Questions tagged [non-linear-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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6 views

Propagating in time a Nonlinear Dynamically Inverted (NDI) System

Background Suppose I have a nonlinear system given by $\dot{x}=f(x)+G(x)u$ $y=Hx$ where $x$ is the state, $y$ is the output, $G$ is a control matrix. This form is identical to how one would ...
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19 views

Solving a nonlinear periodic ODE

Is it possible to solve the below ODE for arbitrary real values of $c$? $$2 (d_x \theta)^2 [\sin(2\theta) -c \cos(2 \theta)]-d_x^2\theta[\cos(2\theta) +c \sin(2 \theta)]=0,$$ where $\theta=\theta(x)...
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Can you solve this cycloid problem? [closed]

A particle is moving on a smooth curve under gravity and its velocity varies as the distance ( measured along the arc) from the highest point. Prove that curve is cycloid.
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65 views

Find nullclines for nonlinear system [closed]

I have a nonlinear system and need find and plot nullclines: $$ \dot{x}=0.1(-x-1.8*10^{-3}Q(y)+1.3*10^{-3})\\ \dot{y}=0.1(-y-2.1*10^{-3}Q(x)+D) $$ here $Q(x)=\frac{100}{1+e^{(0.01-x)/0.003}}$ and D ...
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44 views

Existence of constant of motion

Consider the following dynamical system $$\begin{cases}\dot x = y-\varepsilon (x^2+y^2)x \\ \dot y=-x-\varepsilon(x^2+y^2)y\end{cases} $$ There exists some integral of motion (or constant of motion, ...
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33 views

Itô Integration by parts on a distribution for defaulted banks

The question is based on the McKean-Vlasov problem formulated in the paper "A McKean-Vlasov equation with positive feedback and blow-ups", namely: \begin{equation} \begin{cases} X_t = X_0 + B_t -\...
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22 views

Derivation of internal dynamics of nonlinear system in order to derive Byrnes-Isidori Normal Form

I have a nonlinear system (Ball & Beam) which is described by the following equations of motion: $$ \ddot{y} + \frac{mg}{a} \sin(θ) -\frac{m}{a}y\dot{θ}^2 = 0 $$ $$ \ddot{θ} + \frac{2m}{b}y\dot{...
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40 views

Geometrical methods for studying systems of nonlinear differential equations of high orders, suitable in the computational plan

Recently, I began to study systems of high-order nonlinear differential equations. As an example, I can cite the system of equations from this topic. Phase portrait of n-dimensional state-space ...
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19 views

Is topological semi-conjugacy sufficient for the case described below, or does one need full conjugacy?

I'm taking an undergraduate course in nonlinear dynamics, and the idea of topological conjugacy between (one-dimensional) iterated maps was introduced as follows: Let ${I}$ and ${J}$ be intervals. We ...
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31 views

Linear Stability Analysis question/clarification

I've been stuck on this problem for the past few days, and was wondering if someone could clarify part ii) of the following question. In order to understand the question, I will also include my ...
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24 views

In at 3-dimensional system, if two of my axes are the center subspace, would my center manifold be the plane of these two axes?

I was doing extra problems for my dynamics class and I came across this system after I transformed to Jordan normal form: $$\begin{bmatrix}\dot{u}\\\dot{v}\\\dot{w}\end{bmatrix}=\begin{bmatrix}-1&...
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14 views

While solving the motion in plane problems (dynamics) how to figure out whether the radial accelration is 0 or mgsin(theta)?

There is a problem : A straight smooth tube revolves with constant angular velocity W in a horizontal plane about one extremity which is fixed. If at zero time the tube be horizonal and a particle ...
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12 views

Why does the change of the Lyapunov function with time of a particular orbit necessarily have a minimum value in Chataev's instability theorem?

In the proof of Chataev's instability theorem, it is assumed that there exists an $m>0$ such that m is the minimum value of the change in the Lyapunov function $V$ with time for the given positive ...
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20 views

How to numerically evaluate index of fixed point?

I want to take a brute force approach to determining whether a system of 2 ODEs can support oscillations/closed orbits with different combinations of parameters. After a little analysis, I know the ...
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12 views

What's the difference between indeterminacy and explosiveness in the context of dynamic systems?

One important question to ask is if the model has as unique stable (asymptotically stationary) solution (determinacy) or multiple solutions (indeterminacy). But what's the difference between ...
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56 views

Time Scaling in Nonlinear Differential Equations

Take the following nonlinear differential equations $$ A_1 \ddot x_1(t) + B \dot x_1(t) + C x_1(t) = f(x_1) \tag{1} $$ and $$ A_2 \ddot x_2(t) + B \dot x_2(t) + C x_2(t) = f(x_2) \tag{2} $$ with $$ ...
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15 views

Is there an analytic way to tell if 1D partial differential equation is conservative?

