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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Is there a classical analog of Bloch's theorem?

In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect ...
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12 views

lineariztion of a nonlinear system

Consider a system described by the following nonlinear differential equation $\ddot{y} \ln{x} + \dot{x} (\dot{y})^{3/2} + x^2y = 1$. Let $x_0$ and $y_0$ be positive real numbers. Linearize the ...
2
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42 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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14 views

Definition of trajectory

I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve. Given a dynamical ...
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1answer
20 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
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1answer
44 views

$x'=\cos^5(x) +1$ has unique solution defined for all $t\in \mathbb{R}$

I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show ...
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43 views

Writing 5-dimensional dynamical system as Hamiltonian system

I've got a 5-dimensional continuous dynamical system, i.e., $$ \dot{x}(t)=f(x,y,z,u,w)\\ \dot{y}(t)=g(x,y,z,u,w)\\ \dot{z}(t)=h(x,y,z,u,w)\\ \dot{u}(t)=q(x,y,z,u,w)\\ \dot{w}(t)=p(x,y,z,u,w) $$ Is ...
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1answer
11 views

Prove $\langle x_0\rangle$ has only finitely many elements if and only if there exists $k_1$ and $k_2$ with $k_1 < k_2$ so that $x_{k_1} = x_{k_2}$

Prove that the orbit $\langle x_0\rangle$ has only finitely many (distinct) elements if and only if there exists $k_1$ and $k_2$ with $k_1 < k_2$ so that $x_{k_1} = x_{k_2}$ I know this to be true ...
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1answer
21 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
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2answers
27 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...
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1answer
17 views

On the computation of the Hessian matrix.

I'm trying to compute the Hessian matrix of a data fit of an ODE model to some data. Below is a cut out of the instructions I'm following (which can also be found at ...
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1answer
26 views

Determining the stability of a system using Lyapunov function

Consider the nonlinear dynamical system describing motion of a simple pendulum with viscous damping given by $$\ddot{\theta}(t)+\dot{\theta}(t)+gl\sin(\theta(t))=0,\quad \theta(0)=\theta_0, ...
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1answer
20 views

State transformation for non-holonomic differential equation.

Given a non-holonomic dynamical system, \begin{align*} \dot x = v\cos\theta \\ \dot y = v\sin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < ...
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45 views

Finding when fixed point is hyperbolic

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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20 views

Spectrum of $T^2$

Is it possible that $T^2$ has a discrete spectrum when $T$ is an invertible measure-preserving transformation whose spectrum is continuous or mixed ?
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18 views

Non-linear simultaneous recurrence system

Given a non-linear, non homogeneous, discrete time recurrence system: $a_i^t = f_i(a_1^{(t-1)},a_2^{(t-1)},\ldots,a_k^{(t-1)},C_1)$, for all $i\in [k]$ where $C_1,\ldots,C_k$ are constants and each ...
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16 views

Dermining stable and unstable manifolds - is my result ok?

Determine all stable and unstable manifolds of the equilibria of $$ \dot{x}=x(1+x)(1-x). $$ Are there homoclinic/ heteroclinic solutions? Hey, just would like to know if I am ...
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1answer
30 views

Non-monotonically decreasing flow whose limit is $\vec{0}$

I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but ...
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1answer
13 views

Conditions for invariance under flow.

I am beginning to study dynamical systems. We are given $U \subset \mathbb{R}^n$ open, a vector field $f: U \to \mathbb{R}^n$, and an associated evolution operator for fixed $t \in \mathbb{R}$ ...
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33 views

Lifting of a Diffeomorphism to an Orientable Double Cover

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
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38 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
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49 views

A kind of Sturm-Picone theorem?

My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve \begin{equation} (p(x)u'(x))'=-q(x)f(u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-r(x)g(v(x)) ...
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37 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ ...
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1answer
62 views

When does the same trajectory appear in two dynamic systems from the same point?

Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. ...
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1answer
22 views

Find all the equilibria of the system

Consider the system $$\begin{align} \dot{u}&= v\\ 147\dot{v}&=8150-588v-20000w\sin{u}\\ 330\dot{w}&=-135w+85\cos{u}+61 \end{align}$$ Find all equilibria where each $u,v,w\in[-\pi,\pi]$. ...
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1answer
38 views

Why do we need to find the multiple roots? (bifurcation curve)

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ The task is to give the bifurcation curve for the equilibria. First of all, equilibria are determined by $$ ...
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2answers
32 views

Discontinuous growth rate in a first-order dynamical system

Consider the dynamical system defined by $$\overset{\circ}{x} = x\cdot g(x),$$ where $$g(x) = \frac{r}{\alpha - x};\quad r,\alpha\in\mathbb{R}^+.$$ I am asked for a biological interpretation of ...
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1answer
31 views

Linear transformation of a dynamical system

How can I show that $x'=hx(1-x)$ can be transformed to $y'=r-y^2$ using a linear transformation (i.e. $y=mx+b$)? I tackled the problem by substituting $x'$ with $y'/m$ and after algebraic ...
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1answer
38 views

Drawing the phase portrait of a nonlinear system

Consider the nonlinear system: $$\begin{cases}\dot{x}_1=(x_1-x_2)(1-x_1^2-x_2^2),\\\dot{x}_2=(x_1+x_2)(1-x_1^2-x_2^2).\end{cases}$$ Draw its phase portrait. Solving $\dot{x}_1=\dot{x}_2=0$, we ...
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1answer
35 views

Speed of linear dynamical system trajectory

[warning: biologist asking math question] In a linear dynamical system, what feature of the matrix controls the speed of the trajectory in state space? Say I have a matrix M describing how the ...
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2answers
63 views

$2$-dim dynamical system IVP

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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48 views

Planar system: Analyse the existence of equilibria and determine their bifurcations

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ Analyse the existence of equilibria and determine their bidurcation. The equilibria can be determined by setting ...
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48 views

Solving a system of nonlinear second-order differential equations with initial/boundary conditions.

