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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13 views

ensure marginal stability of system, or center of a nonlinear dynamic system

If I have a coupled nonlinear system, like $$\dot{x}=ax-bxy$$ $$\dot{y}=cxy-dx$$ by using jacobian matrix, I can find that the point ($\frac {d}{c}$,$\frac {a}{b}$) is a center. I think in a ...
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13 views

toplogical entropy of general tent map

Measure theoretic entropy of General Tent maps The linked question made me wonder how to calculate the topological entropy of a general tent map. Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define $T: I ...
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1answer
12 views

Finding normal coordinates of a system

We have coupled oscillators with equations of motion: $$\ddot{x} = -10x+18y$$ $$\ddot{y}=-3x+5y$$ At $t= 0$ we have $x=a$ and $\dot{x}=\dot{y}=y=0$. I found the solution to be $$\begin{pmatrix} x(t) ...
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1answer
18 views

Bifurcation Diagram stability

I'm wondering how you know which branches are stable. For example taking $\alpha$ as the bifurcation parameter $$y'=\frac{y(1500-y)}{3200}-\alpha.$$ So I plot $\alpha=\frac{y(1500-y)}{3200}$ and flip ...
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1answer
30 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
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21 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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36 views

Solving coupled oscillation systems

We have coupled oscillators with equations of motions: $$\ddot{x} = -10x+18y$$ $$\ddot{y}=-3x+5y$$. Initially $x=a$ and $\dot{x}=\dot{y}=y=0$. I found the solution to be $$\begin{pmatrix} x(t) \\ y(t) ...
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16 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
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1answer
29 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
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17 views

Subshifts of finite type; No fixed or period 2 points

I'm working out of Devaney's Introduction to Chaotic Systems, and one of the problems I'm working on is to construct a subshift of finite type in $\Sigma_3$ with no fixed or period two points, but ...
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54 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
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2answers
64 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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1answer
22 views

Is a cascaded chaotic system is still chaotic?

I am curious whether a new system which cascades two individual chaotic systems is always chaotic. My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that $${\rm ...
2
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0answers
23 views

Relationship between Reproductive Ratio and Jacobian in Population Model

In class we defined the Reproductive Ratio, $R_0$ of a population modelled by SIR, SEIR,... as the average number of secondary infections caused by an average infected individual in an average ...
2
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0answers
13 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
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1answer
41 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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1answer
28 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
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0answers
34 views

Fixed point - definitions (asymptotically stable, repelling)

Given the following fixed point $(\widehat{x},\widehat{y} )$. If this point is not asymptotically stable, can I then conclude it is a repellor? Is a repellor/repeller defined to be a not ...
2
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1answer
71 views

Show that system is Transcritical bifurcation

In what ways can you show that transcritical bifurcation occurs? For example take the system $$\dfrac{dx}{dt}=xr+2x^2 $$
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1answer
49 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
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1answer
40 views

Dynamical system with periodic orbit

We are given dynamical system $\phi $ in $R^2$, and know that it has periodic orbit (means $\phi(T,x_0)=x_0$ for some $T>0$ and $x_0 \in R$). We are asked to prove that the system has stationar ...
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1answer
21 views

Question about Kronecker factor

In his paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by ...
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0answers
20 views

Complex Symplectic Matrix

A symplectic $2n\times 2n$ real matrix is a constant matrix $M$ that satisfies $$MJM^T=J$$ where $J=\begin{bmatrix}0 &I_n \\-I_n & 0\end{bmatrix}$. We now that if such matrix is used as a ...
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9 views

I have this question: might the sign of the first lyapunov coefficient be different when once solve with the formula and another in matcont. [closed]

i explain this: suppose we have this ODE system $\dot x_i=f(x_i)$ when i solve it with this formula in Kuznetsov's book with $l_1(0)=(1/2w_0) Re[<p,C(q,q,\bar q)>...]$ the sign be negative, ...
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2answers
57 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
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1answer
96 views

Find values of the parameters in Predator prey model

$$r' = F_1(r, f) = r − cr^2 − drf$$ $$f' = F_2(r, f) = −f/4 + erf + gf^2$$ Consider the case where $g = 0$. For what values of the parameters, $c, d$ and $e$, which are all assumed to be positive, ...
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18 views

Is it possible to apply the Melnikov function to nonperiodic perturbations?

