Questions tagged [nonlinear]

Used to mark questions about nonlinear differential equations, `NonlinearModelFit`, and related to nonlinear dynamics.

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Optimization when actively working with the NDSolve command

Lagrangian of three-mass system with Mathematica Based on the Lagrangian of a mechanical system, we can obtain a system of equations of motion. These equations can be explored numerically using the ...
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3answers
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How solve this nonlinear equations system by 12 unknowns?

We consider the following integral equation of mixed type: $x[t]=\frac{t^4}{6} -\frac{t^3}{3} +t+\Sigma_{\mu=1}^2\int_{0}^t k_\mu[t,s]G_\mu[s,x[s]]ds ,0\leq t\leq 1 $ ; (1) where $k_1 [t, s] = s^3$...
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High-dimensional second-order differential matrix equations

I am working on Lagrangian derived high-dimensional motion equations for a robot in matrix form. The structure of such an equation is known: $M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=0$ where $q=[\dot{\...
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1answer
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Nonlinear frequency response

I am trying to reproduce the result of this paper, namely Figure 10 which depicts the solution of equation 64 given as, qmax is the steady state solution of the oscillatory equation given. I tried to ...
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1answer
138 views

A system of nonlinear ODEs

I am working with Mathematica to plot a system of Nonlinear ODEs, I did a program but it doesn't work. I don't know exactly where is the problem? If someone kindly can help with a remark or a ...
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Problem with Lagrangian and Matrix Calculus in Mathematica

Continuing the questions: Lagrangian of three-mass system with Mathematica Equations of motion for two-mass torsional oscillator with the gear train Derivation of equations of motion for a multi-body ...
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Logarithmic NonlinearFit with 10 parameters? Is Not a Real Number

I am trying to fit a collection of data with a fairly complicated function, and have been running into some errors. I am very new mathematica user, and am not really sure where I am going. This is my ...
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1answer
134 views

Equations of motion for two-mass torsional oscillator with the gear train

This is my first topic and I continue work on that: Lagrangian of three-mass system with Mathematica I found interesting problem here, and try reproduce results. Assumption: $d_1=0$ Algorithm: Write ...
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38 views

Non-linear correlation

This may be a more mathematics question. There is also a coding issue. I would like to test non-linear correlation of two vectors Maybe I have overlooked a function that already does this, although my ...
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1answer
112 views

Incorrect result of DSolve

Let us dsolve that Cauchy problem with 12.3 on Windows 10 Pro: ClearAll[w, z]; sol = DSolve[{w'[z] == -1/2 - Sqrt[1/4 - 3*w[z]], w[1] == -1}, w[z], z] ...
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Output is derivative of state variable

Given simple system of ODE. \begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \end{cases} As an output, I want to use $y=\dot{x_1}$. But when I use the ...
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Bilinearization with Mathematica - where to start?

I tried to bilinearize the two equations: ...
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How to correctly DSolve it?

Solving a PDE with Mathematica, I obtain sol = DSolve[Sqrt[D[u[x, y], x]] + Sqrt[D[u[x, y], y]] == x, u, {x, y}] ...
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1answer
86 views

Asymptotic Output Tracking: Compensator properties

Asymptotic Output Tracking: Code Issues The question is, rather, of a theoretical nature (practical applications can be viewed in the topic at the link). Asymptotic Output Tracking is said to be based ...
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235 views

“Spaghetti”-solutions for ODE nonautonomous system and reduced vector field

Projections of the 3-dimensional phase-space of a non-autonomous ODE system Multidimensional obstacle avoidance in ODE (Visualization) Given simple system of ODE: $\begin{cases} \dot{x}=g \\ \dot{o}=2 ...
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Projections of the 3-dimensional phase-space of a non-autonomous ODE system

Given classical system of ODE: $\begin{cases} \dot{x}=g \\ \dot{g}=t \cdot (-g+\frac{df}{dx}) \\ \dot{h}=-h+\frac{d^2f}{d^2x} \end{cases}$ where $f = e^{-x^2}$ I am constructing a three-dimensional ...
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71 views

