0
votes
0answers
27 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
0
votes
2answers
40 views

Derivative of a time derivative

I wasn't really sure how to phrase the title. Suppose I have a purely time-dependent function $x(t)$, and I want to know its time derivative $\dot{x}:=\dfrac{dx}{dt}$. Then I ask how the time ...
4
votes
0answers
66 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
0
votes
0answers
40 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
6
votes
1answer
139 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
15
votes
1answer
474 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
0
votes
1answer
28 views

Order in taking limits?

So I am reading this physics paper that they define Kolmogorov entropy for dynamical systems as follows: $$K=\lim\limits_{\epsilon\to 0}\lim\limits_{T\to \infty}\frac{I(\epsilon,T)}{T}$$ They ...
1
vote
1answer
251 views

calculus, predator-prey system

The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady states of the ...
0
votes
0answers
189 views

Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
2
votes
1answer
107 views

How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
1
vote
0answers
15 views

How fast should I make a human detector signal?

So I am designing a break beam and I want to be able to catch all people running at whatever speed they can possibly run at.. I need to pulse the signal (ideally between 1-50Hz.. after 50Hz my sensors ...
1
vote
1answer
95 views

Average number of predators and prey in Lotka–Volterra model?

Once again I wouldn't be surprised if this can be found maybe even on Wikipedia but I'm not a native English speaker and unfortunately couldn't find this myself. So assuming standard Lotka–Volterra ...
5
votes
0answers
59 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...
5
votes
2answers
87 views

A continuous function that when iterated, becomes eventually constant

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq ...
1
vote
1answer
1k views

Determining the bifurcation value(s) for a one-parameter family

Let's say we have a one parameter family: $$ \frac{dy}{dt} = y^2 + k $$ I want to find the bifurcation value. What does this mean? It seems like I need to set dy/dt = 0 and then solve for k, but ...
0
votes
0answers
47 views

How do I determine the distance of an object if the acceleration is constantly changing?

I am investigating a pendulum's movement and I am trying to create a proof, or an equation that describes the movement. My problem is that as the pendulum is falling the angle between the force of ...
0
votes
2answers
111 views

Approximate a Discrete time dynamical system by a continuous one

Is there any method by which we can somehow "embed" a non linear discrete time system into a continuous time dynamical system? (Assume discrete time system here is a set of non linear difference ...
1
vote
1answer
104 views

Can any dynamical system be written as a hamiltonian system?

Can I always find a Hamiltonian for any given Dynamical System such that the Hamiltons' equations are satisfied? The hamiltonian may be an extremely complicated function (Possibly containing complex ...
4
votes
1answer
202 views

Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(…\cos x))}}$ exist? [duplicate]

Possible Duplicate: Explaining $\cos^\infty$ Does the following limit exist? $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$ If yes, find the limit. If no, please ...
0
votes
1answer
85 views

How to model a system for tracking a person using kalman filter?

I need to model a system for human motion. The following link shows for to build a system for a plane. I am currently reading the documentation for a kalman filter library ...
1
vote
2answers
96 views

Question about dynamical behavior near point

Let $x' = f(x)$ be autonomous first–order equation differential with an equiliburiium point $x_0$. Suppose $f'(x_0) = 0$ what can I say about the behavior of soluton near $x_0$? If $f'(x_0) ≠ 0$ and ...
-2
votes
2answers
165 views

What kinds of maths to learn for understanding dynamical systems in cognitive science?

A current trend in cognitive science is to view the mind as a dynamical system (e.g., Continuity of Mind by Spivey, in which cognition is understood as a "continuous and often recurrent trajectory ...
0
votes
1answer
95 views

find the critical value of r.

For the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of $r$. $$\dot{x} = rx + \cosh x$$ I seem to understand how to do the first ...
1
vote
1answer
137 views

Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1

Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below. $x'=f(x)=-x$ $x(0)=1$ EDIT / UPDATE: x_n+1=x_n + ...
2
votes
1answer
160 views

Why this vector field $f$ belongs to $C^1({\bf R}^2\times {\bf R})$?

The following system is an example in a book of dynamical system(in the section about Hopf Bifurcation). $$ \begin{align} \dot{x}=\mu x- y-x\sqrt{x^2+y^2} \\ \dot{y}=x + \mu y-y\sqrt{x^2+y^2} ...
1
vote
0answers
193 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
2
votes
1answer
144 views

Minimizing the cost of a path in a dynamic system

So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ ...
1
vote
1answer
867 views

Determine a conserved quantity in a dynamical system Lotka-Volterra

I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are: $\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$ ...