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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17 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
28 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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11 views

“offset” after linearisation

Consider this system: $\text{dx1}=\text{x2}$ $\text{dx2}=-a (\text{x3}-1)+c (\text{c0}+\text{c1}-\text{x1})-d \text{x2}$ $\text{dx3}=\frac{a \text{x2} \text{x3}+\text{dm}}{a ...
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18 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
35 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
30 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
30 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
37 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
34 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
23 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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45 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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45 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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27 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
40 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
48 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
70 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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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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33 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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29 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 ...
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0answers
12 views

Nyquist criterion how check if system is asymptotic stable [closed]

i have G(s) like this $G(s) = \frac{s+1}{s^2 + s + 1}$ I must check if close system G(s) is asymptotic stable, so i must show what (-1,0) are or are not part of nyquist plot and i dont now how to do ...
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49 views
+50

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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42 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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64 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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19 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
12 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
17 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 ...
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91 views

External ballistics: Prove that the range is a concave function of the elevation

Consider a projectile moving in a plane. One of many different models for this problem is the following ordinary differential equation \begin{align} x''(t) &= -Ex'(t), \\ y''(t) &= -Ey'(t)- ...
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1answer
39 views

Numerical integration of a system of stiff ODEs starting at a singular point

Good afternoon, I have a system of $3$ highly non linear differential equations, which I have to integrate form a starting singular point $x^1=[1,1,1]$, and theoretically I have to arrive to an ...
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40 views

Closed formulas for the invariants beyond determinant and trace

Let $A$ be an $n\times n$ matrix and let $\lambda$ be a complex number. I was trying to obtain closed formulas for the invariants appearing in the development of the polynomial $\det(A-\lambda I)$ ...
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1answer
65 views

Phase portrait of a non-linear system

Given the following system of ODEs, it is in my understanding that to sketch trajectories in the $SI$-plane, one has to first find the critical point(s), which is only the point $(0, 0)$ in this case, ...
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22 views

Non-integrability and splitting of separatrices

It is well-known that the (first-order) Melnikov method is the standard technique to detect non-integrability of a perturbed system of ordinary differential equations or maps. Namely, the unperturbed ...
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2answers
39 views

Norm-bound for triangular hurwitz matrix exponential

Let $(-A)$ be a real Hurwitz lower-triangular matrix (this implies that all the eigenvalues of $A$ are real and negative). Since $(-A)$ is Hurwitz, we know that there exist $\alpha,\lambda>0$ such ...
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1answer
39 views

A question on dynamical system's focus values on its center manifold

I have recently come across this problem involving the center focus of dynamical systems of a parameter vector field related to center manifold: We define a vector field on $ R^3 $ given by: ...
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1answer
54 views

Bring system in normal form up to the second order

Bring the system $$ x'=y+xz,\quad y'=x^2+y^2+z^2,\quad z'=-2z+xy $$ to a normal form up to the second order (kill all non-resonant quadratic terms). The equilibrium is $(0,0,0)$ ...
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1answer
38 views

The dimension of splitting subbundles in the definition of a hyperbolic set.

Let $f:M\rightarrow M$ be a diffeomorphism and $\Lambda$ be a hyperbolic set, where $M$ is a compact Riemannian manifold without boundary. In the definition of the hyperbolic set $\Lambda$, the ...
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29 views

When is the critical manifold of a dynamical system an attractor?

In the study of autonomous dynamical systems of the form $\dot{x}=f(x)$ with $f : \mathbb{R}^n \to \mathbb{R}^n$, one has the notion of a hyperbolic equilibrium point $x^e : f(x^e)=0$, $Re(\lambda_i) ...
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1answer
14 views

Normal form of subcritical pitchfork bifurcation.

I'm working the a dynamical system $\dot{x} = r x - \frac{x}{1+x^2}$. I have already worked out that it is a subcritical pitchfork bifurcation. At least, that what my bifurcation diagram shows. ...
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1answer
38 views

Bring the linear part of a ODE system into a diagonal form

Consider the system $$ \dot{x}=y,\quad\dot{y}=z,\quad\dot{z}=-x-y-z+x^2+az^3. $$ Check that the zero equilibrium has a pair of pure imaginary eigenvalues. Make a linear coordinate ...
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1answer
25 views

What can we learn from the sign structure of the Jacobian matrix?

I am studying a $4 \times 4$ Jacobian matrix. I know the sign structure (that is, I know whether each element is positive, negative or zero), but I do not know magnitudes of each elements (i.e. their ...
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2answers
29 views

Determine perturbation around saddles for 2D system

Consider the system $$ \dot{x} = \mu + x^2 - xy\\ \dot{y} = y^2 - x^2 -1 $$ with $\mu \neq 0$ and small. I need to determine the Taylor/perturbation expansion of the two saddles $a^+$ and $a^-$ up to ...
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1answer
44 views

Determine a stable and a center manifold for the rest point

Recently, I dealt with determining stable/ unstable/ center manifold. Here is one task. Determine a stable and a center manifold at the rest point of the system $$ ...
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1answer
45 views

Dynamical Systems: Finding Directions Around Equilibria?

I'm struggling to find the best way to determine directions about equilibria in order to draw a phase plot of this non linear dynamical system: $x' = x − xy,$ $y' = \dfrac{4}{5} − x^2 + x − y.$ By ...
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26 views

Determine and classify all equilibrium points of this non-linear DE

Consider the DE $\begin{cases} \dot{x}=-2x(x-1)(2x-1)\\ \dot{y}=-2y \end{cases}$. Determine all equilibrium points and classify these. Choose between a saddle point, (in)stable nod, center or a ...
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25 views

Stable manifold theorem for fixed point when we have a non-diagonizable linearization matrix or a diagonal matrix with multiple eigenvalues

In literature I found, the stable manifold theorem is usually formulated in case we have n-dim ODE system with a fixed point that is a saddle and in case the linearization matrix is diagonal having k ...
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27 views

How to determine local stable and local unbstable manifold when we have a double negative eigenvalue?

Consider the non-linear system $$ \dot{x}=-x+y+x^2,\quad\dot{y}=-y-x^2. $$ The equilibrium point is $(0,0)$. The linearization matrix in $(0,0)$ is $$ A=\begin{pmatrix}-1 & 1\\0 & ...
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0answers
45 views

Correct determination of local stable and unstable manifold?

Consider the system $$ \dot{x}=-x+xy,\qquad\dot{y}=y+x^2. $$ The task is to determine the equations for the local stable and local unstable manifold $W_{\text{loc}}^{s}$ and ...