How to change Machine Precision digits to meet the tolerances

I am trying to solve for Tcm and Mag by solving nonlinear equations using FindRoot command using following code: ...
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Construction of Navigation Function: Error

https://en.wikipedia.org/wiki/Navigation_function https://www.sciencedirect.com/science/article/abs/pii/S0921889015302451 https://arxiv.org/pdf/1605.00638.pdf - Paragraph III I am trying to create a ...
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Multidimensional obstacle avoidance in ODE. Part II

Multidimensional obstacle avoidance in ODE (Visualization) https://math.stackexchange.com/questions/4146255/multidimensional-obstacle-avoidance-in-ode For some time, I studied this question more ...
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1answer
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Even though the fit seems correct, NonlinearModelFit throws a failed convergence error

Consider the following data ...
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1answer
117 views

Multidimensional obstacle avoidance in ODE (Visualization)

A simple 3-dimensional ODE system is given: $F=\begin{cases} \dot{x}=g+g_{U_{rep}} \\ \dot{g}=-g+\frac{df}{dx} \\ \dot{h}=-h+\frac{d^2f}{d^2x} \end{cases} $ Task: Make the variable $g$ move so that ...
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Tuning the optimal control synthesized according to the Pontryagin maximum/minimum principle and choosing the cost function

I continue to study the topic I started here: Problem with optimal control and Pontryagin's maximum principle A simple ODE system $(1)$ is given: $F=\begin{cases} \dot{x}=g \\ \dot{g}=-g+\frac{df}{...
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1answer
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System of ODE $\rightarrow$ Affine State-Space $\rightarrow$ System of ODE in Cauchy form

Given simple system of ODE: \begin{cases} \dot{x}=G \\ \dot{z}=-z+\dot{f} \\ \dot{g}=-g+z \cdot s+u \\ \ddot{h}+\dot{h}+h=z \cdot m \end{cases} where: $x,z,g,h$ - state-space variables $f=-(x+s)^2$ $s=...
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1answer
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How to fit the following data?

Consider a dataset in the form {mN,tauN,x,value[mN,tauN,x]} (it may be downloaded here): ...
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1answer
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How can I do NonlinearModelFit for only specific parameters?

I have a two dimensional data of the form $$\{x,y,f[x,y]\}$$. For example, ...
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1answer
326 views

Problem with optimal control and Pontryagin's maximum principle

For dynamic system: $\dot{x}=\frac{df}{dx}+u$ where $f=e^{-x^2}$ It is necessary to develop optimal control, minimizing criterion: $J= \int_{0}^{t_f} ((\frac{df}{dx})^2+u^2) \,dt $ Algorithm: We ...
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FeedBack linearization and stabilization error

Consider nonlinear system: \begin{cases} \dot{x_1}=x_4+u \\ \dot{x_2}=-x_2+\frac{(\tanh(k \cdot x_5)+1)}{2}+U(t)+u \\ \dot{x_3}=-x_3+\dot{f} \\ \dot{x_4}=-x_4+x_3 \cdot s(t) \\ \dot{x_5}=-x_5+x_3 \...
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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{...
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1answer
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How does Mathematica solve this system of nonlinear equations?

During a calculation for a physics lab, I ran into the following six non-linear equations with unknowns $x_1,x_2,x_3,x_4,x_5,x_6 \in \mathbb{R}$ ...
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MaxStepSize for NonlinearModelFit?

Is there a way to restrict the step size used in NonlinearModelFit? In our applied problem, NonlinearModelFit, at the first ...
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1answer
99 views

Getting Mathematica to solve a system of two second order nonlinear ordinary differential equations

I tried solving a system of two second order nonlinear ordinary differential equations using the DSolve command. First, I tried like this: ...
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Asymptotic Output Tracking - How to Track?

