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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1answer
35 views

Estimate the Lipschitz constant of the product $xf(x)$, where $f$ is Lipschitz

It seems to me that this is such a simple question, but the answer has eluded me for a month or more. I will give two interpretations. The first is more general. The second is for those possessing ...
1
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0answers
25 views

Find the bounded orbits of the system $x'=y$, $y'=x+x^2-y$

Consider $$ \frac{dx}{dt}=y,\quad \frac{dy}{dt}=x+x^2-y. $$ Describe the set of all bounded orbits. Hint, use $H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}-\frac{x^3}{3}$. The equilibria are $$ ...
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0answers
43 views

Prey predator excercice

Consider the 2-dimensional system of non-linear ODEs, semplified instance of a predator-prey population model $\dot x=\alpha x (1-x)-xy$ $\dot y = y(x-y)-\beta y$ with $\alpha = 1 ,\beta = 1/2$ ...
0
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0answers
21 views

Find all bifurcation values of a function

I need to find all bifurcation values of the function $x' = u + cos(x) + cos(2x)$. How do I find all bifurcation values of $u$? I know the solution is $u < -2, u = -2, -2 < u < 0$, and that ...
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5 views

assume one-side subshift A is a 0.5-chain on shift function.specify the point t in the A which 0.5-shadows the 0.5-chain

specify the point t in the A which 0.5-shadows the 0.5-chain.i think use by shadows definition and shadows theorem and chain.pseudo orbit contains periodic orbit.
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0answers
26 views

Replicator equation for mixed strategies?

The the replicator equation is usually defined for pure strategies. More specifically, the replicator eqn for $n$ strategies is given by: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} ...
2
votes
1answer
43 views

Irrational flow on torus

I have an interesting dynamical systems problem that has had me stumped for a few hours now, so I'm hoping I can get some help. The problem is concerned with flows on the torus. The model is given by ...
0
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0answers
52 views

Periodic solutions of autonomous system in polar coordinates

Let $x = r \cos θ$, $y = r \sin θ$. An autonomous system is expressed in polar coordinates: $$\frac{dr}{dt} = r(1-r)(r-2), \frac{dθ}{dt} = -1$$ (a) Find all the periodic solutions (including critical ...
0
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1answer
20 views

Find the limit cycle/s and fixed points for this dynamical system.

Consider the dynamical system: $$\dot{r}=-ar^4+ar^3+r^6-r^5+r^2-r~;~~\dot{\theta}=1$$ Find all fixed points and limit cycles for: a) $~~a=2$ b)$~~a<2$ c)$~~2<a<2\sqrt{2}$ Attempt For ...
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1answer
18 views

phase space partition and symbolic dynamics

I want to learn the basic theory of phase space partition and symbolic dynamics, can you point to any recent thesis and books containing a good exposition ? Thanks!
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1answer
74 views

Dynamics of Cubic Chaos [closed]

Consider the family of functions $f_{\lambda}(x) = \lambda x − x^3$. Describe the dynamics of this family of functions for all $\lambda < −1$.
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2answers
87 views

conservation law for the trajectories

Is it possible to find the conservation equation as the form of $Q=h(x,y)$, given that $$\dot{x}=x-xy$$ $$\dot{y}=5xy-5y$$ I am not sure how to start with.
0
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1answer
53 views

solve two-dimensional nonlinear system numerically

Given the system $$x' = x(1 − y^2) \\ y' = y^2(1 − x^2)$$ I can plot the trajectories according to the analysis of its fixed points, but I need help with solving it numerically. Can anyone help me ...
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0answers
59 views

Examples of real orbits with irrational period

Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$. More specific at most 2 pieces. Im talking about integer iterations starting at $f(0)=0$ and with ...
1
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0answers
22 views

Floquet Theory : zero equilibirum stability

Wondering if someone could explain exactly what it means when someone says "zero equilibirium is unstable because the floquet multiplier is greater than 1". I understand the FM part but I don't know ...
2
votes
0answers
24 views

Ergodicity under measure-theoretic isomorphism

Suppose we have two measurable dynamical systems $(X_1,\mathcal{B}_1,\mu_1,T_1)$ and $(X_2,\mathcal{B}_1,\mu_2,T_2)$, with $\mu_i(X_i)=1,\ i=1,2$. Suppose they are measure-theoretically isomorphic ...
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0answers
68 views

Compute stable/unstable/center manifold for nonlinear system of ODEs

Given a system: $\dot{x_1} = x_1(x_n - \phi(x))$, $\dot{x_n} = x_n(x_(n-1)-\phi(x))$ where $\phi(x) = x_1x_2 + x_2x_3 + \ldots + x_nx_1$. Let $S_n = \left\{x\in R^n: \sum_{i=1}^{n} x_i = 1, x_i\geq 0\ ...
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0answers
32 views

Why doesn't the frequency of a strategy reach zero under the replicator dynamics?

