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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20 views

Hamilton principle/dynamics teaching in earlier stages.

In finding dynamic motion of particles we use laws of conservation of energy and momentum. It is found the dynamics formulation using action integral $$ \int (T-V)\, dt $$ builds ODEs for dynamic ...
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20 views

Proof that $M_ {\boldsymbol f}$ has a neighbourood diffeomorphic to the product $T^n\times D^n$

I'm reading Arnold's proof of Liouville's theorem and got stuck with the following problem in subsection §50, A. Here the manifold $M_{\boldsymbol f}$ is defined as $\boldsymbol F^{-1}(\boldsymbol f)$,...
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1answer
17 views

Estimation for entropy

Let $T\colon X\to X$ be continuous and $X$ compact and $K\subset X$ compact. By $s_n(2^{-k},K,T)$ denote the maximal cardinality of any $(n,2^{-k})$ separated subset of $K$. Suppose, we know for ...
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1answer
48 views

Constructing a Poincare Map

I need to construct a Poincare Map of the following dynamical system: $\dot x = x-(x+y)(x^2+y^2)$ and $\dot y = y + (x-y)(x^2+y^2)$ I changed the system to polar coordinates which gives me: $\dot ...
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1answer
12 views

Let $\{ y_k \}$ that satisfies $ y_k\le {2^k\over M}y_{k-1}^\beta$ , then $\lim_{k\to \infty}y_k=0$.

Let be a sequence $\{ y_k \}^\infty _{k=0} \subset (0,\infty) $ that satisfies $$ y_k\le {2^k\over M}y_{k-1}^\beta , $$ where $k=1,2,...$, and $\beta\gt 1$ , $M\gt0$. Prove that if $M\gt2^...
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1answer
47 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
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35 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
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1answer
23 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
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24 views

Dynamical system in a square

I am considering a problem that is asking me to explore a deceptively simple dynamical system and discover some of surprising properties. I want to consider the motion of four particles A,B,C and D in ...
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1answer
44 views

Stability of nonlinear ODE linearized on a periodic solution.

I'm reading a paper about the effects of pulsed immunotherapy in cancer treatment. They have system$$ \begin{array}{ccc} \dot{E} & = & cT-\mu_{2}E+\frac{p_{1}EI_{L}}{g_{1}+I_{L}}+s_{1}\\ \dot{...
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1answer
18 views

Show that the following functions are flows on the spaces indicated

Show that the following functions are flows on the spaces indicated. Find the vector field for each flow $$\phi_t(x)=\frac{x+\tanh t}{1+ x \tanh t}, x \in [-1,1]$$ Solution so far So $\phi_0(x)= \...
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1answer
61 views

How to convert dynamical system to polar coordinates? [closed]

I have a dynamical system on the plane given by $$\dot{x}=-y+x\left(1-\sqrt{x^2+y^2}\right)\\ \\ \dot{y}=y+x\left(1-\sqrt{x^2+y^2}\right)$$ I want to convert this into polar coordinates as it will be ...
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0answers
25 views

Long-time behavior of a diffusion-like equation

I would like to known the long-time behavior of $\mathbf{u}_{t}(n)$ obeying the equation $ \mathbf{u}_{t+1}(n) = T_l\mathbf{u}_{t}(n-1) + T_0\mathbf{u}_{t}(n) + T_r\mathbf{u}_{t}(n+1) $ where $T_l = ...
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0answers
24 views

Determine the stability of the equilibria

I'm given a discrete dynamical system in which $f$ is given by $$f(x,y)=(1+y-ax^2,bx). a,b \in \mathbb{R}, b\neq 1$$ I'm asked to find the equilibria which I have done below: $$f(x,y)=(x,y) $$ $$\...
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1answer
48 views

Bifurcation Problem

I am trying to classify the type of bifurcation for the dynamical system given by: $\dot x = x^2+y^2-2my$ $\dot y= mx-y$ with m as a varying parameter The fixed points are at (0,0) and ($2m^2 \...
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2answers
182 views

How to know whether an Ordinary Differential Equation is Chaotic?