In the case of planar Hamiltonian system, the classical Melnikov function deals with the periodic perturbation. Is it possible to apply the Melnikov function to nonperiodic perturbations?
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32 views

Question about Singer's theorem

I recently studied Singer's theorem, but every proof I have read does not detail one important step that I still don't understand. This step can be written as the following: We have $h:I\subseteq ...
3
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1answer
65 views

Compact space, continuous dynamical system, stationary point

I'm having trouble proving that if $X$ is a compact metric space and every continuous function $f : X \rightarrow X$ has a fixed point, then every continuous dynamical system $ \varphi $ on $X$ has a ...
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29 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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62 views

Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire? [migrated]

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
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1answer
46 views

Differential Equation: Periodicity of a circle with zero radius in polar coordinates

I am given the following diff. equation in polar coordinates: $$\dfrac{dr}{dt} = r(1 + a~\cos \theta - r^2) \\ \dfrac{d \theta}{dt} = 1$$ where $a$ is a positive number and is less than $1$. I am ...
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0answers
64 views

Predator prey system question?

QUESTION 5. 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 ...
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15 views

How to Build a Foresight System? [migrated]

For a research project, I'm asked to find ways to build an economic foresight system. For example, for the production of cheese. We will have data about the market indicators, like price, demand etc. ...
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1answer
20 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
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1answer
25 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 ...
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1answer
29 views

Reference about the Conley index thoery

I'm reading "Isolated invariant sets and the Morse index" by Charles Conley.But I'm lost in some of the concise description or definition.Could you recommend me some references or textbooks for the ...
1
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2answers
64 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
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0answers
24 views

Ask about a definition [on hold]

Is there any definition about holographic space in mathematics? I searched in network but I didn't find
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60 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
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1answer
56 views

Equilibria and stability

Find all equilibria for the following system and determine their stability: $$x'=y^2-4$$ $$y'=x^2-1$$
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1answer
38 views

Jordan canonical forms and diagonalizing.

In my dynamical systems, we are asked to find the Jordan Canonical form of the Jacobian in order to analysis the linear stability at fixed points in a second order system. I believe that even for one ...
1
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1answer
200 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 ...
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1answer
26 views

Period 3Dynamical Systems [closed]

Let $x^{+}=f(x)$ be a scalar dynamical system for $x\in\mathbb{R}$, with $f$ a strictly decreasing function. How can we prove that a solution of period 3 does not exist?
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1answer
54 views

Ergodic Rotation of the Torus

Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel ...
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33 views

Historical study of dynamical system

I am currently doing a historical study on my school project 'study of ODE' which slowly shift to the study of dynamical system as I am interested in pursuing my study of ode from linear system, phase ...
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2answers
69 views

How do I Linearize

How would I solve the following problem? Linearize around the fixed points $$\left\{\begin{align}\frac{\text{d}x}{\text{d}t}&=y-x^2\\\frac{\text{d}y}{\text{d}t}&=y-x\end{align}\right.$$ I ...
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0answers
18 views

Numerically calculate the boundary of a basin of attraction for a high dimensional dynamical system

I am looking for an efficient, non-exponential time algorithm to calculate the boundary of a basin of attraction for a stable fixed point in a high dimensional nonlinear dynamical system. The naive ...
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2answers
32 views

Simple proof of invariant sets

Let How to prove the unit circle is an invariant set? My way is that: At t1, x1(t1)^2 + x2(t2)^2 = 1, so the eqs become: Since both x1 and x2 are functions of 't', so solve it and obtain: ...
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1answer
28 views

Question on diffeomorphisms

Suppose that we are given an autonomous ode $\dot{x} = f(x)$ where $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$. My (elementary question) is that is the time one map for the ode above a local ...