I had similar question about system of ordinary differential equation, where answer is yes. I am not sure what conservative should mean in the field of PDE, again. Maybe it is again conservation of (...
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1answer
16 views

Distance of perturbed flow from the unperturbed stable manifold

Consider the following system $ \dot x= v\\ \dot v= x - x^2(1+\varepsilon cost)$ Let $\phi_\varepsilon(x,v)=\psi_\varepsilon ^{2\pi}(x,v)$ the flow of system with initial condition $(x,v)$ at time $...
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37 views

The stability of a fixed point, given that the one of the eigenvalues of the linearised system is zero and the other it negative?

I have the following dynamical system $$\frac{d x}{d \tau}=\gamma x(1-x)-\alpha x y$$ $$\frac{d y}{d \tau}=y\left(1-\frac{y}{x}\right),$$ where $\gamma$ and $\alpha$ are constant parameters. I am ...
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26 views

Does stability along axes imply stability of the fixed point?

Let say I have a 2D-dynamical system $$\dot x = f(x,y,\alpha_i)$$ $$\dot y = g(x,y,\alpha_i)$$ $i=1,2,...,n$ where $\alpha_i$ is a constant parameter. Let $(x_0, y_0)$ be a fixed point, of which we ...
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13 views

what is the geometric representation of Lyapunov stability

Take a look at the theorems below, in a dynamical systems linear or nonlinear, we construct a function usually the energy-function of a physical system, compute its derivative and apply the theorem to ...
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17 views

Bounded rationality: Least-squares learning: understanding the technique

This method is presented in Evans and Honkapohja (2001) I don't understand the formula used by the least squares learning technique to form expectations in economic models. This formula is given by, ...
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42 views

What is the correct name for this kind of growth expression, is it quadratic growth?! (it's not linear or exponential)?

I've constructed a growth scenario for an energy model and rather than use linear or exponential growth, I'm applying a linear increase rate to the increment amount, not to the base amount. It's a ...
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32 views

Solve coupled nonlinear differential equation - truck towing car problem

Imagine a truck towing a car using a taught rope of constant length $l$. The trucks position at any time is $(x,y)$ and is known in parametric form: $$x=f(t)$$ $$y=g(t)$$ The position of the car $(X,...
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17 views

Argument variable vs. indexed variable in Fourier transform.

Why is the transformed variable in some fourier transforms written as an index whereas sometimes it is written as an argument ? For example, $$G_{\bf k}({\bf{v}},t)= \frac{1}{(2\pi)^3}\int g({\bf ...
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22 views

Poincaré sections and one dimensional maps

New to dynamical systems and chaos theory: many textbooks start the discussion on chaos with one-dimensional maps and its associated orbits. Some more math-heavy textbooks begin the discussion with ...
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12 views

Passivity in State Space Models

Suppose I have the state space system $a\dot{x} = -x + \frac{1}{k}h(x) + u$, and output $y = h(x)$, where $a,k > 0$. The only information I'm given about $h(x)$ is that $h\in[0,k]$. I want to show ...
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26 views

If $t_{\max}<\infty$ then ${\displaystyle \limsup_{t\to t_{\max}}|x(t)|=\infty}$.

Consider the following differential equation \begin{equation} \begin{cases} y'(t)=f(t,y(t))\\ y(t_{0})=x_{0}\in\mathbb{R}^{n} \end{cases}\label{eq:EDO} \end{equation} where $f:[t_{0},\infty)\times\...
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19 views

Are there descriptions of nonlinear dynamical systems other than ODEs?

The canonical way to describe a dynamical system is to write it into the state-space representation, i.e., $\dot{x} = f(x, u), y = h(x, u)$. For a linear dynamical system, we can also use the transfer ...
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28 views

Calculation of Symmetry generator of cylindrical KdV equation

I have calculated the generators of the cylindrical $KdV$ equation $$u_t+(u/2t)+uu_x+u_{xxx}=0,$$ but I got three generators, $$X_1=\partial_x,\\ X_2=2t^{1/2}\partial_x+\left(1/2t^{1/2}\right)\...
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23 views

Backstepping control of second order nonlinear system

$\dot{x_{1}}=x_{2}^2-3\sin(x_{1})x_{2}$ $\dot{x_{2}}=x_{1}^3-3x_{2}\cos(x_{1})+u^{1/2}$ Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make the ...
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30 views

Strogatz Exercise 8.5.3: Logistic equation and Poincarè maps

I am going through Strogatz's Nonlinear Dynamics and Chaos and am stuck on one exercise from chapter 8 (exercise 8.5.3). The first part of the problem says: Consider the logistic equation $\dot{N}...
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30 views

existence of a circular limit cycle and its uniqueness for a nonlinear oscillator equation

Show that the nonlinear oscillator governed by $$\ddot{x}+a\dot{x}(x^2+\dot{x}^2-1)+x=0$$ where $a>0$, has a circular limit cycle, and find its amplitude and period. Give an argument to show ...
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question about saddle-node bifurcation in a $2$D-system of vector fields

