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Questions tagged [non-linear-dynamics]

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Zero eigenvalue with deficient jacobian

Let $\dot{x} = f(x)$, $x \in \mathbb{R}^N$ be a system of ODEs with an equilibrium at the origin. Assume that the linearization at the origin has $K<N$ zero eigenvalues but only $K-1$ linearly ...
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How do Static Loop-Transformations Work

Loop-Transformations (input feedforward or output feedback) can be used for passivity analysis, also in the case when considering a static (memoryless) nonlinearity. This is described for example in ...
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In billiard systems, why are birkhoff coordinates needed to create area preserving maps?

Birkhoff co-ordinates, when used to obtain Poincaré sections of a billiards dynamics are often referred to as 'area preserving'.. why ?
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An first integral of nonlinear differential equation as like forced pendulum nonlinear diff. eq.

I'm trying to face this nonlinear differential equation: $$ y''(x)+\omega^2\sin\,y(x)=a\,x \,\;(1)$$ and I'm interested to found the solution of $ y'(x)$ (an first integral) The homogeneous part of ...
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1answer
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Checking stability of a fixed point

If the fixed point is hyperbolic, then it is said that linearisation gives the correct result . Is there an intuitive way of understanding why this is so ? And for marginal cases, when the fixed ...
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Phase trajectory must always enclose a fixed point

I found this problem in strogatz nonlinear dynamics . The theorem says, A closed phase space trahectory must enclose a fixed point . The question is asked as, is this true for phase surfaces other ...
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Center manifold and projection onto center eigenspace

Consider a system $\dot{x} = f(x) \in \mathbb{R}^N$ with an equilibrium at $x_0$ for which the Jacobian has a zero eigenvalue and all other eigenvalues have negative real part. By the Reduction ...
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1answer
45 views

Is chaos a topological property for continuous dynamical systems?

Following the definition of chaos given by Devaney, a continuous map $f$ on $(X,d)$ separable metric space with no isolated point is said chaotic if it is topologically transitive, that is for any ...
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What is the source of symmetry in this recurrence plot of the Circle map?

I was reading about the Circle map in this Wikipedia entry. This is a dynamical map with a single variable $\theta$, and dynamics defined by $$\theta_{n+1}=\theta_n + \Omega -\frac{K}{2\pi} \sin (2\...
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How to determine the stability of a fixed point if the derivative at the point is equal to one? ($\,\left\lvert\, f'(x^{\ast})\right\rvert = 1$)

Context: I am learning about 1-dimensional maps: For instance the logistic model of population growth. Suppose I have the map $x_{n+1} = f(x_n)$. The point $x^{\ast}$ is called a fixed point of ...
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Nonlinear map having conserved quantity

I am reading the following discussion:                           Does this simple 2D dynamical system have a conserved quantity?    Does this dynamical system have another conserved quantity? ...
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Tight upper bounds for a monotonically increasing non-linear recurrence

I have the following non-linear recurrence: $$y_{n+1} = \sqrt{\frac{2}{1+y_n}}y_n,\quad y_0 \in[0,1]$$ Some basic thought shows that $0$ and $1$ are fixed points of this, and that $0$ is repelling ...
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The level sets of integral are invariant sets (Wiggins' textbook)

I am reading the following book: Introduction to applied nonlinear dynamical systems and chaos, Stephen Wiggins On p. 77, for a general vector field $$\dot{x} =...
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What are some good recommendations of nonlinear equations/functions?

I have a project for my matlab course. I need to find a nonlinear equation to use to find the roots of it using various root-finders. I then have to write up a paper talking about the these different ...
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Revisit “example of an unstable fixed point for which the linearized dynamics are stable”

I am reading the following discussion: example of an unstable fixed point for which the linearized dynamics are stable The above discussion is for the vector field (continuous time). Is there an ...
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Moser and Smale-Birkhoff homoclinic theorems

I have found in J. Moser "Stable and Random Motion in Dynamical Systems" the theorem about the topological conjugacy to the Bernoulli shift on a symbol space, and then again very well summarized and ...
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Imitation Dynamics: Rest Points and Jacobian

Questions: Show that for a given matrix $A$ the imitation dynamics in the following equations: $$\dot{x}_i=x_i\sum_j x_j\psi((A\mathbf{x})_i-(A\mathbf{x})_j) $$ $$\dot{x}=x_i((A\mathbf{x})_i-\mathbf{x}...
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Plot of the stable/ unstable manifold in case of complex eigenvectors of the Jacobian matrix.