Assuming we have an ordinary differential equation (ODE) such as Lorenz system: $$ \dot x=\sigma(y-x)\\ \dot y=\gamma x-y-xz\\ \dot z=xy-bz $$ where $$ \sigma = 10\\ \gamma = 28\\ b = \frac{8}{3}\\...
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58 views

A question about the “state-transition-matrix” of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
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1answer
56 views

2D Bifurcation Classification

Given the system with m as a varying parameter: $\dot x = mx^2-y$ and $\dot y = m+y - x$ Determine any bifurcations that occur Attempt: x nullcline $y=mx^2$ y nullcline $y=x-m$ Fixed ...
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56 views

Find imaginary part of complex expression

Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im m\{\frac{1}{...
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1answer
42 views

Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where $-\frac{1}{4}&...
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1answer
31 views

Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ [duplicate]

Basically let $S(x)=\sin(x)$ such that $S^2(x)=\sin(\sin(x))$ and $S^3(x)=\sin(\sin(\sin(x)))$ and so on until $S^n(x)=\sin(\sin(\ldots\sin(x)\ldots))$ Prove that $S^n(x)\rightarrow 0$ as $n\...
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3answers
43 views

Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted ...
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1answer
38 views

Invariance of sets for systems of ODEs

Given the system of ODEs $$x' = x(1-y)$$ $$y'=y(x-1),$$ let the set $Q=\{(x,y):x\ge 0, y\ge 0\}$. Explain why $Q$ is invariant for this system of ODEs. My explanation: If $x > 1$ and $y<1$ then ...
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1answer
40 views

Why c>1/4 is not in Mandelbrot set

As title: $f_c(x)=x^2+c$ I got to the step: $f_c(x)>x$ (for all x) But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$ Thanks,
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1answer
33 views

Solve a system of second order differential equations

I have a system of second order differential equations which is $$m_1x_1''=-k_1x_1-k_2(x_1-x_2)\\m_2x_2''=-k_2(x_2-x_1)$$ where $(x_1(0),x_1'(0),x_1(0),x_2'(0))=(1,0,2,0)$ and $(m_1,m_2,k_1,k_2)=(1,...
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23 views

Stability of LTI systems under saturation

Consider the saturation function $$ \sigma(u)=\max(\min(u,1),-1) $$ for $u\in\mathbb{R}$. With slight abuse of notation, if $u\in\mathbb{R}^n$ let $\sigma(u)$ also denote the same function applied ...
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1answer
52 views

behavior of the Linear system of an ODE model

I am working on a predator-prey model and the linearization about and equilibrium point $(0,e_2)$ has Jacobian matrix as follows $$\mathcal{J} = \begin{pmatrix} 0 & 0\\ b& -b \end{pmatrix},$$...
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1answer
16 views

How to show that $x_{n+1} - x_* = M(x_n) (x_n - x_*)$ dynamics converges?

I am trying to find conditions of convergence (or non-convergence) of a system that behaves in the following manner (quasi-linear since the matrix is not stationary): $$ x_{n+1} - x_* = M(x_n) (x_n - ...
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1answer
29 views

Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
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1answer
17 views

Entropy of factor map

Let $A$ and $B$ be two compact metric spaces with $B\subset A$- Moreover, let $T\colon A\to A$ continuous and let $S\colon A\to B$ a continuous surjection with $S\circ T=T\circ S$. Moreover assume ...
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1answer
51 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
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26 views

Center manifold reduction

This a verbatim copy of an example on center manifold reduction on nonlinear dynamical system I found on some lecture note: Consider the system \begin{align*} \dot{x}&=x^2y-x^5\\ \dot{y}&=-y+...
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32 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
2
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2answers
38 views

What is the period of the harmonic oscillator?

