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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28 views

Existence and uniqueness of solutions with respect to λ of $\dot x(t) = λx(t) + cos(t)e^{λt}, t > 0$

Consider ODE $\dot x(t) = λx(t) + cos(t)e^{λt}, t > 0$, $x(0)=x_0,\lambda \in \mathbb{R}$. Investigate existence and uniqueness of solutions with respect to λ and find a closed representation of ...
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25 views

Determine for which values of $(a, b, c, d, e, f, g, h)$ is the following vector field Hamiltonian

The vector field is given as: $x' = −\mu y + ax^3 + bx^2y+cxy^2+dy^3$ $y' = \mu x + ex^3 + fx^2y + gxy^2 + hy^3$ So what steps do I need to take. Do I need to compute the Hamiltonian function ...
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4answers
126 views

Ambiguities on Solving a First Order PDE by Method of Characteristics

I am having difficulties to understand that how the method of characteristics works. I will clarify my question with an example. Consider a real valued function of two real variables $u(x,y)$ which ...
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27 views

KAM theorem for symplectic maps using Generalized Implicit Function Theorems

I've been studying KAM theory for a while and as many of you surely know, there exist many methods in proving "KAM theorems" for different settings. Most of the literature deal with the persistence of ...
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25 views

Eigenvalues of Hyperbolic Fixed Point

Definition: My book defines Hyperbolic Set of a diffeomorphism $f:M\to M$ as a compact, $f$-invariant set $\Gamma\subseteq M$ (where $M$ is endowed with a Riemannian metric) where there exists ...
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41 views

Understanding an argument about $\lim \sup x(t)$

I am reading a paper and I am stuck understanding some argument. I have the following system of equations (different that the paper one). all parameters are assumed to be positive. $$x' = x(1-x) - ...
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9 views

Conley Index for a Singularity in a Tetrahedron in a linear vector field

For a linearly interpolated vector field on a tetrahedron $T$, with vertices $a_0=(1,1,1)$, $a_1=(1,-1,-1)$, $a_2=(-1,-1,1)$, $a_3=(-1,1,-1)$, and corresponding vector values $v_0=(1/2,-1,-2)$, ...
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24 views

Dynamical Systems Time Dependent Matrix

Consider the time dependent matrix x'=Ax with $A = \begin{bmatrix}1&t\\0&-1\end{bmatrix}$ Find the general solution. We were given a hint to solve the second component first. I know how to ...
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17 views

Transfer function in dinamical systems

i need some help. I have this kind of system $\dot{x} = Ax(t) + Bu(t)$ $y(t) = Cx(t)$ $x(0) = 0$ $A = \begin{bmatrix} 2 && -1 \\ 2 && -5\end{bmatrix}$ $B =\begin{bmatrix} 2 ...
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20 views

Integrability and area-preservation property of maps

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
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40 views

Rotation map on $S^1$ preserves measure

I'm having a little trouble understanding following the example in my book as to why the rotation map $R_{\alpha}$ preserves Lebesgue measure. We have $R_{\alpha}([x])=[x+\alpha]$ and ...
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17 views

Global Solutions on Compact Space

My textbook has the following example: Let $f,g:\mathbb{R}^2\to\mathbb{R}$ be $C^1$ functions such that $f(x+k,y+l)=f(x,y)$ and $g(x+k,y+l)=g(x,y)$ for $x,y\in\mathbb{R}$ and $k,l\in\mathbb{Z}$. Then ...
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30 views

Question about dense trajectory on $k$-dimensional torus under rotation map

Today when I was doing ergodic theory problems I faced with following problem: Assume rotation map on $k$-dimensional torus under $\alpha=(\alpha_1,...,\alpha_n)$ then orbit of all $x$ in ...
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22 views

Definition of Poincare Section

Def: Let $X\subset Y$ with $\phi_t:Y\to Y$ a semiflow. $X$ is called a poincare section of $\phi_t$ if the first return time $\tau(x):=$inf$\{t>0:\phi_t(x)\in X\}\in\mathbb{R}^+$ for every $x\in ...
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31 views

What is the meaning of flow in this problem?

