Questions tagged [complex-systems]

This tag is for questions regarding to complex system, a system composed of many components which may interact with each other.

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What are the latest advancements with regards to complex systems?

I am an endocrinologist, 90 years old, interested in finding out the latest developments in the field of complex systems. I am not aware if this concept is still valid in today's modern mathematical ...
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Variation of graphs with respect to "number of nodes" vs fixed "number of nodes"

I am having a hard time finding mathematical discussion on variation of graphs with respect to number of "nodes". It seems graph theoretical/network analysis is very much focused on networks ...
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Can you modeling complicated dynamics without using differential/difference equations?

Let's imagine there is a phenomenon I want to understand. I have a few multivariate time series about the phenomenon but not a lot. I don't know how the variables are related to each other but from ...
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complex system dynamic model

I have a complex system composed of $n$ sub-systems. Each sub-system have dynamic equation as below: $${\overline{p}}_{i,t+1}=\frac{1}{\left|N_i(t)\right|}\sum_{j\in N_i(t)}{{\overline{p}}_{j,t}} $$ $$...
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Can this problem be treated in purely mathematical way other than the graphical way?

Problem Statement: I attempted all the parts. For part (a), $E_1 = P + R + S$, so $ \dot E_1 = \dot P + \dot R + \dot S$. Plugging these values and rearranging the terms would get the desired ...
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Calculating the Lyapunov exponents for the seir epidemic model

I am trying to numerically calculate the Lyapunov exponents for the seir epidemic model given as: $$ s^{'} = b - bs - \beta si \\ e^{'} = \beta si - (\alpha + b)e \\ i^{'} = \alpha e - (\gamma + b) i ...
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What's the relationship between these processes?

A snow ball rolling down a mountain: It's surface area gets bigger, thus the ball collects more snow until it can't hold it's own weight anymore. Two hot dog stands are next to each other: If one has ...
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How much algebra is preferable for studying/doing research in dynamical/complex system and networks

This question seems quite broad to ask... The situation here is that I'm a second-year undergrad student majoring in math and statistics. I'm really interested in fields like complex system and ...
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Hessian Matrix with infinite Eigenvalue

I'm dealing with an optimization problem, so I need to determine if I get maximum, minimum, or saddle points in a function of two variables. However, When evaluating the eigenvalues o my Hessian ...
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Is the set of all grammatically correct words a lattice and is there grammar for which this set is not lattice?

Let G be a formal grammar of any Chomsky type (as defined by the production rules) and let the production rules establish the order among words of this grammar: words a and b are a<=b if a can be ...
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Strange fixed point in state space

I'm studying the following dynamical system, \begin{align} \dot{x} &= y \,\, , \\ \dot{y} &= \frac{\left(-4 x^3+33 x^2-78 x+54\right) y^2+(x-3) (2 x-6)^2}{(3-2 x) (2-x) x (2 x-6)} \,\, , \end{...
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Can synchronized chaotic systems be periodic or stochastic?

I am confused regarding the terms chaotic, periodic, and stochastic in the context of coupled chaotic systems: In general, if chaotic systems are coupled together and synchronized then does the ...
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Conservation of the Hamiltonian

I'm struggling with the following calculus of variation problem. For an autonomous problem, it is often said that the Hamiltonian is constant along an extremal trajectory. However, the proofs of that ...
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Lyapunov-Stability - global asymptotic stability

sadly I wasn't able to find a 'control systems' community, so I'm posting here. I'm working through the book of H. Khalil - Nonlinear Systems and I'm stuck at the following part: He states the ...
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How to determine Markov partitions?

Given: \begin{equation} f_1(x)= 4x(1-x)\\ \end{equation} \begin{equation} f_2(x)=\begin{cases} 1-2x , & \text{if $ 0 \leq x \leq 1/2$}.\\ 2(x-1/2)/3, & \text{if $ 1/2< x ...
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The Evolution of inverted pendulum is on which Smooth Manifold?

The Dynamical System (Cart& Inverted Pendulum) does it evolves on smooth manifold or Non-smooth manifold? If so, what is the order for this smooth manifold($C^0,C^1,C^n,C^{\infty}$). Please give ...
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How to choose a set of boolean functions with a specified probability of getting a 1

So let's say I have a boolean function $f(x)$ that takes in a size k binary vector and outputs a binary scalar. Each function is defined as a $2^k$ vector. For example $f((0,0)) = 0, f((0,1)) = 1, f((...
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How do I perform this complex integration?

Evaluate the complex integration $(z^2 + 3z)$ wrt $z$ along the circle $|z| = 2$, from $(2,0)$ to $(0,2)$ in a counterclockwise direction. As far as I understand, this can be solved by taking $x = 2 \...
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Need Help in solving the following Darboux system of non-linear equations

I am working on a personal math project of mine, and in order for me to continue I need to know the solution to this following system of nonlinear equations. $$ \frac{d\omega_1}{d\eta} = \omega_2\...
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How do you obtain the fixed points and stability of a piecewise function?

Hi so I'm trying to work out how to find the fixed points of a piecewise function and do a stability analysis. One of the past exam papers in this complex systems course I'm doing has this question: ...
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Do a complex system's attribute changes always exhibit depedance?

Say a complex system C has an attribute A. Can I assert that a change of A is always – to some degree – dependent upon A’s past? My reasoning is the following: Any attribute of C is influenced by C....
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Stability of delay differential equations

I have encountered a 2-dimensional system of differential equations. One of them is a delay differential equation (DDE). Can anybody explain to me how to analyze the stability of a DDE?
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Staff icebreaker - is stasis ever attained?

Yesterday at work we had a staff day, where we were asked to play an interesting game as an icebreaker. We (50 or so people) were told to stand in a circle and choose 2 people at random out of the ...
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