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.

Filter by
Sorted by
Tagged with
-1
votes
1answer
52 views

Dynamics of the gain/attenuation $k(t)$ in the ODE-system

Given system of equation: \begin{cases} \frac{dx_1(t)}{dt}= \frac{df}{dx_1} \\ \frac{dx_2(t)}{dt}+x_2(t)=k(t) \cdot \frac{d^2f}{d^2x_1} \\ \frac{dk(t)}{dt}=??? \end{cases} where $x_1(t),x_2(t),k(t)$ - ...
0
votes
0answers
9 views

Convergence proof of iterative linear quadratic regulator (iLQR)

Background The trajectory optimization problem can be expressed as: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{1}, \ldots, \mathbf{u}_{T}} & \sum_{t=1}^{T} g(\mathbf{x}_{t}, \mathbf{u}_{t})\\ ...
0
votes
1answer
30 views

How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)

Picture of Question + solution Hi, In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$). How do we jump from this to knowing ...
1
vote
0answers
31 views

Nonlinear system with time-optimal control

Given nonlinear system: \begin{cases} \dot{x_1}=x_3+u \\ \dot{x_2}=-x_2+\dot{f} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) \\ \dot{x_4}=-x_4+x_2 \cdot (\frac{16}{\alpha^2}(\sin(\omega t)-\frac{...
0
votes
0answers
47 views

Numerical methods for non-linear (and second order) differential equation

I would like to hear some hint in solving this differential equation, where $\xi$ is constant: $-\frac{6 (1-6 \xi ) \xi \varphi (t) \varphi '(t)^2}{36 \xi ^2 \varphi (t)^2-6 \xi \varphi (t)^2+1}+\...
0
votes
0answers
6 views

Convergence of non-linear vector fixed point iteration

Let $A$ be an $d\times d$, non-negative, symmetric matrix. Let $\mathbf{a}\in\mathbb{R}^d$ be a positive vector and $\mathbf{x}_0 = [1,1,\cdots,1]^T$. Under what conditions on $A$ and $\mathbf{a}$, ...
0
votes
0answers
20 views

Zero overshoot criterion from the initial point $x_0$ to the final $x_*$, $x_*$ unknown in advance

Let's say there is an ODE: $\dot{x}=f(t,x)+u$ Condition: variable $x$ passes from initial state $x_0$ to final state $x_*$, that do not know in advance. Is it possible to make a transient in such a ...
0
votes
0answers
43 views

Asymptotic Output Tracking - Where to Place the Input Control Signal?

I ask for help from specialists in differential equations, dynamical systems, optimal control and general control theory; I have the following system of differential equations: \begin{cases} \frac{dx(...
0
votes
0answers
22 views

How can I find the fixed points of this Duffing oscillator differential equation?

The problem is to find the fixed points for the equation: $$\ddot{x}+ \dot{x}- x + ax^3=b \cos(ct)$$ where $a,b$ and $c$ are constants. The Duffing oscillator is a 2nd order differential equation and ...
2
votes
1answer
47 views

Dynamical System that exhibits a fold bifurcation of Limit Tori?

Fold Bifurcations of a fixed points (i.e. saddle node bifurcations) and Fold bifurcations of limit cycles (i.e. when a stable limit and unstable limit cycle annihilate) are observed in plenty of ...
0
votes
1answer
27 views

Linearization of a fixed point (dynamical systems)

Just looking at the following piece of math from Strogaz's dynamic / chaos book. What I don't understand is the last part, where he claims that O($Ƞ^{2}$) is negligible if $f^{'}(x^{*})!=0$. I guess ...
0
votes
0answers
16 views

Convexity of next state with respect to control input when executing zero-order hold (ZOH) control input

I'd like to know whether the resulting next state of a continuous-time dynamical system is convex with respect to control input when executing zero-order hold (ZOH) input. For simplicity, suppose that ...
3
votes
0answers
24 views

Why are continuous partial derivatives up to order two (rather than one) of nonlinear autonomous (2D) systems sufficient for linear approximation?

In Boyce and Diprima's ODE's and BVP's (10th edition page 522), it says that for the nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y) \qquad\qquad\qquad (10),$$ "The system ...
1
vote
1answer
45 views

Problem with the continuous equivalent of Newton's method optimization

In the arcticle Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization I found an interesting formula and its properties. The screenshot of the page from the article I was led ...
1
vote
0answers
38 views

Rate of convergence in nonlinear system with unknown steady-state

I ask for help in mathematicians, specialists in the theory of differential equations. $\frac{dx}{dt}=\frac{d}{dx}f(x)$ where $f(x)$ - unknown unimodal function. For example: $f(x)=−(x−x_∗)^2$ or $f(x)...
0
votes
0answers
29 views

Stabilization of gradient systems for finite time

I ask respected experts to help in solving the next task. We have the following differential equation: $\frac{dx}{dt}=\frac{d}{dx}(f(x))$ where $f(x)$ - arbitrary unimodal function (like $-x^2$,$-(x-1)...
0
votes
0answers
19 views

Assume $f(x)$ is a 1D map exhibiting a period-3 orbit P. Then is P a period 3 orbit of $f(f(x))$?

