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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About Duffing equation

Is there a relation between Duffing equation and Van der Pol equation? My second question is what is the application(s) of stochastic Duffing equn. in practice ?
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77 views

Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
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28 views

Fixed points of $u'= u(1-u+ \alpha v)$, $v' = \rho v(1-v + \beta u)$

Hopefully a simple question however it has me stumped... I have the following system: $\frac{dN_1}{dt} = r_1N_1(1- \frac{N_1}{k_1} + b_{12} \frac{N_2}{k_1})$ $\frac{dN_2}{dt} = r_2N_2(1- ...
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21 views

Proof involving Poisson bracket

Not being able to understand how each term has been simplified to get from the third step to the fourth step. So how did 1/2m become 1/m and {qj,plpl}pk become {qj,pl}plpk and how did k/4 become ...
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22 views

How to do this Poisson bracket proof

For the proof of the above equation, I understand the first step which has been obtained from the definition but in the second step I don't understand why they are summing over $j$ first ...
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19 views

Poisson bracket proof

For this question I understand the first line of the solution which they have obtained from the definition but how have they simplified each term to get to the second line from the first line? The ...
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21 views

Parameter estimation - Holt's Two parameter Linear Exponential Smoothing

The reference for the below equations can be found in the Link . Note that $k$ is the timestamp and $i$ is the $i^{th}$ entry of a vector or $(i,i)^{th}$ entry of a matrix, $F$ in this case Equation 1 ...
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24 views

Is a gradient system considered an ODE or PDE?

When you have a system of the type $$\dfrac{dx(t)}{dt} = \nabla V(x)$$ Is this considered an ODE or a PDE? Because you have a single derivative with respect to $t$ on the lefthand side, whereas on ...
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49 views

Nuclear operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
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51 views

Finding periodic points of diffeomorphism of the circle

I want to find all the periodic points of the following diffeomorphism of the circle: $f(x) = x + \frac{1}{4} + \frac{1}{10} \sin(8 \pi x) \mod 1$ Where a periodic point is $p$ such that $f^n(p) = ...
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32 views

Poisson bracket proofs

I understand the first sentence you wrote for the need of a different summation index. However, i'm still not able to understand the individual steps. Like how in the first line we have four ...
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27 views

Sketching a dynamical system

Sketch the dynamical system \begin{align} \dot x_1 = x_2 \\ \dot x_2 = 1 \end{align} Firstly we may integrate this to find $$x_2(t) = t + A$$ $$x_1(t) = \frac12t^2+At + B$$ How do I then change ...
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27 views

How to model this recurrence?

I'm having some problems on how to model this situation correctly, using difference equations. Say there's a medicine that has a half-life of 12 hours (every 12 hours, the amount of it on your blood ...
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16 views

How does Melnikov function for a Hamiltonian change if one considers an augmented symplectic manifold?

Suppose we have a nonautonomous nearly integrable Hamiltonian system, periodic in $t$ with period $2\pi / \omega$ $$H_{\epsilon}(x,y,t)=H_{0}(x,y) + \epsilon H_{1}(x,y,t)$$ with $(x,y,t) \in ...
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27 views

How to discretize state space with uniform grid

Let us consider a general continuous time stochastic differential equation represented by *dx* = A(x)dt + B(x)udt + $\sigma$ dw where A(x) represent the ...
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34 views

On “bounded” in intuition for a theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
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272 views

Prov that function is eventually periodic to origin.

Let $f:\mathbb{Z}^4 \rightarrow \mathbb{Z}^4$ by $f(w,x,y,z) = (\mid w-x \mid,\mid x-y \mid,\mid y-z \mid,\mid z-w \mid)$ Prove that for any $(w,x,y,z) \in \mathbb{Z}^4$ there is $n>0$ such that ...
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30 views

Two trajectories $x_1(t)$ and $x_2(t)$ of the six dimensional system

Two trajectories $x_1(t)$ and $x_2(t)$ of the six dimensional system $\dot{x}=A_{6x6}x$ are given by $$x_1(t)=t^2e^{-2t}v_1+te^{-2t}v_2+e^{-2t}v_3+v_4 ,\> \> x_2(t)=v_5+e^{-t}v_6 \> \> ...
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1answer
45 views

