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3
votes
2answers
209 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?
9
votes
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 ...
10
votes
2answers
433 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 $$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$ ...
10
votes
1answer
1k views

Which areas in physics overlap with those of social network theory for the analysis of the graphs?

I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked a lot in this field. This causes me to assume that there are sub-fields in ...
5
votes
1answer
567 views

Physics-oriented books on fractals

I'm looking for some good books on fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis ...
3
votes
2answers
780 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 ...
1
vote
1answer
130 views

How to find the value of the parameter $a$ in this transfer function?

I am given a transfer function of a second-order system as: $$G(s)=\frac{a}{s^{2}+4s+a}$$ and I need to find the value of the parameter $a$ that will make the damping coefficient $\zeta=.7$. I am not ...
2
votes
2answers
1k views

Characteristic length, characteristic time and complexity of the process

Different physical processed (starting from elementary particles or even below to the universe itself) have different length scales $L$ and different characteristic times $T$. Larger processed tend to ...
6
votes
4answers
105 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 ...
6
votes
2answers
506 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 ...
3
votes
1answer
73 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 ...
2
votes
1answer
319 views

Scale invariance and self organized criticality

On wikipedia i have found this statement: In physics, self-organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their ...
1
vote
0answers
22 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 ...
1
vote
2answers
388 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 ...