Consider a two dimensional continuously differentiable vector fields in $\mathbb{R}^2$ $$\dot{u}=a(1-u)-uv^2$$ $$\dot{v}=uv^2-(a+k)v$$ where $a,k>0$ are parameters. Show that saddle-node ...
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18 views

stability of a nonlinear system with an without absolute value

I have a discrete-time nonlinear system in the form of \begin{equation} x(t+1)=A(x(t))x(t)+B(x(t))u \end{equation} where $u$ is a constant vector with $0<u_i<1$. I can show that $\lim_{t\to\...
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37 views

Eigenfrequencies of Hamiltonian dynamical systems

Consider the Hamiltonian $$ H=H(x,y,p_x,p_y) $$ which generates the dynamical system $$ \dot{x}=+\frac{\partial H}{\partial p_x} $$ $$ \dot{y}=+\frac{\partial H}{\partial p_y} $$ $$ \dot{...
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23 views

finding a closed orbit for an oscillator equation

Consider the oscillator equation $$\displaystyle\ddot{x}+F(x,\dot{x})\dot{x}+x=0$$ where $F(x,\dot{x})<0$ if $r\leq a$ and $F(x,\dot{x})>0$ if $r\geq b$ where $r^2=x^2+\dot{x}^2$. Show ...
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28 views

Is this a correct control input for this nonlinear system

Take a look at this system $$ \begin{align} \dot{x}_1 &= \cos x_2 + (x_2+1)x_3 \tag{1}\\ \dot{x}_2 &= x^3_1+x_3 \tag{2}\\ \dot{x}_3 &= x^2_1+u \tag{3}\\ y&=x_1 \tag{4} \end{align} $$ ...
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Creating a SIIR (susceptible, infected, isolated, recovered) model using differential equations.

I wasn't too sure of where to post this since it's a mix of physics (dynamical systems), medicine, and mathematics but here it goes. I am trying to model the current outbreak of Covid 19 using a more ...
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1answer
75 views

How to convert a nonlinear coupled system of equations to a linear system of equations?

Suppose we want to convert, \begin{align*} y_1’’ &= t^2-y_1’-y_2^2 &\quad \quad y_1(0)=0\quad y_1’(0)=1\\ y_2’’ &= t-y_2’-y_1^3 &\quad \quad y_2(0)=1\quad y_2’(0)=0 \end{align*} into ...
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23 views

How to express nonlinear transformation in the matrix form

I was trying to find the eigenvalue of a nonlinear system. As we know linear transformation can be written down matrix form, $r(t+\Delta t)=r(t)+v(t)\,\Delta t$, $v(t+\Delta t)=-\omega^2_0\,\Delta t\,...
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1answer
23 views

Continuous dependence of duffing oscillator solution on forcing term

Consider the Duffing oscillator $\ddot{x} + 2 \gamma \rho \dot{x} + \rho^2 (\dot{x} + \alpha x^3) = F(t)$ for some forcing term $F(t)$. Are there any results that state that $x$ depends continuously ...
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54 views

How to solve the nonlinear partial integro-differential equation by the finite difference method?

How to solve the following nonlinear partial integro-differential equation? Suppose the following equation: $m \ddot{v}+c_{1} \dot{v} + D\left(v^{\prime \prime \prime}+v^{\prime} v^{\prime \prime 2}+...
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49 views

Non-linear ode solution with integral formula

I am trying to figure out how the ode $\dot{x}=x^2, x(0)=c>1$ has the solution $x(t)=(\frac{1}{c}-t)^{-1}$ using the formula $x(t) = x_0+ \int^{t}_{t_0}f(s,x(s))ds$. I tried the substitution ...
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34 views

Two interpretations of Chaos?

Broadly speaking, I cannot pin down what is meant by Chaos. I understand that (informally) if a dynamical system is highly sensitive to initial input data then this system is said to be chaotic. Eg ...
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54 views

Lyapunov Spectrum of the Lorenz System

Consider the Lorenz System $${\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\[6pt]{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\[6pt]{\frac {\...
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80 views

A first order non-linear differential equation?

I'm trying to solve this non-linear differential equation : $$ \frac{dy}{dx}= \frac{y^3}{(y+1)^2(y+2)^2} $$ with the boundary condition $y(x_0)=x_0$, $x>0$, and $y(x)$ being a positive function. ...
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31 views

Is there a general solution to this system of coupled nonlinear first-order differential equations relating to enzyme kinetics

In a system of irreversible enzyme kinetics for binary binding reactions, one often comes across a system of equations of the form $$\frac{dc_i}{dt}=\sum_{j}\sum_{k} \beta_{ijk}c_jc_k$$ where $c_i$ ...
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67 views

How to solve nonlinear second order differential equation?

I'm currently doing my master thesis. Please refer to the attached image. It is a Deformable Linear Cable which is represented as a two-link manipulator that the joints have consisted of springs ( $...
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41 views

Poincaré-Bendixson applied to region enclosed by homoclinic orbit

Let $p$ be a saddle point of the planar ODE $x' = f(x)$ with $f$ smooth. Suppose $\gamma$ is a homoclinic orbit starting and ending in $p$. Define $\Gamma := \gamma \cup p$ and let $\mathcal{U}$ be ...