I have developed a set of $n$ equations, $n$ variables for my dynamic system. The derivatives are second and first order in terms of $\theta$ (angle) of different components of the system (basically a ...
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33 views

What does $D(f(\textbf{x}))$ mean

If we have a nonlinear dynamical system with $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ what do we need to do to find $D(f(\textbf{x}))$? Is it ...
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1answer
42 views

Product of Two systems with the same asymptotically stable fixed points

I am trying to figure out the nature of a new dynamical system that is equal to the product of two dynamical systems with the same asymptotically stable fixed point. For instance, if i have $x' = ...
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1answer
51 views

small amplitude oscillation of rotating system.

I've solved the euler-lagrange equation for a frictionless bead on circular vertical loop of radius a where the loop is rotating at $\Omega$ to get the equation of motion for the bead as ...
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1answer
72 views

How to see that this equilibrium is a center?

Consider the system $$ \dot{x}=y,\quad \dot{y}=-x+y^2. $$ Obviously, $(0,0)$ is an equilibrium. The linearisation matrix at zero has purely imaginary eigenvalues. So, at least we know that zero is no ...
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0answers
14 views

FIR estimator for IIR system

Suppose that we have a dynamical system of which the impulse responses are infinite (IIR). Now I found methods on papers (http://dx.doi.org/10.1109/9.839942) estimating states or outputs of such a ...
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48 views

Limits of two fixed points of $E_\mu(x) = \mu e^x$

Please let me know if this proof is OK. Problem statement: Given that $E_\mu(x) = \mu e^x$, where $0 < \mu < 1/e$, show that if $q_\mu < p_\mu$ are fixed points, where $q_\mu$ is attractive ...
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30 views

Show that $E_\mu$ has no periodic points that are not fixed points

Problem statement: Consider $E_\mu(x)=\mu e^x$, where $0<\mu<1/e$. Show that $E_\mu$ has no periodic points that are not fixed points. It is in my understanding that what we need to show is ...
3
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91 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
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43 views

Understanding Hopf's theorem

Hopf's Theorem Suppose, we have a family of systems which depend on a parameter $\varepsilon$ and suppose that at $\varepsilon=0$, $(x,y)=(0,0)$ is an equilibrium that undergoes Andronov-Hopf ...
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67 views

Show that there is a limit cycle in the dynamical system

I have the dynamical system \begin{align} \dot{x}_1 & = -x_2+x_1(1-x_1^2-x_2^2), \\ \dot{x}_2 & = x_1 + x_2(1-x_1^2-x_2^2) \end{align} With the initial conditions $x_1(0)=x_{10}$ and ...
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32 views

How to determine the nature of this equilibrium point?

If we find the equilibrium point and linearize the system $x'=-x+ay+x^2y\\ y'=b-ay-x^2y,$ we get that the point is $(b, \frac{b}{a+b^2})$ and the matrix associated with the linearilized system is ...
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21 views

Show that if $\langle x_0\rangle$ is a $n$-cycle and $n$ is a prime number, then $\langle x_0\rangle$ is a prime $n$-cycle.

Show that if $\langle x_0\rangle$ is a $n$-cycle and $n$ is a prime number, then $\langle x_0\rangle$ is a prime $n$-cycle. Recall: a natural number $n\neq 1$ is a prime number if there are no ...
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1answer
13 views

Explain why the eventual behavior of $\langle y_0\rangle$ is the same as the eventual behavior of $\langle x_0\rangle$

If $y_0\in\langle x_0\rangle$, from some orbit $\langle x_0\rangle$, then the eventual behavior of $\langle y_0\rangle$ is the same as the eventual behavior of $\langle x_0\rangle$. In fact, if ...
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1answer
18 views

Repelling or attracting spiral phase portrait in canonical basis

If matrix $\bf{A}$ of a system $\bf{x}'=\bf{A}x$ (*) has only complex eigenvalues and eigenvectors with non-zero real parts, and we make the substitution $\bf{y}'=\bf{B}x$ (**), where ...
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1answer
50 views

Understanding the dynamics (“jumps”) of a system

In my following question, I am referring to an article dealing with reaction-diffusion equations. I'll give screenshots of the parts of the text I am dealing with. The setup is the following: ...
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1answer
16 views

Properties of minimal $\mathbb{Z}$-actions on infinite compact spaces

How does one prove that (1) a minimal $\mathbb{Z}$-action on an infinite compact Hausdorff space is free? (2) for such an action, we can find a nonempty open subset $U$ of the space such that ...
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3answers
45 views

Differential System with Initial Conditions

So I have this system $$\frac{dx}{dt} = x y$$ $$\frac{dy}{dt} = 2 y$$ $$(x(0),y(0)) = (1,1)$$ Although I'm not too sure where to start. I know one method you have to take the derivative with ...