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

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
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15 views

A question about fundamental matrix of periodic system $x'=A(t)x$

$X(t)$ is a fundamental matrix of linear differential equation $x'=A(t)x$ where $A(t)$ is a periodic matrix with period $T$ . Show that there exist a non-singular matrix like $C$ such that for ...
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Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
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20 views

Dynamical Systems Periodic Orbits existing

Consider the nonlinear dynamical system $(1)$ : $x' = y(1 + x−y^2)$, $y' = x(1 + y−x^2)$, where $(x,y)\in\mathbb{R}^2$. (i) Determine the equilibrium points of $(1)$ (ii) Classify the ...
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13 views

Prove that the acceleration in a circular movement is $a=v^2/R$

I don't understand the part when we find out that two triangles are similar because they have 2 angles congruent and 2 sides perpendicular/normal. Firstly I can't find out why the two angles are ...
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18 views

What is symmetric differential equation? [on hold]

What is the meaning of Z2-symmetric differential equation? and genericaly What's the meaning of symmetry about differential equation?
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66 views

Question about solution of $y''+(w^2+b(t))y=0$ .

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dx <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{x\to\infty} ...
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19 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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20 views

Existence of minimal sub-systems

A topological dynamical system is a topological space $X$ together with a continuous function $f : \ X \to X$. In the following, I will assume that $X$ is compact and Hausdorff (in other words, I work ...
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2answers
46 views

Do solutions of $\dot{x} = \frac{x}{t^2} + t$ exist satisfying $x(0) =0$

Suppose we have the 1-dimensional ODE \begin{equation} \dot{x} = \frac{x}{t^2} + t \end{equation} Do there exist solution curves with initial condition $x(0)=0$? If you proceed in a standard way ...
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2answers
49 views

Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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22 views

Using matlab to plot periodic orbits of a map

I am trying to plot periodic orbits of a map upto period 10 and I need help with the matlab code (not an expert in matlab). I talked about it to my advisor and this is what I have until now: In this ...
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2answers
32 views

Eigenvector multiplication

I don't understand how multiplying eigenvetors by an expression like $e^{-2t}$ works, and results in this graph. Can someone explain this to me?
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0answers
38 views

Nullclines for differential equations

Consider the system of differential equations $$\dot {x}=y-x^2$$ $$\dot {y}=x-y$$ a. Determine the fixed points (1,1) (0,0) b. Determine the nullclines and the signs of $\dot {x}$ and $\dot {y}$ ...
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0answers
13 views

Alpha and Omega Limit Sets in Polar Coordinates [duplicate]

I guess here I am not sure how to get started, I know the definitions: The $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the ...
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0answers
11 views

Alpha and Omega Limit Sets for Linear Systems [duplicate]

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I know how to calculate the eigenvalues and the ...
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1answer
22 views

Expressing σ as a binary shift map.

Having shown that the only fixed point of $\sigma$ is $x=0$, I've now got the show that the fundamental period-$2$ points of σ are of the form $x=0 . ababab \ldots$ where $a,b \in \{0,1\}$ and $a\neq ...
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16 views

Intertialess Gear Train transducer differential equation setup

The system below shows two rotational inertias, I1and I2, connected by a gear train. An ideal gear train is inertialess. Notice that there are no rotational springs and dampers, just the two inertias ...
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1answer
16 views

Differential equations, stability of fixed points

Consider the differential equations: $$\dot{x}=x^2-9$$ $$\dot{x}=x(x-1)(2-x)=-x^3+3x^2-2x$$ a. Find the stability type of each fixed point. (I am not sure about the stability of the points. Do I ...
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0answers
15 views

How to Compute the Poincare Map for a non-autonomous dynamical system?

So I am studying Poincare maps (this isn't homework or anything), and I'm a bit confused about how to find the ideal poincare section, and then how to compute the general poincare map. Can someone ...
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2answers
29 views

Polar coordinates, Differentiation

Can someone clarify this step for me please, "The polar coordinate r satisfies $r^2=x^2+y^2$, so by differentiating with respect to t we get $r\cdot\dot r=x\cdot\dot x+y\cdot\dot y$" I am totally ...
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1answer
28 views

Theorem on existence and uniqueness

Consider the differential equations and: $x'=x^2$ with initial condition $x(0)=x_0$≠0 $x'=x^2-1$ with initial condition $x(0)=x_0$ $x'=x^2+1$ with initial condition $x(0)=x_0$ a. Verify that the ...
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0answers
17 views

Why hyperbolic fixed point has modulos 1? [closed]

I'm trying to prove that {p} is a hyperbolic fixed point if and only if it has not eigen value of modulos 1.
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1answer
19 views

Partial Fractions to solve Logistic Equation

I am not really understanding how my book is getting $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x}+\frac{x'}{K-x}$$ so $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ ...
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28 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
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1answer
30 views

Wronskian of a fundamental set of solutions

(instead of the dot above, i used ' and ", am I correct in thinking that these are equivalent?) Consider the system of equations, $$x'_1=x_2$$ $$x'_2=-q(t)x_1-p(t)x_2$$ where $q(t)$ and $p(t)$ are ...
2
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1answer
35 views

Trace of a matrix

"the preceding 2 scalar solutions correspond to the vector solutions $x^1(t)=(t,1)^T$ and $x^2(t)=(t^2,2t)^T$ which have the Wronskian $$W(t)=\det\left[\begin{array}{lr} \mbox t & t^2\\ \mbox 1 ...
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1answer
25 views

Fixed Point Summary

I know someone has given me resources for this before but I can't seem to find them... Would someone please summarize stable vs unstable, attracting vs repelling, and node, saddle,etc fixed points? I ...
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0answers
34 views

Conjugacy map can be chosen Lipschitz

An exercise in Katok and Hasselblat's Introduction to the Modern Theory of Dynamical Systems (Section 2.1, exercise 2) goes as follows: Let $f$, $g$ be $C^1$ maps defined in a neighborhood of the ...
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1answer
25 views

Linear approximation to a system in the neighbourhood of the origin?

