Questions tagged [nonlinear-dynamics]

This tag is for questions relating to nonlinear-dynamics, the branch of mathematical physics that studies systems governed by equations more complex than the linear, $~aX+b~$ form.

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Does a spiral fixed point going from stable to unstable always indicate Hopf bifurcation?

Does a spiral fixed point going from stable to unstable (with change in some parameter) always indicate Hopf bifurcation? Could it be a homoclinic bifurcation instead? (Homoclinic bifurcation keeps ...
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Pontryagin Maximum Principle with terminal and initial conditions

Consider a control problem with Lagragian $L(t,x,u)$ (where $u$ is the control, $x \in \mathbb{R}^d$ the state) and dynamics $\dot{x}=f(x,u,t)$. I have mostly seen problems in which the dynamical ...
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Show the system has one equilibrium point

I was wondering how we would show that the system: dx/dt=-x^3+2x-4y dy/dt=-y^3+2y+4x has only one equilibrium point. I have seen cases where the system is, ...
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If steady states of a dynamic system exist only as limits, are they actually steady states?

I have a nonlinear dynamic model in discrete time. A simplified version of my dynamic system is: \begin{equation} x_{t+1} = \frac{1}{1 + \exp(f(x_t))} \end{equation} where $$f(x_t) = −\beta \left(2d \...
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Prove that the periodic solutions of a periodically driven system with least period $T$ can only have periods $kT$ for integers $k$.

I am currently having trouble with some properties of periodic solutions of periodically driven dynamic systems. It is an exercise (1.5.3) from Guckenheimer and Philip Holmes. Nonlinear oscillations, ...
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I think there is a significant issue with Strogatz's working definition of attractors.

In Strogatz's book, Nonlinear Dynamics and Chaos, he gives the following working definition of attractors (page 690 of second edition): "More precisely, we define an attractor to be a closed set $...
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Feedback linearization with integral action - How?

Assume that you know sort of the dynamics of the system. It's not 100% perfect, but it's at least 90% perfect. $$\dot x = f(x, u)$$ I want to find a control law that suits this system. I have been ...
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Adaptive step size for Euler Method - How to create?

I think Euler's Method is a great way to simulate ODE:s. It's not the most accurate, but it's the fastest and simplest. Euler's Method is usaly used with fixed step size, where ...
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If the cross product result is known, then how to calculate the factors vector $A$ and vector $B\,$?

Assume that $$\nabla H_1\times\nabla H_2 \:=\: V\quad\text{and}\quad V \:=\: \big(σy, x(r − z), xy\big)\,.$$ My question: If $V$ is given, is there any way to find out what $\,\nabla H_1\,$ and $\,\...
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Limit Cycle Analysis - State Space Rep. to Polar Coordinates Question

I am trying to follow an example that does not show how a set of dynamics equations is converted to polar coordinates: $\theta=\tan^{-1}\frac{x_2}{x_1}$ $\frac{d}{dt}\tan\theta=(\frac{1}{\cos\theta})^{...
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Dynamical systems with control input

Please I have been trying to write the mathematical formulation of my nonlinear dynamical system for quite some time and I will appreciate any input. ** Problem Description** Assuming, I am traveling ...
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How to use the solution of an homogeneous non linear DE when plugging a discretized input.

there's a certain class of non-linear (NL) DE (non linear in $f$) $$ x' = f(x)+g(t), $$ the homogeneous form of which admits an analytic solution. Suppose the solution to such an homogeneous form $x'=...
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clarification for the tangential and normal components of acceleration

please help me understand this bit from the book: "Vector Mechanics for Engineers Statics and Dynamics by Beer, Johnston(12th edition)"(page 692). why or how is ${\it \Delta}e_t = 2\sin(...
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Prove the $k$-th power of the logistic map with parameter $\mu = 4$ has $2^k$ fixed points

I'm trying to solve an exercise in which I need to prove that the logistic map with parameter $\mu = 4$, $F_4:[0,1]\to[0,1]$, $F_4(x) = 4x(1-x)$, satisfies that for every positive integer $k$, $F_4^k$ ...
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Non-hyperbolic disease free equilibrium

There are compartmental models that have constant total population therefore zero eigenvalues in their jacobians such as https://www.nature.com/articles/s41591-020-0883-7 , https://www.nature.com/...
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Does finite-duration solutions to differential equation could be thought as the reciprocal of a system with finite-time blow-up?

Does finite-duration solutions to differential equation could be thought as the reciprocal of a system with finite-time blow-up? Introduction Recently I have found on the papers by Vardia T. Haimo ...
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Under what conditions does convergence of DEs imply convergence of solutions?