I was trying to plot the unstable manifold of a fixed point of a two dimensional map $f(x,y)$. I have the fixed point say $(x^*,y^*)$. Next, I found the Jacobian matrix at the fixed point from which I ...
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1answer
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Solving a 2nd order non-linear ODE

First off, please correct me if my title is wrong. I want to solve an equation which has the following form: $f'' + Af^3 + Bf = 0$ The closest I have gotten to such a form was when looking as the ...
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When Linearised Model of Finite Depth Waves is Not a Sufficient Model

I have been investigating the linearised model of water wave motion in a finite depth fluid. In my particular case the flow is Inviscid, Irrotational and Incompressible and surface tension effects are ...
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1answer
51 views

Help with first order non-linear differential equation

I've been trying to solve this one for a while, but I still can't make it. Here's the problem. I have a $f_1(t;\rho,\nu)$ that for $t\to\infty$ and for $\rho>\nu$ goes as $f_1\sim t^{\nu/\rho}.$ ...
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Squared-derivative PDEs

Is there a general theory for equations of the type $ f_y^2 = A(x,y) f_x$? where one first derivative is expressed as a multiple of the other one. Concretely, I'm interested in the equation $$ ( x+...
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1answer
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Derivation of Polyanin's formula for Abel's ODE

There is a general solution from the Polyanin textbook for the equation, $y\cdot\frac{dy}{dx}-y=Ax+B$ The solution in parametric form is $x = C \cdot e^{-\int \frac {t \cdot dt}{t^2-t-A}} $ and ...
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Is this non-homogeneous partial differential equation analytically solvable and if not what would be an appropriate numerical method?

The equation is: $$\frac{\partial h}{\partial t} = \frac{\nu}{(Ft)^2}\frac{\partial^2 h}{\partial \theta^2} + \eta(\theta,t)$$ where $\nu$ and $F$ are constants and $\eta$ is a function of $\...
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Limit Map of Discrete Dynamical System

Let $f:H\rightarrow H$ be a countinuous map from the separable hilbert space into itself, for every $x\in H$ define the discrete dynamical system $$ \xi_x^{n+1}\triangleq f(\xi^n_x);\qquad \xi^0_x\...
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1answer
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Variation of the Diffusion Equation

Let $\mathcal{L}=D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$ be a differential operator describing diffusion ($D$) with drift ($v$) and a source ($\beta$). As part of a ...
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tracking error state space, non-linear control example

I am trying to understand an example from [1]. In detail I do not understand how the equation for the dynamic of the tracking error is chosen. I am not a mathematician so please forgive me if I may ...
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Lyapunov Indirect Method

I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium ...
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2answers
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Partial differential equations that involve an infinite “continuum” of variables: “Each point in space contributes additional degrees of freedom”?

Page 11, Nonlinear Dynamics and Chaos, by Strogatz, says the following: This is the domain of classical applied mathematics and mathematical physics where the linear partial differential equations ...
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1answer
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Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...
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How to calculate the averaged equations for the weakly nonlinear oscillator $\ddot x+x+\varepsilon (x\dot x^2)=0$?

This is Strogatz exercise $7.6.5:$ For the system $\ddot x+x+\varepsilon h(x,\dot x)=0$, where $h(x,\dot x)=x\dot x^2$ with $0 < ε << 1$, calculate the averaged equations and if possible, ...
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How to solve this non-linear second order difference equation

Is there a direct method to analytically solve this non linear second order difference equation problem ? \begin{equation} \begin{split} \left \{ \begin{array}{ll} \frac{\beta\nu}{2\sqrt{\Phi_{t+1}}...
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Nonlinear system of equation in real life

Good night (from my country) for all Mathematicians here. Last week my teacher gave me an indivudal assignment to do during the holidays. He asked us to provide either nonlinear or nonlinear system of ...
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Improved model of a fishery: $\dot N=rN(1-\frac{N}{K})-H\frac{N}{A+N}$

Strogatz exercise $3.7.4.a:$ An improved model of a fishery is: $$\dot N=rN\left(1-\frac{N}{K}\right)-H\frac{N}{A+N}.$$ a) Give a biological interpretation of the parameter $A$; what does it ...
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Fundamental matrix of ODE system $\dot{x} = (At+B)x$

Suppose I have a system of ODEs $\dot{x} = (At+B)x$, where $x(t)$ is a, say, $n \times 1$ vector, and $A$ and $B$ are constant $n \times n$ matrices. What is the fundamental matrix of this system? I ...
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Solve sine and exponential nonlinear differential equation?