Asymptotic Output Tracking: Code Issues Abstract state-space: \begin{cases} \dot{x_1}=x_3 \\ \dot{x_2}=-x_2+\frac{df}{dt} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) + u \\ \dot{x_4}=-x_4+x_2 \...
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1answer
221 views

Numerically solving 2 nonlinear PDEs of 2nd and 1st order

I want to compute flux coordinates $\{\psi,\theta,\chi\}$ as functions of cylindrical coordinates $\{r,\theta,z\}$ in the problem of ballooning mode instability in mirror traps (also called open traps)...
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3answers
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ODE problem using DSolve

I would like to use DSolve (or NDSolve) to verify that the solution to the ODE problem -4(v''[t]+(2/t)v'[t])-2*v[t]*Log[v[t]]-(3+(3/2)Log[4 Pi])*v[t]==0, for $t\...
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Mathematica returns some warning and errors instead of Output

I have a set of three coupled PDEs. Using NDsolve to get an output plot of my PDE's with the given parameters. As I'm new to Mathematica, I have almost spent two days, but cannot locate the problem ...
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1answer
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Problem with the continuous equivalent of Newton's method optimization

In the article 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 ...
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1answer
34 views

AsymptoticOutputTracking for output with boundary condition

I want to try asymptotic output tracking, but with inequality. There is a differential equation: $\frac{dx}{dt}=\frac{d}{dx}(-x^4)$ With output $y=\frac{d}{dx}(-x^4)$, The output should strive for $0$,...
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NDSolve big system non linear coupled ODE's Method Residual Stiff System

I'm trying to develop a continuous time unscented Kalman filter for a dynamic system inspired to this article: "On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear ...
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How does NonlinearModelFit work? [closed]

Suppose I have a model $ f (x) $ with some free parameters $ a, b $ and I want to fit my model to a data set called Dx. How does NonlinearModelFit find the value of ...
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1answer
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Finding fixed points of a system of difference equations

Can anyone help me in finding the fixed points of the systems satisfying all the parameters are real, positive? $$x_{n+1}=x_{n-d}e^{(r(1-\frac{x_{n-d}}{k})-\frac{\beta y_{n-d}}{(x_{n-d}+\gamma)}+x_{n-...
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1answer
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AsymptoticOutputTracker to Matlab-Simulink

What is the structural scheme for AsymptoticOutputTracker in Mathematica? I need this in order to transfer the feedback signals received with the help of ...
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0answers
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How to get a frequency equation from limited power expansion of differential equation solution?

I am trying to extract frequency that is variable depended from nonlinear coupled differential equation. I managed to get a solution in form of power series expansion up to 8th term and possible more....
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2answers
114 views

Solving a non-linear system of equations symbolically [closed]

How would I solve this systems of non-linear equations symbolically: $$ \mu N -\frac{\beta S I}{N} - \nu S = 0 \qquad (1)$$ $$\frac{\beta S I}{N} -\gamma I - \nu I = 0 \qquad (2)$$ $$\gamma I - \nu R =...
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ParameterTable from NonlinearModelFit returns error messages

I am doing some nonlinear regression using NonlinearModelFit function. Here in the code I am currently working on. ...
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1answer
112 views

Finding Chi^2 of a Non-Linear Fit Model

Question: I have some data, with a y=Ae^(-kt) model applied (It's a radioactive decay), thanks to answers on another question, the model is applied and returns the ...
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1answer
60 views

FindRoot not obtaining solution for nonlinear system of log equations

There is a system of 9 non-linear equations, eq containing logarithms that I think it should be solvable numerically. There are a total of 9 variables with two of ...
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1answer
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Comparison of NDSolve and Asymptotic Output Tracking results: Problem identified

My question is a continuation of the topic: Asymptotic Output Tracking: Code Issues Edit: Take system of ODE for example: $\begin{cases} \frac{dx}{dt}=H \cdot \alpha \sin(\omega t)+\alpha \omega \cos(\...
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Fitting nonlinear model to intensity plot with Gaussian weights?

I am trying to fit my model $\omega(\vec{k})$ to the following experimental data to extract three parameters $J_x$, $J_y$, and $J_z$. The data that I am trying to fit to is: ...
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77 views

Constraint of a variable with an expression in nonlinear model fitting - independent variable constraint?

I am attempting to fit a set of experimental binding data: ...
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38 views

Affine state-space: Nonlinear output

I am using a system of equations to experiment: $\begin{cases} x_1'=x_2 \\ x_2'=x_1^2-x_2+u \end{cases} $ As an output, I want to use the following non-linear output: $y=e^{-x_1^2}$ ...

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