Consider a continuous time nonlinear dynamical system $$\dot X = f(X, t), \quad X = (x_1, \dotsc, x_N),\tag{1}$$ and a particular solution $X(t)$ with initial value $X(0) = X_0$. Moreover, suppose ...
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It is $x(t) = e^{c_1-t}\cdot\theta(c_1-t)$ a solution to $\dot{x} = -|x|$ with $\theta(t)$ the unitary step function?

It is $x(t) = e^{c_1-t}\cdot\theta(c_1-t)$ a solution to $\dot{x} = -|x|$ with $\theta(t)$ the unitary step function? I am trying to understand solutions of finite duration to differential equations. ...
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How can I find closed loop dynamics of this system?

There is controller, where $a>0$. Now, if $v>0$, then $u = \ddot\theta(t)$ and if $v<0$, then $-u=\ddot\theta(t)$. However, how can I detect whether $v$ is positive or negative?
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Theory for solving nonlinear matrix equation

I'm trying to solve a system of equations for $\theta\in \mathbb{R}^m$ of the form $$ X V_\theta (X^\top \theta - b) = 0 $$ where $X \in \mathbb{R}^{m \times n}, V_\theta \in \mathbb{R}^{n \times n}, ...
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Dimensionless form of the ODE for a simple pendulum with forcing and damping

I'm tasked with analysing the behaviour of a simple pendulum with driving and damping, which has the equation of motion: $$mL^{2}\ddot{\theta} + k\dot{\theta} + mgL\sin{\theta} = FL\cos{\Omega}t$$ For ...
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Is there any way to solve this second order Riccati equation?

I encountered this problem while studying the value of a firm which faces uncertainty over the price of its output good. Consider the following nonlinear ODE for $f(x)$ where all Greek letters are ...
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Understanding Uniqueness of solutions of differential equations - nonlinear ODEs - pendulum example

Understanding Uniqueness of solutions of differential equations - nonlinear ODEs - pendulum example I am trying to understand If the nonlinear ODE of the classical equation for the pendulum with ...
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Fourier transform of a nonlinear Schrödinger soliton

The nonlinear Schrödinger equation $$ i\frac{\partial\psi}{\partial t} = -\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}- |\psi|^2\psi $$ has the single-soliton solution $$ \psi(x,t)=A \frac{e^{iv(x-...
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Solving steady state diffusion with non-linear decay

I want to solve steady state diffusion with constant production term (in the source $[-L_s, 0]$) and a non-linear degradation term, where degradation takes place over the whole domain $[-L_s, L]$, but ...
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How to find the correct upper and lower bound of a matrix

I am trying to find a correct upper and lower bound for an equilibrium point of the dynamics of matrix R. The equilibrium point is given as: $$ R = ARA' + C $$ and I have been able to solve this ...
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How to render soliton solutions of nonlinear Schrödinger equation exponentially unstable?

The nonlinear Schrödinger equation in $\mathbb{R}^{3+1}$ is $$\phi_t + \nabla^2 \phi + f(|\phi|^2)\phi + V(x)\phi= 0, \quad \phi(x,0) = \phi_0(x).$$ (I require $\phi$ to be square-integrable, and also ...
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nonlinear differential equation for the hypocycloid

I've been trying to tackle a nonlinear first order differential equation that appears when trying to solve the brachistochrone problem through earth's gravitational field. Although being a long and ...
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How Achieve This Specific Non-Linear Mapping

Let's say in a computer program, we have a Slider whereby a user selects a value. The slider widget itself produces a value, X, from 0.0 to 1.0. This value then maps to some other range. Most people ...
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Can a nonlinear system give same solution with distinct set of parameters?

This question comes after experimenting with learning the parameters of a nonlinear system $\dot{\mathbf{x}} = f(\mathbf{x}), f: \mathbb{R}^n \rightarrow \mathbb{R}^n$. If I take a system that has $f =...
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Using Jury Conditions to Show Instability

If I want to force an equilibrium point of a discrete dynamical system to be unstable can I just violate one of the conditions for stability stated in the jury conditions $$|\mbox{Trace} (J)| < 1 + ...
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4 votes
1 answer
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Instability of a parameter varying system whose parameters belong to a compact set

Suppose, there is a system $$\dot{x}=f(t, \gamma_p(t), x)$$ with $x\in\mathbb{R}^2$. For my specific case, parameter vector $\gamma_p$ is a scalar and known monotonic function with a compact image set ...
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Is this Partial Differential Equation likely to be unsolvable?

Consider the following partial differential equation (PDE) of $f(x, y)$: $$\alpha x^{2} f_{xx} + \beta x f_{x} - \gamma f - \delta y f_y + \eta f^{2} + \theta f f_y y + \lambda + \kappa x^v y^w = 0$$ ...
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Can anyone determine whether there is a solution to this partial differential equation (PDE)? Has anyone encountered a PDE like this before?