Background The replicator equation with $n$ strategies is given by the differential equation: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} a_{ij}x_{j} - \phi \right) \qquad i = 1, ...
0
votes
0answers
26 views

Linearization, coordinate transformation, origin of coordinate system

Assume, we have a non-linear 3d ODE system. Now I want to consider what happens near the point $U=(u,v,w)$. I linearized the system at point $U$. To do so I made the coordinate transformation $y:x-U$, ...
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0answers
21 views

compltely integrable vector field-Definition

I have a question about definition. when I have some ode system : $x'=f(x)$ where $x\in \mathbf{R^n}$ and $f$ is some smooth vector field. What does it mean that the system is completely ...
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0answers
14 views

Why do we call the $(y,z)$-plane non-leading here and the $x$-axis leading?

Consider an equilibrium state of a three-dimensional linear system, namely assume the case where the three eigenvalues $\lambda_i, i=1,2,3$, are real and $\lambda_3 < \lambda_2 <\lambda_1<0$. ...
0
votes
2answers
31 views

Solving quadratic recurrence inequality

Let $(n_k)$ be a sequence of natural numbers starting at $n_1=2$ and growing as follows $$n_k\leq n_{k+1}\leq n_k^2+2$$ As far as I know this sort of dynamical system may have no closed-form solution, ...
2
votes
1answer
41 views

Show that the system $\ddot x+x\dot x+x=0$ is reversible

Show that the system $\ddot x+x\dot x+x=0$ is reversible My attempt was to use a property from the book that if the system $\dot x = f(x,y)$ $\dot y = g(x,y)$ is reversible then $f(x,-y)=-f(x,y)$ ...
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0answers
16 views

Thms on Dynamical Systems: Cont. functions on Compact Spaces (sources)

I've recently started taking a discrete-time dynamical systems perspective on a topic. I've been able to introduce a reasonable metric on a set, obtaining a compact space. Under that metric, a nice ...
2
votes
1answer
32 views

New coordinates for an ODE system (eigenvectors)

When $n=2$, the general form of a linear ODE system is $$ x'=a_{11}x+a_{12}y,~~~~y'=a_{21}x+a_{22}y. $$ Assume we have two eigenvalues $\lambda_1$ and $\lambda_2$ and both are real and negative. When ...
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0answers
20 views

Any parametrization $h\colon S^1\to \Gamma$ possible?

Consider the following model, and, in particular, the second picture given in the link, namely the blue line which represents a trajectory which I call $\Gamma$. Is it possible to give any ...
2
votes
2answers
33 views

why topological conjugacy does not preserve ergodicity?

I want to know why topological conjugacy does not transfer ergodicity from one dynamical system to another?and if we change the maps from continous to C1-differentiable will the problem solve or not?
2
votes
1answer
30 views

Topological entropy of $C^1$ function on compact manifold is finite

I am trying to solve Exercise 2.5.7 from Introduction to Dynamical Systems by Brin and Stuck: Prove that the topological entropy of a continuously differentiable map of a compact manifold is ...
1
vote
2answers
33 views

How to write a discrete dynamical system into first order system

I need guidance on how to solve this here. $$x_{n+1} + 3x_n - 4x_{n-1} = (\sqrt{2})^n cos \left(\frac{n\pi}{6}\right)$$ I am required to transform the above equation into a first order finite ...
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0answers
49 views

Stable, unstable and centre subspace: How to draw the orbits in this space?

Let the eigenvector $X_1$ span the unstable subspace, the eigenvector $X_2$ span the stable subspace and the eigenvector $X_3$ span the central subspace. The three vectors are shown in the picture ...
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0answers
10 views

Each $(x,y)\in M$ is hyperbolic equilibrium

Hopefully, my question makes sense. Let $M$ be a manifold such that each point $(x,y)\in M$ is an hyperbolic equilibrium of $\dot{x}=f(x,y,0)$. Does this imply that the directions that are normal to ...
2
votes
1answer
50 views

whether $t=0$ is missed in the proof?

