In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical ...

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203 views

Chaos (and logistic functions); what is it and is it truly chaotic?

I'm currently studying discrete dynamic models and I am now reading about the logistic function $x_{n+1} = ax_n(1-x_n)$. Below there is a picture what happens with different values of a: These are ...
3
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1answer
102 views

Repeated Eigenvalues 2

Two problema from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney). Examples (pages 112-113): If $$A= \begin{pmatrix} ...
2
votes
1answer
100 views

Repeated Eigenvalues

Two problems from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney), examples page 112,113: If $$A= \begin{pmatrix} 2 ...
3
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1answer
35 views

Need further help and verification regarding a dynamic model problem

We have the model $y= ax(1-x)$ and we want to find the period 2 solutions such that $X_{n} = X_{n+2} $ and $X_n \neq X_{n+1}$. My teacher told us to do this problem with the quadratic formula. This is ...
2
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1answer
177 views

How to find periodic solutions using a graphing calculator

We have the model $X_{n+1} = 4\left(X_n - \dfrac{1}{2}\right)^2$ with a given $X_0$ on the domain $[0,1]$. We have the following question: Use your graphing calculator to figure out if there are ...
3
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0answers
58 views

temporal operators: interpreting them topologically in a dynamic topological system

There's a paper that i've been reading recently called "Dynamic Topological Logic" which can be found at: http://individual.utoronto.ca/philipkremer/onlinepapers/DTL.pdf. I have a question about ...
2
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0answers
91 views

Does this type of bifurcation exist?

I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed ...
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vote
1answer
26 views

Discrete dynamic models

We have the equation $$x_{n+1} = ax_n(1-x_n) - v_n$$ Why are there only fixed points for $(a-1)^2 - 4av_0 \geq 0$? Show that if $ 1<a<4$, there are 2 fixed points with $0<p_1 < p_2 ...
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2answers
397 views

What programming languages are used in (chaotic) dynamical systems and nonlinear phenomena research?

I'm currently considering pursuing postgraduate studies in the field of chaos/dynamical systems/nonlinear phenomena, and was wondering whether there are particular programming languages that are ...
5
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0answers
186 views

Proving that $f$ has points with prime period 3

Exercise: Let $f:\mathbb{R}\longrightarrow\mathbb{R} $ be continuous, $n>3$, and $x_1,x_2,\cdots,x_n$ be points such that $x_1<x_2<\cdots<x_n$. Show that if $f(x_i)=x_{i+1}$ for ...
3
votes
1answer
97 views

Path of particle under gravity

If a particle is subjected to gravity then $$\frac{∂^2 u}{∂\theta^2} +u = \frac{GM}{h^2} $$ where $$ u = \frac{1}{r}$$ and $$h = r^2\dot{\theta}.$$ If you solve this you get ...
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0answers
87 views

Dynamics of solutions close to $x(0)$ of $\dot{x}=\sqrt{x}+f(t)$ for $f(t)$ small when $t \ll 1$

I was looking at the dynamics of the real solutions close to $x(0)=0$ for the non-autonomous ODE \begin{equation} \dot{x}= \sqrt{x} +f(t) \end{equation} where $f(t)>0$ is `small' for $t \ll 1$ ...
0
votes
1answer
106 views

which book teaches analysis of nyquist, bode and rlocus diagram

would like to use knots to get a formula for nyquist diagram however, no crossing, and have no experience in analysis of graph related to control, as i have no books mentioning this and i observe ...
0
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2answers
46 views

Under which circumstances are there fixed points?

Consider the following equation: $$x_{n+1} = ax_n + b$$ Under which circumstances is there a fixed point solution? Under which circumstances is there a period 2 solution? So for the first question I ...
2
votes
2answers
52 views

Linear reccurence relation

We have the linear reccurence relation $$x_{n+1} = \dfrac{3}{2}x_n - 20$$ with $n = 0,1,2...$ and $a,b$ are constants. Does this equation have a fixed point? Does the equation have a period 2 (a ...
4
votes
1answer
541 views

Attractive and repulsive fixed points

Consider the function $f(x) = ax(1-x)$. I have to show that if $ 1 < a < 3$ that the fixed point $p_2 = \dfrac{a-1}{a}$ is attractive, and if $ 3 < a < 4$ it is repulsive. I actually have ...
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vote
3answers
77 views

Find the fixed points

We have $x_{n+1} = ax_n +b$ with $x_0$ given. We have to find the fixed points of this function, and decide for which values of $a$ they are stable. So I looked it up and found that a fixed point is ...
6
votes
1answer
901 views

How to prove Mandelbrot set is simply connected?

In this lecture note of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is ...
4
votes
0answers
102 views

What is this bifurcation?

I have a discrete dynamical System $x_{n+1}=f(x_{n},x_{n-1},x_{n-2},x_{n-3},x_{n-4},\lambda)$ with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger ...
1
vote
1answer
66 views

Dynamic system, fixed point

Given the dynamic system $u^{k+1}=g(u^{k})$, and a vector $v ∈ \mathbb{R}^n$ that satisfies $v=g(v)$, which is said to be a fixed point of the system. i) Suppose that the solution for a dynamic ...
2
votes
1answer
222 views

Help with brute force method of producing bifurcation diagrams of discrete-time systems

I have a homework question concerning a brute force method of creating bifurcation diagrams. This seems really abstract for me and would like a clearer description of how the method works. Can someone ...
5
votes
1answer
171 views

Delay-differential equation

Consider the equation $$ f'(t)=\frac{f(t-b)}{t-b}$$ $f'(t)=\frac{df(t)}{dt}$ and $b$ is a constant. Does anyone know if this equation has a name, an analytic solution and how to find the solution? ...
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1answer
73 views

Implementing logistic map understanding

For the following project I'm supposed to complete, I am not fully understanding what it is exactly asking for. Am I supposed to run a code using the first equation or the second? What is the ...
3
votes
2answers
570 views

How do I find transfer function of a discrete-time system when its state-space form is given?