Consider a two dimensional vector field $X$ defined by $x'(t)=p(x,y), y'(t)=q(x,y)$, where $p$ and $q$ are differentiable functions. Specifically I am trying to show that ...
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43 views

Understand a proof concerning a theorem about topologically equivalent systems

Two systems are topologically equivalent if there exists a homeomorphism of the phase space which takes phse curves of one system to phase curves of another. Theorem Let $f\colon ...
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14 views

From impulse response to state space representation

I am provided with an impulse response as following: $$h(t) = t^2 e^{-2t} + \sin(t)$$ I have to present a state space representation of for this (in fact: state space representation in real Jordan ...
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44 views

Analyzing Dynamical System

Any suggestions on analyzing the long-term behavior of this system? I've computed the Jacobian, but I'm not sure how to gain much insight about the behavior from this. The eigenvalues appear to be ...
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1answer
29 views

Computing Lyapunov Exponents

Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map ...
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98 views

Find necessary and sufficient conditions so that $(0,0)$ is stable.

Suppose we have the system $$ \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) = \left(\begin{array}{c} f(x) + y \\ g(x) \end{array}\right). $$ Here $f,g: \mathbb{R} \to \mathbb{R}$ are ...
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17 views

The orbit map and the restriction of the orbit map both are open.

Let $\phi: G \times X \to X$ (Often denoted by $(X,G)$) is a topological group action. We know that for any $x\in X$, the orbit map $\phi_x:G \to X$ is a continuous mapping and the restriction ...
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26 views

normal form theory

My question consider the reason for calculating normal form. given vector field: $$\dot{x}=F(x),\ \ x\in \mathbb{R^n}$$ we try to bring the system to normal form by a sequence of transformation but ...
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42 views

Why does using orthogonal matrices as change of basis produce decoupled system of equation?

Suppose I have the following set of differential equations $\dot x_1 = -x_1 - 3x_2 \quad \dot x_2 = 2x_2$ This is the phase portrait of our system But let's take a change of coordinate using $P = ...
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133 views

Obtaining Positive Solutions by the Method of Characteristics for a First Order Linear PDE

Consider the function $u(x,y):{\mathbb{R}^2} \to {\mathbb{R}}$ and $u(x,y) \in {{C}^1}({\mathbb{R}^2})$. The function satisfies the following boundary value problem $$c_1 u_x + c_2 u_y = f(x,y)$$ ...
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22 views

Need help proving that a matrix with a specific structure is non-singular

I am working on an engineering problem that requires finding the solution to the following system of linear equations: $$ \underbrace{\left(AB+C\right)}_M\boldsymbol{k}=\boldsymbol{y}$$ where ...
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1answer
40 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: ...
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25 views

Explanation for the following bifurcation diagrams.

I am asked to plot the bifurcation diagrams of $x'(t)=ax+3$, $x'(t)=x^3-x+a$, and $x'(t)=x^2-ax$ respectively. The solutions (from an instructor) are as follows. Can anyone explain how to do this ...
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41 views

Understanding Proof of Poincare Recurrence Theorem

I'm trying to follow a proof in my book of the Poincare Recurrence Theorem, but I have three questions about this proof: Theorem Let $(X,\Sigma,\mu$ be a finite measure space, $f:X\to X$ be a ...
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9 views

Almost periodic system means uniformly lyapunov stable

Let $X$ be a compact metric space, $T$ be homeomorphic on $X$. Suppose that the system $(X,T)$ is almost periodic, i.e. for each point $x\in X$, for every $\epsilon>0$, the set of n such that ...
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50 views

Periodic solutions of an ODE on a circle.

I am trying to mark some homework solutions. The question is: suppose $\dot{x}=f(x)$, with $x \in \mathbb{S}^{1}$, and $f(x) \neq 0$. Assume existence and uniqueness theorem is satisfied. Is it ...
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96 views

Phase portraits

I have problem and need your help. I must draw phase portraits of dynamical system which looks like this: $$\dot{x}_{1}(t) = -x_{1}(t) + x_{2}(t)$$ $$\dot{x}_{2}(t) = -x_{2}(t)$$ I know that the ...
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1answer
102 views

Prove that $\dot{x} = Ax + f(t)$ has all bounded solutions

Given an $n\times n$ matrix $A$ has all eigenvalues with negative real parts. Prove that if the system $\dot{x} = Ax + f(t)$ has a bounded solution, then all the solutions are bounded. My attempt: ...
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1answer
70 views

Why are periodic solutions impossible for a differential equation on $\mathbb{R}$?