What would a linear approximation to the following system near the origin be? $${dx \over dt}=-y-x(x^2+y^2), {dy \over dt}=x-y(x²+y²)$$ I have no idea how to find this... I'm looking at this as an ...
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0answers
30 views

What exactly do the eigenvalues of a Jacobian matrix mean intuitively in a dynamical system? [closed]

I read that if they are negative the system is stable but I do not understand why.
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33 views

Hamiltonian Dynamics and the canonical symplectic form

1- What kind of advantages does one have by having a canonical symplectic form on $T^*M$ apart from the form being exact? Would it for instance provide any advantage to studying Hamiltonian dynamics ...
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1answer
17 views

A sequence of numbers' question. (From Krengel's book on Ergodic Theorems).

On page 136 of Ergodic Theorem's by Ulrich Krengel, in the proof he sets: $c_n= \sup_j (n)^{-1} \sum_{i=0}^{n-1}x_{i+j}$ then he argues that for any $k,m$ positive integers one has: $$c_{km}\leq c_m$$ ...
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1answer
42 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
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1answer
48 views

Fixed points and stability of them

Find the fixed points and classify them for the system of equations: $$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$ $$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$ is the only fixed point (0,0,0)?? jacobian: ...
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1answer
27 views

System of differential equations, phase portraits and stability of fixed points

Consider the system of differential equations: $$x'=-x-y+4$$ $$y'=3-xy$$ a. Find the fixed points. $x'=-x-y+4$ $x+y=4$ $x+3/x=4$ x=3,x=1 $y'=3-xy$ $y=3/x$ fixed points: (1,3), (3,1) b. ...
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23 views

Are they stable or unstable limit cycles?

I am using cl_matcont to perform a bifurcation analysis of a dynamical system of ten equations (equations are identical in two blocks, thus 8 of them and 2 of them are the same) During the ...
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0answers
36 views

Proving an attractor (i.e set with self similarity) is connected

Let $K$ be an attractor for iterating function system of two similarity maps i.e $$K=f_1(K)\cup f_2(K)$$ A similarity map is defined to be $f_i:\mathbb{R}^d\to \mathbb{R}^d$ s.t $$\forall x,y\in ...
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0answers
66 views

System of differential equations, phase portraits

Consider the system of differential equations: $$x'=y-x^2$$ $$y'=x-y$$ a. Determine the fixed points. So setting both equation equal to 0, I get: $y=x^2$ and $x=y$ So the only fixed points would ...
0
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0answers
29 views

Limit sets for autonomous ODEs

Consider the autonomous ODE $x'=f(x)$. Let $\gamma$ be an trajectory of the system. If $\gamma$ is periodic, it is trivially true that $\gamma\subset\omega(\gamma)$. When $\gamma$ is not periodic, ...
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1answer
37 views

System of differential equations, phase portrait

Consider the system of differential equations: $$x'=-2x+y-x^3$$ $$y'=-y+x^2$$ a. Determine the fixed points. (Am I correct in thinking that to determine the fixed points, I must set x' and y'=0? I'm ...
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1answer
54 views

How to prove symmetry of the following Maxwell-Bloch equations?

I have the following Maxwell-Bloch equations: $\dot{E}=-\alpha_{1} E+ k_{1}P$ $\dot{P}=-\alpha_{2}P+ k_{2}ED$ $\dot{D}=-\alpha_{3}(D-\lambda) -k_{3}EP$ In this system ...
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1answer
15 views

Sampling a matrix of an AR model

Let us consider a dynamic system $x_t = A x_{t-1}+v_t$ where $v_t$ is multivariate normal noise with zero mean, i.e. $v_t\sim\mathcal{N}(0,\Sigma)$ and $A$ is a matrix. As far as I know, for some $A$, ...
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0answers
47 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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1answer
52 views

Alpha and Omega limit sets (dynamical systems)

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
1
vote
0answers
51 views

Alpha and omega limit sets [duplicate]

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
0
votes
2answers
25 views

System of equations, limit points

This is a worked out example in my book, but I am having a little trouble understanding it: Consider the system of equations: $$x'=y+x(1-x^2-y^2)$$ $$y'=-x+y(1-x^2-y^2)$$ The orbits and limit sets ...
2
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1answer
37 views

Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't ...
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0answers
18 views

Nullclines and regions of positive value

To get an idea how the solutions of an autonomous ODE', i.e. $$\dot{x}=f(x,y) \\ \dot{y}=g(x,y)$$behave, one approach is to sketch the nullclines and then picking for each nullcline points $(x_0,y_0)$ ...
0
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2answers
33 views

Preserving orbits by multiplication with a non-vanishing function

I'm reading through some notes from a past course of mine, where a system of ODE's of the form$$ \begin{array}{c} x'=h(x,y)f(x,y)\\ y'=h(x,y)g(x,y) \end{array} $$ appears, such that ...