Index of circle containing fixed point

Given $\dot x = x^2-y^4 $ and $ \dot y = y^2 -x^4 $ Find the index of the circle $x^2 + y^2 = a^2$ with $a < 1$ Attempt: I employed linear analysis by finding all the fixed points. There are ...
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1answer
45 views

On corollary and theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
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26 views

Invariant Subspaces Dynamical Interpretation

Consider the linear system $\dot{x}=A_{nx n}x$ ; $x(0)=x_0$. Recall that a subspace $\mathcal{U}$ s (dynamically) invariant to the flow if for all initial conditions $x_0 \in \mathcal{U}$ the ...
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Nonhomogenous Linear System

You are given the time dependent linear system in $\mathbb{R^n}$ $\bf{\dot{x}}$=$A$$\bf{x}$ where $A$ is a fixed $n\times n$ matrix and $b : \mathbb{R} \rightarrow \mathbb{R^n}$ is continuous. $i)$ ...
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28 views

stability of the equilibrium at the $\bf{x}=0$

Consider the system $\bf{\dot{x}}$=$A$$\bf{x}$ in $\mathbb{R^6}$. You are given that the null-space of $A$ has dimension 2 and the stable subspace of $A$ has dimension 2 and that is an ...
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43 views

Root mean square function (RMS) in the ODEs of a dynamical system

I am trying to obtain the differential equations of a dynamic system in which one of the blocks calculates the RMS of a signal. When arranging the ODEs I see that the differential version of the RMS ...
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31 views

Stability of the equilibrium at the origin

Consider the system $\bf{\dot{x}}$=$A_{3x3}$$\bf{x}$ You are given that $A$ is nonsingular and the unstable subspace of $A$ is trivial. Is this information sufficient to infer the stability of ...
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43 views

Dynamical System in Polar Coordinates

I have a dynamical system defined by : $ \dot x = {(x+iy)^n + (x-iy)^n \over2}$ and $\dot y = {(x+iy)^n - (x-iy)^n \over2i}$ Converting the system to polar coordinates gives the system: $\dot r = ...
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48 views

System of ODEs - from Cartesian to polar

Given the system of ODEs, $$\dot{x}=x^2+3y^2-1$$ $$\dot{y}=-2xy$$ How does one transform it into polar coordinates $(\rho, \theta)$? Here's my line of reasoning: let $x=\rho \cos(\theta)$, $y=\rho ...
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modal truncation of state space system while preserving certain eigenvalues

Given a state space system $(A,B,C)$: $$\dot{x}=Ax+Bu\\y=Cx$$ Is there any method to obtain a reduced system $(A_r,B_r,C_r)$, where $$\dot{x}=A_rx+B_ru\\y=C_rx,$$ such that the eigenvalues of $A_r$ ...
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37 views

How to determine whether dynamical systems are dissipative, conservative or expanding?

I have the dynamical systems: $dx/dt=y$, $dy/dt=-w^2x$ $dx/dt=y$, $dy/dt=-by-w^2x$ $dx/dt=a(y-x)$, $dy/dt=x(b-z)-y$, $dz/dt=xy-cz$ with $a,b,c,w\in\mathbb{R}$ How do I ...
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1answer
96 views

Show that $\bf{x^*}$ is an equilibrium point.

Let $\dot{\bf{x}}=f(\bf{x})$ be a dynamical system with $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ being continuous and locally Lipschitz. Suppose that a particular solution trajectory satisfies ...
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1answer
67 views

How many completely invariant domains can there be for a rational function?