Can anyone determine whether or not this PDE for $f(x,y)$ has a solution? Has anyone encountered a similar PDE before? $$\alpha x^2 f_{xx} + \beta x f_x - \gamma f - \delta y f_y + \frac{1}{\eta} \...
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Question About Similarity Solution Method

The similarity solution method is supposed to be able to solve at least some nonlinear partial differential equations, in addition to linear ones. After playing around in Mathematica to find a ...
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Is there more than one way to derive an energy function from differential equations?

I don't know how this energy function (screenshot below) comes from the oscillator equation. I know you can get it from $E =\frac{{\dot x}^2}{2} - \int \ddot x (x)dx$, which is conservative (meaning $...
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Inequality between class $\mathcal{K}_{\infty}$ functions

A function $\alpha: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ is said to belong to class $\mathcal{K}_{\infty}$ if it is continuous, strictly increasing, $\alpha(0) = 0$, and $\lim_{s \rightarrow +\infty}...
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How to convert a non linear diffirential equation to a difference equation?

Take a differential equation of the form $$ \frac{da(t)}{dt} = -\gamma e(t)r(t)$$ for $\gamma\in\mathbb{R}^+$. Let $e(t)$ and $r(t)$ be two time varying signals. How do we convert this differential ...
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Lyapunov Exponent of Nonlinear Schrödinger Operator

I am interested in ideas for how to compute the maximal Lyapunov exponent of the nonlinear Schrödinger-Newton system given by $$ \partial_t s(t, \vec{x}) = L(s(t,\vec{x})) $$ where $$ L(s(t, \vec{x})) ...
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Calculate time from one state to another using 6DOF state space modeling of a quadcopter

I'm new here. I have the 6DOF state space representation of a quadcopter given below, \begin{align} \dot x(t) &= Ax(t) + Bu(t) \end{align} Where, $x(t)$ = State Vector, $u(t)$ = Input (or control)...
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Do some nonlinear PDE steady state solutions depend on initial conditions (non unique)?

I was told by a colleague that for some nonlinear PDEs the initial conditions can change the steady-state solution. So can a stable steady solution depend on the initial conditions for nonlinear PDEs? ...
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Fourier series for periodic and bounded solutions (very theoretical question - HELP)

I have bounded periodic solutions to a non-linear differential equation. Drawing a phase portrait shows that they form closed curves (slightly irregular separate circles) with the center at the origin....
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5 votes
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What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$?

What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$? It could be proben that there exists some solutions? Are these solutions unique? and obviously, which are these solutions? (...
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Approximation of coupled nonlinear system

I have the following system and I wish to approximate it analytically: $$\dot{A}=\frac{P}{4A}(1-A^2)sin(2\chi) $$ $$\dot{\chi}=\alpha(\frac{1}{A}-\frac{1}{A+1})-0.5\sigma-0.25Pcos(2\chi) $$ And $A(t),\...
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Non-linear initial value problem

Problem Consider the following Initial Value Problem (IVP) \begin{equation*} \begin{aligned} \dot{\xi}(t) &= v(t)\,\cos[h(t)]\\ \dot{v}(t) &= a_{\parallel}\\ \dot{h}(t) &=\omega \end{...
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Non-linear differential equations where vector field does not change sign

I am working on nonlinear dynamics wherein I have encountered a class of autonomous non-linear dynamical systems where the elements of the underlying Jacobian matrix of the vector field with respect ...
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4 votes
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It is possible to find a solution to $y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0,$ $\,y'(0)=0,\,y(0)= 1/4$?

It is possible to find an exact solution (hopefully in "close form") to $$y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?... There ...
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Nature of solutions to 2nd order ODE with 2 functions, initial and asymptotic conditions

Given the ODE $$(\frac{a''}{a})^2 + 2(\frac{a'b'}{ab})^2 + 2(\frac{b''}{b})^2 + (\frac{1-b'^2}{b^2})^2 + 4\frac{a''a'b'}{a^2b} - 8\frac{a'b'b''}{ab^2} + 2\frac{a''(1-b'^2)}{ab^2} = 0$$ with $a$ even, $...
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3 votes
1 answer
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Given $(a_n)$ such that $a_1 \in (0,1)$ and $a_{n+1}=a_n+(\frac{a_n}{n})^2$. Prove that $(a_n)$ has a finite limit. [duplicate]

Given $(a_n)$ such that $a_1 \in (0,1)$ and $a_{n+1}=a_n+(\frac{a_n}{n})^2$. Prove that $(a_n)$ has a finite limit. Clearly $a_n$ are increasing. Also, $$\frac{1}{a_{n+1}}=\frac{1}{a_n\left(1+\frac{...
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6 votes
1 answer
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How does this expression follow algebraically from the last one?

I was reading this paper: Global stability for an HIV/AIDS epidemic model with different latent stages and treatment Everything is understood apart from on page 7 of the pdf (page 1486 in the document)...
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