I read this and this Wikipedia pages, but both of them are explaining continuous-time systems. My question is about discrete-time case. For example, given the state-space equations of the second ...
2
votes
2answers
221 views

Milk and Coffee will they ever finish

Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass ...
1
vote
1answer
41 views

Limiting value of iteration $x(k+1) = A x(k) + B u(k)$ for summable $u(k)$

A matrix $A$ is known to converge such that $\lim_{k\rightarrow \infty} A^k = \bar{A} \neq 0$. We have an iteration defined as $$x(k+1) = A x(k) + B u(k), \ \ k\in \mathbb{Z}_+.$$ $\{u(k), ...
2
votes
2answers
132 views

Solve system of first order differential equations

I have to solve differential systems like this: $$ \left\{ \begin{array}{c} x' = 3x - y + z \\ y' = x + 5y - z \\ z' = x - y + 3z \end{array} \right. $$ Until now I computed the eigenvalues $k = ...
2
votes
1answer
85 views

Normal form of a vector field in $\mathbb {R}^2$

Edited after considering the comments Problem: What is the normal form of the vector field: $$\dot x_1=x_1+x_2^2$$ $$\dot x_2=2x_2+x_1^2$$ Solution: The eugine values of the matrix of the linearised ...
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votes
2answers
163 views

example of an unstable fixed point for which the linearized dynamics are stable

What would be an example of an unstable fixed point for which the linearized dynamics are stable? Thanks in advance
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vote
1answer
37 views

possible dynamics on $\mathbb{R}^2$

A linear map is non-hyperbolic if $|\lambda_i|=1$ for a least one eigenvalue $\lambda_i$. Catalogue the possible dynamics of a non-hyperbolic linear map on $\mathbb{R}^2$ For something like this ...
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0answers
54 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
10
votes
1answer
254 views

Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are ...
2
votes
1answer
76 views

Exp. Stability of perturbed system with temporally vanishing perturbation

I have a perturbation problem for which I can't find a fitting theorem in Khalil's Nonlinear Systems. Maybe someone can point me in the right direction: Given a nominal system $\dot x(t) = ...
2
votes
0answers
64 views

How to compress a linear operator and have the lossless composition property.

Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
3
votes
1answer
439 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
0
votes
1answer
53 views

Complex Dynamics of periodic points

How do I show that e^2PIi/5 as a periodic point for the function of f(z)= z^3. Also what is it's prime period?
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vote
1answer
384 views

How do you find the fixed and period-$2$ points of $f(x)=x^2-3x+3$?

I am trying to do this question using the Fixed Point Factor Theorem. I keep getting an answer $>0$ at the end of my long division of $f(x)-x$ into $f^2(x)-x$ therefore I must using the wrong ...
2
votes
1answer
80 views

Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic

Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
1
vote
3answers
53 views

Fixed points and iterates of an invertible function

Suppose that $g : [0,1] \rightarrow [0,1]$ is a continuous and strictly increasing function such that $g(0)=0$ and $g(1)=1$. Under these hypotheses $g(x)$ has an inverse function $g^{-1} :[0,1] \to ...
0
votes
1answer
33 views

Range of stability for iterative map

Using linear stability analysis, I would like to compute the range of stability of the fixed points and the $2$-cycles of the following iterative map: $x_n = x_{n-1}^{2} - 3\mu$. Setting $x = x^{2} - ...
1
vote
1answer
50 views

Strictly invariant sets of the rotation transformation on a discrete space.

Fix an integer $n>0$. Consider the space $X=\{a_0,a_1,...,a_{n-1}\}$ with transformation $T:X\to X$ defined by $T(a_i)=a_{i+1(\text{ mod n})}$. What are the strictly invariant sets of this space? ...
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1answer
53 views

A good reference to learn the concept of partition function

I am looking for a good reference and easy to learn the concept of partition function in mathematics. Can anyone help me?
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2answers
143 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
3
votes
2answers
109 views

Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$

From numerical test, I know $x=1$ is an attractive fixed point of the function $$ f(x)=\frac12 \left(x+\frac{1}{x}\right), $$ on $(0,\infty)$. Is there a way to prove it? Since $$ ...
3
votes
4answers
877 views

PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
3
votes
2answers
167 views

Ergodic theory question about the support of a measure.

I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question: ...
3
votes
1answer
90 views

Examples of systems conforming the Lorentz Attractor

Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor? It can be any kind of system, just a little list. It can be a system ...
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1answer
67 views

How write these two motions into one system

f describe the horizontal motion g describe the vertical motion have error when put m(t) in x() Maple code ...
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2answers
188 views

Conditions that Roots of a Polynomial be Less than Unity

Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
5
votes
1answer
247 views

Strength of attraction of fixed points

Consider a smooth map $f: \mathbb{R} \rightarrow \mathbb{R}$ with an attracting fixed point $F$. Then, we have if $f'(F) \ne 0$, $F$ is a "simple" attracting fixed point, if $f'(F) = 0$, $F$ is a ...