Assume $f(x)$ is a 1D map exhibiting a period-3 orbit P. Then P is a period 3 orbit of $f(f(x))$? I know this is a somewhat elementary question but I'm only asking it to confirm if my thought process ...
1
vote
0answers
25 views

Can $x(t) = Acos(at^2+b)$ be interpreted as a generic chaotic trajectory of some dynamical system?

Can $x(t) = A\cos(at^2+b)$ be interpreted as a generic chaotic trajectory of some dynamical system? We have just been introduced to the background theory behind chaotic systems but haven't worked ...
0
votes
0answers
27 views

Does the map on [0,1] x [0,1] with $x_{n+1} = (x_n + y_n) \bmod 1$ and $y_{n+1} = (x_n - y_n) \bmod 1$ admit a unique inverse for any $(x_n,y_n)$

My lecturer posed this interesting question for us to dwell on before the next class and but I am unsure how to get started. My inclination is that it does but I would like to be able to formalise ...
2
votes
1answer
49 views

Is it true that the a generic trajectory of the logistic map ${x_{n+1}}=4 x_n(1- x_n)$ can come arbitrarily close to $(\frac{\sqrt{5}-1}{2})$

I know that at r=4, the map exhibits chaotic behaviour for almost every trajectory except for very few which lead to fixed periodic behaviour. For this reason, I am inclined to say that the above ...
0
votes
0answers
14 views

Radius & height of hollow cylinder that keeps a particle with initial velocity pressed against the internal surface due to centrifugal force (cyclone)

I'm trying to determine the required radius, $R$, and length of a hollow cylinder, $L$, to keep a particle that enters tangentially with an initial velocity $V_{t,i}$ at the top of the cylinder "...
0
votes
1answer
22 views

Obtain non linear solution using neural network

The function f(x)=theta·x where theta is a row vector and x is a column vector, is a linear function. How can I obtain a non-linear function g(x) using a multi layer network, that also takes in x as ...
0
votes
0answers
16 views

How to understand the linear disturbance model?

There is a linear disturbance model given as: $\dot{d}=Md$ where $d \in R^{2 \times 1}$,$M=[0\,1;-1\,0]\in R^{2 \times 2}$ $d_{x} = Nd$ where $N=[1,2]$ By setting any random initial values(non-zero) ...
7
votes
1answer
105 views

What is the most general Carathéodory-type global existence theorem?

I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$ $$ \begin{equation} \left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \...
0
votes
0answers
34 views

Which nonlinear Observer to study to estimate the speed of the Plant?

I have designed the mathematical model of the plant with nonlinear hystersis function $f(x_1)$ and is validated using simulation. Now I want to design the nonlinear observer to estimate the speed (...
0
votes
0answers
6 views

Inversion of nonsmooth coefficients on derivatives in first order ODEs

Consider the the following non smooth matrices $M \in \mathbb{R}^{N \times N} $ and $K \in \mathbb{R}^{N \times N} $ with respect to $q \in \mathbb{R}^{N \times 1}$ and $ v \in \mathbb{R}^{N \times 1}$...
0
votes
0answers
42 views

Is there an analytic solution for the following non-linear equation $x'(t) = a(t) + b\, x(t) + c\,x^3(t)$?

Here $b, c$ are constants. Of course there exists locally a solution by the Picard-Lindelöf theorem but I'm looking for an explicit expression. In fact, this question comes from here but I changed the ...
1
vote
1answer
67 views

Is the phrase “Weakly/strongly Non-linear” qualitative or quantitative?

When it comes to differential equations, I notice that "Weakly Nonlinear" seems to be code for 'Relatively well behaved, easy to qualitatively predict, even if the solution requires an ...
1
vote
0answers
11 views

Jacobian of curvilinear dynamics, simplifying to avoid division by 0

This comes as a follow-up to other question. The original was about discretizing a nonlinear curvilinear dynamics equation, this follow up is about taking the Jacobian and simplifying such as remove ...
2
votes
2answers
67 views

Solving first order non-linear ODE with logarithm

I want to solve for $t \in \mathbb{R}, u'(t)=-u(t)\ln \lvert u(t) \rvert$. I defined two cases: $\mathbb{R^*_+}$ and $\mathbb{R^*_-}$. For $\mathbb{R^*_+}$: $$\frac{du}{u}=-\ln(u(t))dt$$ And by ...
1
vote
1answer
84 views

Expanding a PDE in powers of a small parameter?

I'm working on an assignment for my quantum mechanics class and I've arrived at a nonlinear inhomogeneous partial differential equation for a complex function $S:\mathbb{R}^2\to\mathbb{C}~;~S:(x,t)\...
0
votes
1answer
39 views

Discretization of nonlinear curvilinear dynamics equations

I am considering a tracking model of the form $$ \dot{x} = v\cos\psi\\ \dot{y} = v\sin\psi\\ \dot{v} = a\\ \dot{a} = 0\\ \dot{\psi} = \omega\\ \dot{\omega} = 0 $$ which is considering 2D position in $...
0
votes
0answers
26 views

Solution or approximations for ($3\times3$) system of differential equations?