I am considering rational functions $R:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ of degree $d\geq 2$. A completely invariant set $U$ is a set for which it and its complement are ...
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31 views

Systems of second order differential equation

i'm following a course in Hamiltonian systems and regarding the part of linear systems I found this exercise from a book and need to solve it. My ideas are just after the test of the exercise. ...
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40 views

Equilibria for a dynamical system

This question is from a question in which I'm trying to find the equilibria of a discrete dynamical system. I need to solve $$f(x,y)=\begin{pmatrix} 1+y-ax^2 \\ bx \end{pmatrix}=\begin{pmatrix} x \\ y ...
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1answer
59 views

Arnold's proof of Liouville's Theorem on integrable systems

My question happens to be almost identical to the one left unanswered/closed here, which gives a bit of background information - it may not be necessary. I hope the reason it was closed on ...
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1answer
56 views

Hamiltonian system; breakdown for different level set values

I have a system of differential equations defined by the hamiltonian of the scalar function $H=y^2+e^{-xy}-c$, for some $c>0$. I am asked to describe what happens for $c=1$. I can tell there is a ...
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1answer
44 views

Poincaré Index of a periodic orbit

I am trying to formalize the following proof on Perko's Differential Equations and Dynamical Systems, which says that a periodic orbit has index +1. My only problem is trying to prove that the map ...
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44 views

Stability of a time-varying system that converges to a time-invariant system

Consider a system $\dot x = A(t)x$, where $A(t)$ is a matrix satisfying $A(t)\rightarrow A$ exponentially fast. For each fixed $t$, $A(t)$ is Hurwitz (all the eigenvalues in the open left half plane). ...
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1answer
17 views

How to transform a coupled differential equation into a system with diagonal linear part

Consider the system given by $$iu_t +u_{xx}+2|u|^2u = -v+iu$$ $$iv_t +u_{xx}+2|v|^2v = -u-iu$$ I am trying to transform the system into a system with diagonal linear part. I can solve a problem like ...
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2answers
37 views

Oscillatory solutions if derivatives are are independent of each other?

Definition: An oscillatory solution is one where $(x(t), y(t))$ is a trajectory and $x(t)$ and $y(t)$ are not constant. Further, for any $n \in \mathbb{N}$ we have $x(t+nt) = x(t)$ and $y(t+nt) = ...
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1answer
26 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
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1answer
36 views

Understanding explicit computation of a stable manifold in $\mathbb{R}^2$

After introducing and proving the stable manifold theorem in $\mathbb{R}^2$, my instructor gave an example of a system for which it was possible to explicitly compute the stable manifold. I am trying ...
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278 views

Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). ...
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1answer
69 views

The solutions of $y^{\prime \prime}+y=g$ are bounded

Suppose that $g$ is a continuous differentiable, increasing and bounded real function. How can one prove that the solutions of the differential equation $(E)$ $$y^{\prime \prime}+y=g$$ are bounded? ...
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1answer
17 views

Topological conjugation map for 1d flows

I am working on an example of 1d flow conjugation from class. For a vector field on $\mathbb{R}$, we have a flow $\phi_{t}(x)=xe^t$, with vector field $F(x)=x$ and another flow $\psi_{t}(x)=xe^{2t}$ ...
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73 views

straight-line solutions of dynamical system

Given the dynamical system $\dot x = f(x,y)$ and $\dot y = g(x,y)$ where f and g are homogeneous of degree n (i.e. $G(\alpha x,\alpha y)=\alpha^n G(x,y))$ show that straight line solutions of the ...
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1answer
49 views

Definition of the dynamical ball Bowen Walters

I'm learning continuous flows and I found this definition: Let $(X,d)$ be a compact metric space and $\phi:\mathbb{R}\times X\rightarrow X$ be a flow continuous. Denote by $\mathcal{H}$ the set of ...
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1answer
63 views

How to decide whether it is a system of Differential Algebraic Equations or a System of Ordinary Differential Equations?

I am struggling to name some of my dynamic models right. To be specific, I am not sure whether I should call it a system of Differential Algebraic Equations (DAEs) or a System of Ordinary Differential ...
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26 views

Stability of Extrema of Conserved Quantities

The following problem occurred on a homework assignment last week, and I wanted to know how to do the analysis to prove it in the nontrivial case. Consider a two dimensional system with ...
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2answers
48 views

How to write a system of equations as a dynamical system?

I am having a lot of trouble understanding how to move from a system of ODE's to a dynamical systems point of view (that will allow me to make a phase-plane analysis). Assume I want to write the ...
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2answers
135 views

Uniqueness of a periodic solution for nonlinear pendulum

I am working with the system of ODE's or second order differential equation representing the nonlinear pendulum with constant torque and damping. \begin{equation*} \theta'=v \end{equation*} ...