I am trying to find the elapsed time $T$ (or transit time) over one cycle of the harmonic oscillator $$\ddot{x} + \omega^2x=0.$$ I worked this out to be \begin{align} T &= \int_{R}^{-R} \...
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46 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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2answers
49 views

Stability criteria for linear systems with auxiliary variables

Classical texts for control theory show the linear system $\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative. Does the same criteria apply for a system of the following form:...
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14 views

What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits

I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ...
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1answer
73 views

Trapping Region for Dynamical System

Show that the dynamical system contains a closed orbit $\dot x = xf(x,y)+yg(x,y)$ and $\dot y = yf(x,y)-xg(x,y)$ Given Information: f(x,y) and g(x,y) are single valued functions and differentiable ...
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1answer
51 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
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1answer
56 views

proving asymptotic stability dynamical system

I want to show the origin of the dynamical system \begin{align} \dot{x}_1 &= -2x_1+x_2+x_1^3x_2^2\\ \dot{x}_2 &= -x_1-2x_2+x_1^2x_2^3 \end{align} is asymptotically stable over an invariant ...
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1answer
27 views

Differentiation of unit force vector

I was reading a paper and don't know how the following was derived. Given that $f = \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} + g \\ \end{bmatrix}$ and $...
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1answer
28 views

Two ODE planar systems that are orthogonal to each other

Given two planar systems X'=F(X) and X'=G(X) (so F and G are both $C^1$). Assume the dot product of F(X) and G(X) is always zero on $R^2$. Now if F has a closed orbit, prove that G has a zero. My ...
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2answers
32 views

Computing eigenvectors for an eigenvalue of a dynamical system

$\dot{x}=x\left ( 3-x-2y \right )$ $\dot{y}=y\left ( 2-x-y \right )$ In matrix form: $\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix}$ $=\begin{bmatrix} \left ( 3-x-2y \right ) &0 \\ 0& \...
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3answers
22 views

solving coupled equation

$\dot{x}=x\left ( 3-x-2y \right )$ $\dot{y}=y\left ( 2-x-y \right )$ The above is a coupled equation. The fixed point condition requires all x,y for which $x\ast =0$ and$ y\ast=0$ solving, I arrive ...
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2answers
31 views

Time shifted function in differential equation from a control systems problem

Suppose we had the following differential equation with zero initial conditions: $$\ddot{x}\left(t\right)+2\dot{x}\left(t\right)+x\left(t\right)=0.5u\left(t\right)$$ where $u\left(t\right)$ is given ...
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0answers
36 views

'Fast' and 'slow' Eigendirection?

Can someone give an intuition and a definition of what a "fast" and "slow" eigendirection means? A reasonable google search reveals nothing that would help. Thanks in advance.
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1answer
49 views

Proving solutions to the anisotropic kepler system that meet certain constraints lie on the position axes of configuration space

The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{3/2}} \end{equation*} \begin{equation*} y''=\frac{-y}{(\mu x^2 + y^2)^{3/2}} \end{equation*} With $\mu>1$ a constant ...
1
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1answer
38 views

Fixed points of dynamical systems

I haven't been able to find an answer to this question anywhere, so I thought I should post it here. In doing a fixed-points/stability analysis, one is required to find the fixed points of a dynamical ...
0
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1answer
10 views

Discrete-time variation of constant formula

The difference equation is given \begin{align*} x(k + 1) = Ax(k) + Bu(k) \end{align*} with an initial condition $x(0)= x_0$. Inductively, I derived the solution $$x(k) = A^kx_0+\sum_{j=0}^{k-1} A^{k-...
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2answers
33 views

Finding a circle in polar coordinates

I have converted the system of ODEs, $$x'=x-y-x(x^2+5y^2)$$ $$y'=x+y-y(x^2+y^2),$$ to polar coordinates and got this: $$ r' = r-r^3(1+4\sin^2(\theta)\cos^2(\theta))$$ $$\theta'=1+4\cos(\theta)\sin^3(...