When I try to compute $g^{ij}g^{pq}\nabla_{i,j}^2h_{pq}=\Delta tr_gh$, I need $g^{ij}=\delta^{ij},\nabla_i g_{kl}=0$,i.e, it's normal coordinate. But $g(t)$ will change with $t$, only for a $t_0$, I ...
0
votes
1answer
40 views

Steady state of a Dynamical System

Why here can we say that $x=x_s$ increases monotonically, when it depends on time?
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0answers
22 views

If $\gamma$ is a periodic orbit with a saddle point in its interior, then there is another equilibrium point in the interior

I am trying to use this result but want to have some justification of it. I would like to explain in terms of limits sets of the saddle's stable and unstable manifolds.
1
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2answers
31 views

Positive equilibria for a system of eqautions

I have the following system of equations \begin{align} \frac{dx}{d \tau} &= x \left(1-x-\frac{y}{x+b} \right) \\ \frac{dy}{d \tau} &= cy \left(-1+a\frac{x}{x+b} \right) \end{align} I am ...
3
votes
1answer
80 views

Limiting behaviour of a real Möbius sequence

Consider the fractional linear transformation of the real variable $t$, transformation $$f(t)=\frac{at+b}{ct+d}$$ where $a,b,c,d\in\mathbb{R}$. Define $t_{n+1}=f(t_n)$ where $t_0\in\mathbb{R}$. There ...
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0answers
16 views

Absorbing domain is a ball, then associated attractor contains bounded orbits

If we have an ODE $x'=f(x)$ and there is a global Lyapunov function, then there is some large enough $Q$m such that the Ball $B_Q$ is an absorbing domain and the associated attractor of it contains ...
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0answers
10 views

Understanding the definition of maximal attractor

Definition 1 Let an open, bounded region $U\subset\mathbb{R}^n$ have the following property: there exists $T>0$ such that for each point $x_0$ in the closure $cl(U)$ of the domain $U$ the ...
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vote
2answers
22 views

For linear continuous dynamical systems, is the only possible equilibrium point 0?

For linear continuous dynamical systems, is the only possible equilibrium point 0? In all the examples I have seen, only the 0 equilibrium point is considered. I know this is not true for nonlinear ...
2
votes
0answers
89 views

Qualitative properties of nonlinear oscialltor system

Given $\dot{x} = y, \dot{y} = -f(x)$ where $f\in C^{1}$. A point $(x_0, y_0)$ is an equilibrium iff $f(x_0) = y_0 = 0$. (a) Find the necessary and sufficient conditions on $f'(x_0)$ for the ...
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0answers
21 views

Dynamical Systems Maximal Interval of Existence

Find the explicit solution and the maximal interval of existence for the initial value problem $x' = tx^3$ , $x(0) = x_0$. Recall that the maximal interval of existence depends on $x_0$, is open, and ...
2
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0answers
25 views

Poincare Theorem [closed]

The Poincare Theorem is, "If a limit cycle exists in the second order autonomous system $\dot{x}=f(x)$, then $N=S+1$". Does this theorem gives only a necessary condition for existence of a limit cycle ...
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0answers
26 views

How to find the $x(t)$ and $y(t)$ functions in the Lotka Volterra Equations?

I'm doing this math project involving the Lotka-Volterra equations. My goal was to be able to graph both $x(t)$ and $y(t)$ on the same axis against time. So far, I was able to find the combined ...
3
votes
1answer
45 views

What are good literature recommendations to understanding attractors?

I'm a PhD student in Neuroscience working in human physiology. Recently I'm trying to get a better understanding of computational neuroscience as it applies to my field. It is in this context that I ...
1
vote
2answers
98 views

What is the correct approach for studying bifurcations?

Probably a trivial question. Let's say I have the following system of equations: \begin{cases} f\left(x,y,p\right)=0\\ \\ y=g\left(x\right) \end{cases} where $p$ is a parameter, and I want to study ...
4
votes
1answer
52 views

Show that if a trapping region Q is path connected, the basin of Q must also be path connected

Consider a smooth bijection $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$. A trapping region is a bounded subset $Q$ of $\mathbb{R}^2$ with the properties int$Q\neq \emptyset$, $F(Q)\subseteq Q$, and ...
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0answers
6 views

The normalizer of measurable full groups

I'm following Kechris' book Global aspects of ergodic group actions. Let $(X,\mu)$ be a standard measure space (i.e., isomorphic to $[0,1]$ with the Lebesgue measure on Borel sets) and $E$ an ...
0
votes
1answer
35 views

Symmetric difference positive measure.

The following is a lemma from Viana's "Lectures on Lyapunov Exponents". Lemma 9.9 Let $f:M\to M$ be an invertible transformation and $\mu$ be an invariant aperiodic probability measure. For any ...
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votes
1answer
48 views

Cannot find Hopf bifurcation in this problem, graphically

I have a vector field $\frac{dx}{dt}=y, \frac{dy}{dt}=(-x^3+x^2-2x)+y(b-x^2)$. I have found that when $b=x^2$, the Jacobian of the system at $(x,0)$ has pure imaginary eigenvalues, so there should ...
0
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0answers
29 views

3D system of ODEs - an idea to find exact trajectories

Suppose I've got the following system of differential equations in $\mathbf{x}(t) = \left( x(t), y(t), z(t) \right)$: $\frac{dx}{dt} = f_{1}(x,y,z) \\ \frac{dy}{dt} = f_{2}(x,y,z) \\ \frac{dz}{dt} = ...