I have the following set of differential equations that I would like to solve: $$\begin{aligned} \frac{d E_{p}}{d z} &=i \tilde\alpha_p\left(\chi^p_p E_{p}+\chi_{a,s}^{p} E_{a} E_{s}\right) \\ \...
0
votes
0answers
44 views

Linearized input affine system

For a control problem given by the nonlinear dynamics $\dot x=f(x,u)$, we may express the state space by declaring $z=(x,u)^⊤$ with $\dot u = v$ and writing $\begin{bmatrix}\dot x \\ \dot{u}\end{...
1
vote
0answers
66 views

Which way of solving from nonlinear control to choose?

I have a nonlinear system: \begin{cases} x'=f(x)+u \\ y=f(x) \end{cases} where $f(x)$ - gradient of some one-extremal function (for example $f=e^{-(x)^2}$), i.e. $\frac{df}{dx}$. Task: I want ...
1
vote
0answers
51 views

Solve nonlinear, forced and damped Duffing oscillator

I solve Duffing equation with Van Der Paul's method: \begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align} with $$x(t) = Re[A(t) \exp(i \omega_0 t)]$$ and ...
0
votes
0answers
19 views

Does Poincaré-Bendixson theorem and Bendixon's criterion hold for this class of differential inclusions?

Poincaré–Bendixson theorem and Bendixson criterion are known to hold for dynamical systems $\dot{x} = f(x)$ in the plane (i.e. $x\in\mathbb{R}^2$). My question is: Are this results still valid in some ...
0
votes
0answers
8 views

Uniqueness of a differential system with a linear subspace

I've encountered a system of differential equations, which takes the following form; \begin{equation} \begin{split} \dot{L}_1 &=\alpha L_1,\\ \dot{L}_2 &=\beta L_2,\\ \dot{N}& =\gamma N+\...
1
vote
0answers
21 views

Is formal topological conjugacy of polynomials decidable?

This is in some ways a refinement of this question about specific maps. That got me wondering about the question of topological conjugacy of polynomials by polynomial maps: if we have polynomials $f, ...
1
vote
1answer
51 views

How to calculate fixed points and plot bifurcation diagram for non-linear ODE system

I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points ...
1
vote
0answers
27 views

Controlling the dynamics of nonlinear systems with an unknown steady-state

I have repeatedly raised this, in my opinion, an extremely accurately formulated question here, but I have not received a qualified answer to it. Let's take a simple gradient dynamical system: $\frac{...
0
votes
0answers
36 views

Extension of an ODE to dynamical system with certain properties

We have a gradient ODE: $\frac{dx}{dt}=\frac{df}{dx}$ where $f=-x^2$ I want the condition to be met on a given system: $x''+x'=0$ I.e. initial ODE turns into a dynamic system: $\begin{cases} \frac{dx}{...
1
vote
0answers
36 views

Very hard quasilinear PDE

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ on some domain of the plane. Now, numerically I can ...
0
votes
0answers
38 views

Understanding dynamics on Lie Groups

I'm trying to understand the definition of "Group Affine" systems (from Theorem 1 in this paper). I'll restate it here: Let $\frac{d}{dt}X_t = f_{u_t}(X_t)$ be a vector field describing the ...
0
votes
0answers
34 views

Systems that Display Chaotic Behavior

I take a course in 'nonlinear dynamics and chaos'. For our final project, we have to choose a dynamical system in that is nonlinear and specifically one that displays chaotic behavior. I know that ...
0
votes
0answers
14 views

Criterion for the exponentiality of the solution of a gradient differential equation

We have gradient ODE: $\frac{dx}{dt}=\frac{df}{dx}$ where $f = -x^2$, for example. In such a system, transient processes will be exponential, i.e. the condition will be met: $x''+x'=0$ The advantage ...
0
votes
0answers
9 views

Changing the quality of the transient process in a nonlinear system (Part III)

My question is a continuation of the topics: Changing the quality of the transient process in a nonlinear system (in Mathematica) Changing the quality of the transient process in a nonlinear system (...
1
vote
1answer
45 views

Do first-order dynamical systems have any periodic solutions?

From Strogatz's Nonlinear Dynamics and Chaos (2nd ed), Section 2.6: If a fixed point is regarded as an equilibrium solution, the approach to equilibrium is always monotonic — overshoot and damped ...
0
votes
0answers
25 views

How does one find all the fixed points of the Hodgkin-Huxley Model?

I'm trying to find the fixed points of the Hodgkin-Huxley model, given by the system of equations: $$ dV/dt=f_V(V,m,h,n)=\frac{1}C_M[I_{ext}-\bar{g}_{Na}m^3h(V-V_{Na})-\bar{g}_Kn^4(V-V_K)-\bar{g}_l(V-...
1
vote
1answer
26 views

How do I find the equilibrium points of a system with a free variable

For the following system, $$\dot{x_1} = 3(x_1-x_2) \\ \dot{x_2} = x_1(k-2-x_3) \\ \dot{x_3}=x_1x_3$$ I want to determine its equilibrium points together with their stability. By the third equation I ...

1
2 3 4 5 6