?

I see in literature very different (and chaotic) descriptions of what is deterministic chaos. Can you explain to me based in a type of formal definition, which principles need to be exactly fulfilled ...
10
votes
2answers
462 views

Hamiltonian or not?

Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example $$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$ ...
2
votes
0answers
102 views

Book reviewing current state of research on complex networks [closed]

Can anybody recommend a book reviewing the current state of knowledge and active research on complex networks? Not primarily a textbook but a true review of the field - ideally with references to ...
6
votes
2answers
615 views

What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
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3answers
2k views

What are some of the best books on complex systems?

I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory ...
3
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0answers
78 views

Kolgomorov entropy issues

I am long been confused by these entropy terms. Would be obliged if an explanation is provided in less technical jargon What are the differences between Shannon's entropy, topological entropy and ...