# Tagged Questions

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 ...

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 ...
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 $\boldsymbol F^{-1}(\boldsymbol f)$,...
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 ...
48 views

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 ...
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 ...
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!
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 ...
44 views

48 views

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 \... 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 $$\sigma = 10\\ \gamma = 28\\ b = \frac{8}{3}\\... 0answers 58 views ### A question about the “state-transition-matrix” of a physical system, Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ... 1answer 56 views ### 2D Bifurcation Classification Given the system with m as a varying parameter: \dot x = mx^2-y and \dot y = m+y - x Determine any bifurcations that occur Attempt: x nullcline y=mx^2 y nullcline y=x-m Fixed ... 0answers 56 views ### Find imaginary part of complex expression Given the system of ODEs,$$x'=x^3-3xy^2y'=3x^2 y-y^3,$$it can be shown that the system may be written as z'=z^3, where z=x+iy. However, I don't seem to get how to show that \Im m\{\frac{1}{... 1answer 42 views ### Compactness and positive invariance of set under flow of ODEs Given a system of ODEs,$$x'=yy'=x-x^3-yx(0)=x_0y(0)=y_0,$$also given a set S=\{(x,y):V(x,y)\le k, x>0\}, V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}, where -\frac{1}{4}&... 1answer 31 views ### Prove that the iteration of \sin(x) goes to zero as n goes to \infty [duplicate] Basically let S(x)=\sin(x) such that S^2(x)=\sin(\sin(x)) and S^3(x)=\sin(\sin(\sin(x))) and so on until S^n(x)=\sin(\sin(\ldots\sin(x)\ldots)) Prove that S^n(x)\rightarrow 0 as n\... 3answers 43 views ### Positive invariance of a set under a system of ODEs Given the system of ODEs,$$x'=x(1-x-y)y'=y(x-1),$$Q=\{(x,y):x\ge 0, y\ge 0\}, and S=(x,y)\in Q:x+y\le k, k>1, I need to show that S is invariant under this system of ODEs. Attempted ... 1answer 38 views ### Invariance of sets for systems of ODEs Given the system of ODEs$$x' = x(1-y)y'=y(x-1),$$let the set Q=\{(x,y):x\ge 0, y\ge 0\}. Explain why Q is invariant for this system of ODEs. My explanation: If x > 1 and y<1 then ... 1answer 40 views ### Why c>1/4 is not in Mandelbrot set As title: f_c(x)=x^2+c I got to the step: f_c(x)>x (for all x) But what's next? How to show that after k iterations, f^k_c \to \infty as k \to \infty Thanks, 1answer 33 views ### Solve a system of second order differential equations I have a system of second order differential equations which is$$m_1x_1''=-k_1x_1-k_2(x_1-x_2)\\m_2x_2''=-k_2(x_2-x_1)$$where (x_1(0),x_1'(0),x_1(0),x_2'(0))=(1,0,2,0) and (m_1,m_2,k_1,k_2)=(1,... 0answers 23 views ### Stability of LTI systems under saturation Consider the saturation function$$ \sigma(u)=\max(\min(u,1),-1) $$for u\in\mathbb{R}. With slight abuse of notation, if u\in\mathbb{R}^n let \sigma(u) also denote the same function applied ... 1answer 52 views ### behavior of the Linear system of an ODE model I am working on a predator-prey model and the linearization about and equilibrium point (0,e_2) has Jacobian matrix as follows$$\mathcal{J} = \begin{pmatrix} 0 & 0\\ b& -b \end{pmatrix},$$... 1answer 16 views ### How to show that x_{n+1} - x_* = M(x_n) (x_n - x_*) dynamics converges? I am trying to find conditions of convergence (or non-convergence) of a system that behaves in the following manner (quasi-linear since the matrix is not stationary):$$ x_{n+1} - x_* = M(x_n) (x_n - ... 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 ... 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 ... 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$K>0$.$\frac B Ais minimum phase and a ... 0answers 26 views ### Center manifold reduction This a verbatim copy of an example on center manifold reduction on nonlinear dynamical system I found on some lecture note: Consider the system \begin{align*} \dot{x}&=x^2y-x^5\\ \dot{y}&=-y+... 0answers 32 views ### Perburb the Monodromy of Lefschetz fibration over a disk Suppose that\pi:E \to D$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 ... 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} \... 0answers 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 ... 2answers 49 views ### Stability criteria for linear systems with auxiliary variables Classical texts for control theory show the linear system\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative. Does the same criteria apply for a system of the following form:... 0answers 14 views ### What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ... 1answer 73 views ### Trapping Region for Dynamical System Show that the dynamical system contains a closed orbit$\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 ... 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? 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 ... 1answer 27 views ### Differentiation of unit force vector I was reading a paper and don't know how the following was derived. Given thatf = \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} + g \\ \end{bmatrix}$and$...
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 ...