I am trying to understand a simple property of odes defined on a real line (or a subset of the real line). Consider an ordinary autonomous differential equation where the dependent variable $x$ is ...
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1answer
19 views

differential equation system order

my question is very easy: I started study dynamic systems and I have this system: $Mz_1''(t) +\beta z_1'(t) = F(t)$ $Mz_2''(t) - \beta z_2'(t)+kz_2(t)=\beta z_1'(t)$ where $ M, \beta, k $ are ...
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1answer
98 views

Unique solutions to a nonlinear system of ODEs

Given $A$ is a $n\times n$ matrix, and $f: R\rightarrow R^n$ is continuous and bounded. (a) If $A$ has no eigenvalues on the imaginary axis, prove that $\dot{x} = Ax + f(t)$ has a unique solution ...
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54 views

frictionless planar pendulum with pivot on a rotating disk

I'm trying to find the system of equations describing a frictionless planar pendulum (of known mass and length) that has it's pivot point fixed to the edge of a rotating disk (angular speed $\omega$). ...
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87 views

Linearization around a limit cycle versus stability of Poincare map

Thank you in advance for answering the following silly question. I am currently studying 2D limit cycles, say of the system $\dot{x} = f(x)$. Assume that the periodic solution $\gamma(t)$ is a ...
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1answer
48 views

The linearization of a system and the derivative of operator.

Firstly, in red line 1 of the picture below,whether it means make a variable substitution $\widetilde {g}_{ij}=f(x)g_{ij}$? Because in my opinion, the linearization of something is to make a ...
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0answers
74 views

Difference between $\mathbb{T}^n$ and $S^n$ and applications to dynamics (Euler angles and configuration manifolds)

I am struggling to see the difference between the $n$-sphere and the $n$-torus. We define $\mathbb{T}^n = S^1 \times S^1 \times \cdots \times S^1$, where the Cartesian product is taken $n$ times. I ...
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1answer
40 views

The meaning of dependent and independent variables in ODEs

This thought has occured to me a few days ago, and now I am puzzled about some fundamental properties /definitions in the theory of differential equations. Suppose I have an ode ...
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31 views

Drawing Trajectories in a State Space (is Energy Conserved?)

There's something I'm slightly confused about regarding drawing the trajectory of a particle in state space $(x,v)$, where $v:=x'$. Here, I'm only working with $x\in\mathbb{R}$. Suppose a particle ...
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1answer
21 views

Point with a dense trajectory

Let's consider a map $\varphi: [0, 1] \rightarrow [0, 1]$ so that $x \mapsto \{2x \}$. I would like to find a point $x$ so that its trajectory is everywhere dence in $[0,1]$. Firstly, the basic idea ...
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29 views

Determine Slow manifold

The coupled system in two variables $x(t)$ and $y(t)$ $$\frac{dx}{dt}=-xy,\ \ \ \frac{dy}{dt}=-y+x^2-2y^2$$ has the exact slow manifold $y=x^2$ on which the evolution is $$\frac{dx}{dt}=-x^3.$$ ...
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27 views

How to iterate a function 8 times about a given interval of x in a Discrete Dynamical System

This is Dynamical Systems, specifically a discrete system. We are using L and R as in Left and Right such as: L=[0,0.5] R=(0.5,1] and LL=[0,0.25] LR=(0.25,0.5] and so on like that. We keep ...
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1answer
42 views

Question about phase portrait and invariant subspaces

I am trying to understand why the eigenvectors are the $A$ invariant subspaces of a phase portrait. An A-invariant subspace is defined by the relation $AV \subseteq V$ where $V$ is a subspace and $A$ ...
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31 views

For which value of $r$ is the nonlinear dynamical system dissipative?

I could really use some help with the following: When I have input $u$ and output $y$ of the following nonlinear dynamical system, How can I determine for which values of $r$ this system will be ...
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1answer
31 views

Basin of attraction

Given $f : [a,b] \to [a,b]$ and $|f(x)-f(y)| < |x-y|$ for all $x,y \in [a,b]$, I have shown so far that there exists a unique fixed point in $[a,b]$ and that fixed point is attracting. How can I ...
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1answer
59 views

Find and classify the bifurcations that occur as $\mu$ varies for the system

Find and classify the bifurcations that occur as $\mu$ varies for the system \begin{align}\frac{dx}{dt}&= y-2x \\ \frac{dy}{dt}&=\mu +x^2 -y\end{align} What I have so far: The ...
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21 views

System realizability: What definition is correct?

Known that, sys realizability is given by: A system G is realizable if and only if the transfer matrix G(s) is a proper rational matrix. Out of the two statements below, which one is correct and ...
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65 views

Cluster Points of $S_{\Omega}$

Let $T: X \to X$ a continuous map on a compact set $X\subset \mathbb{R}$. A point $x \in X$ is non-wandering if for any open set $U \ni x$ there exists $n>0$ such that $T^{n}(U)\cap U \ne ...