Is it possible to solve this kind of differential equation with forward Euler? $$\ddot y^2 + sin(\ddot y ) + \dot y + y = u$$ I haven't even write this ODE on the first order form. If I would do ...
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Geometric significance of a bifurcation point with algebraic multiplicity $2$?

This is part of Strogatz exercise $3.2.3:$ This is the process by which I found the bifurcation point/points for $\dot x=x-rx(1-x)$: By the method of tangential intersection we have: $$x=rx(1-x)$$ $...
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Nondimensionalization of the logistic equation.

In the process of studying nonlinear dynamics by Strogatz, I saw how he did simplify the model for an insect outbreak with the use of nondimensionalization. So as an exercise I picked an equation and ...
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Classify the bifurcation that occurs at $\mu$ =0

$ dx/dt=\mu x+y+x^2+x^3 , dy/dt=-x+\mu y+x^2y$ What I have done so far is getting the matrix A with $A_{11}=\mu,A_{12}=1,A_{21}=-1,A_{22}=\mu$ at $(0,0)$.I can see the bifurcation is Hopf ...
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Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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Nonlinear Gaussian State Space with Linear Observation Matrix Derivation

For the following State Space Model: $x_k = f(x_{k-1}) + v_k ~~ \sim ~ \Bbb N(0_{n_{v \times 1}}, \sum_v) \\ y_k = Cx_{k} + w_k ~~ \sim ~ \Bbb N(0_{n_{w \times 1}}, \sum_w)$ where $f: \Bbb R^{n_{...
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Describe discrete-time Langevin dynamics by its stationary distribution

Let's consider stochastic dynamics with discrete time step $t$ and states $x_t\in\mathbb{R}^d$ that evolve as $$ \mathrm{p}(x_{t+1}|x_t)=\mathcal{N}(x_t+f(x_t), \Sigma). $$ Moreover, we assume the ...
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System of nonlinear ODEs remains in positive cone

Given a system of three first order nonlinear ODEs $ \begin{align*} x_1' &= f(x_1,x_2,x_3) \\ x_2' &= g(x_1,x_2,x_3) \\ x_3' &= h(x_1,x_2,x_3), \end{align*} $ I'm hoping to ...
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Simple pendulum: How to solve the following second order nonlinear ODEs?

I'm trying to find equation of motion the following pendulum. To do this, let $$\vec r (t) = (l \sin (\theta (t)), -l \cos(\theta(t))).$$ then $$m D^2 \vec r(t) = ( -l \sin(\theta(t))\cdot \dot{\...
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Lotka-Volterra equation (predator-prey): given any initial condition, how can one know the steady-state behavior?

I am trying to find out if it is possible to determine the steady state behavior of the predator-prey system defined by the nonlinear equations: $$\begin{eqnarray} \frac{dx}{dt}&=&ax-bxy \\ ...
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How to deal with non-equilibrium operating point

Given the nonlinear system $$ \begin{align} \dot{x}_1 &= -4x_1 + 10x_2 + u \\ \dot{x}_2 &= -x_1 - 2x_2 - \log(1 + x_1^2) \\ y &= x_1 + x_2 \end{align} $$ Assume the system should be ...
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Phase portrait of a nonlinear system

I have an Lotka Volterra type of system as seen below; $$\begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align}$$ I would like to plot the phase portrait of the ...
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linearizing dynamics about non fixed point for LQR implementation.

I am trying to implement LQR control for the cart pole system. I am curious if I can maintain a constant non-zero pole angle. So, I need to linearize my dynamics about my goal state. I know we can use ...
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Differential equations - why we do care so much about (non)linearity?

This is a very simple question, but I feel I'm missing the bigger picture. Authors will talk of the horrors of nonlinear differential equations and that they're very difficult to solve, but why are ...
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Does this dynamical system have another conserved quantity?

For the 3D system of ODEs: $$\begin{eqnarray}\dot{x} &=& -\beta x y \\ \dot{y} &=& \beta x y + \hat{\beta} z y - \delta y \\ \dot{z} &=& -\hat{\beta} z y + \delta y, \end{...