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The study of nonlinear dynamical systems in lattices is an area of research with continuously growing interest.The first systematic studies of these systems emerged in the late 1930 s,thanks to the work of Frenkel and Kontorova on crystal dislocations.These studies led to the formulation of the discrete Klein-Gordon equation (DKG).Specifically,in 1939,Frenkel and Kontorova proposed a model that describes the structure and dynamics of a crystal lattice in a dislocation core.The FK model has become one of the fundamental models in physics,as it has been proven to reliably describe significant phenomena observed in discrete media.The equation we will examine is a variation of the following form:
The process described involves approximating a nonlinear differential equation through the Taylor method and simplifying it into a linear model.Let's analyze step by step the process from the initial equation to its final form.For small angles,
can be approximated through the Taylor series as:
can be approximated through the Taylor series as:
We substitute 
in the original equation with the Taylor approximation:
in the original equation with the Taylor approximation:
To map this equation to a linear model,we consider the angles 
to correspond to displacements 
in a mass-spring system.Thus,the equation transforms into:
to correspond to displacements in a mass-spring system.Thus,the equation transforms into:
We recognize that the term 
expresses the nonlinearity of the system,while β is a coefficient corresponding to this nonlinearity,simplifying the expression.The final form of the equation is:
expresses the nonlinearity of the system,while β is a coefficient corresponding to this nonlinearity,simplifying the expression.The final form of the equation is:
The exact value of β depends on the mapping of coefficients in the Taylor approximation and its application to the specific physical problem.Our main goal is to derive results regarding stability and convergence in nonlinear lattices under nonlinear conditions.We will examine the basic characteristics of the discrete Klein-Gordon equation:
This model is often used to describe the opening of the DNA double helix during processes such as transcription.The model focuses on the transverse motion of the base pairs,which can be represented by a set of coupled nonlinear differential equations.
% Parameters
numBases = 50; % Number of base pairs
kappa = 0.1; % Elasticity constant
omegaD = 0.2; % Frequency term
beta = 0.05; % Nonlinearity coefficient
% Initial conditions
initialPositions = 0.01 + (0.02 - 0.01) * rand(numBases, 1);
initialVelocities = zeros(numBases, 1);
Time span
tSpan = [0 50];
>> % Differential equations
odeFunc = @(t, y) [y(numBases+1:end); ... % velocities
kappa * ([y(2); y(3:numBases); 0] - 2 * y(1:numBases) + [0; y(1:numBases-1)]) + ...
omegaD^2 * (y(1:numBases) - beta * y(1:numBases).^3)]; % accelerations
% Solve the system
[T, Y] = ode45(odeFunc, tSpan, [initialPositions; initialVelocities]);
% Visualization
plot(T, Y(:, 1:numBases))
legend(arrayfun(@(n) sprintf('Base %d', n), 1:numBases, 'UniformOutput', false))
xlabel('Time')
ylabel('Position')
title('Dynamics of DNA Base Pairs')
% Choose a specific time for the snapshot
snapshotTime = 10;
% Find the index in T that is closest to the snapshot time
[~, snapshotIndex] = min(abs(T - snapshotTime));
% Extract the solution at the snapshot time
snapshotSolution = Y(snapshotIndex, 1:numBases);
% Generate discrete plot for the DNA model at the snapshot time
figure;
stem(1:numBases, snapshotSolution, 'filled')
title(sprintf('DNA Model Displacement at t = %d', snapshotTime))
xlabel('Base Pair Index')
ylabel('Displacement')
% Time vector for detailed sampling
tDetailed = 0:0.5:50;
% Initialize an empty array to hold the data
data = [];
% Generate the data for 3D plotting
for i = 1:numBases
% Interpolate to get detailed solution data for each base pair
detailedSolution = interp1(T, Y(:, i), tDetailed);
% Concatenate the current base pair's data to the main data array
data = [data; repmat(i, length(tDetailed), 1), tDetailed', detailedSolution'];
end
% 3D Plot
figure;
scatter3(data(:,1), data(:,2), data(:,3), 10, data(:,3), 'filled')
xlabel('Base Pair')
ylabel('Time')
zlabel('Displacement')
title('3D Plot of DNA Base Pair Displacements Over Time')
colormap('rainbow')
colorbar
Lots of students like me have a break from school this week or next! If y'all are looking for something interesting to do learn a bit about using hgtransform by making the transforming snake animation in MATLAB!
Code below!
⬇️⬇️⬇️
numblock=24;
v = [ -1 -1 -1 ; 1 -1 -1 ; -1 1 -1 ; -1 1 1 ; -1 -1 1 ; 1 -1 1 ];
f = [ 1 2 3 nan; 5 6 4 nan; 1 2 6 5; 1 5 4 3; 3 4 6 2 ];
clr = hsv(numblock);
shapes = [ 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 % box
0 0 .5 -.5 .5 0 1 0 -.5 .5 -.5 0 1 0 .5 -.5 .5 0 1 0 -.5 .5 -.5 0 % fluer
0 0 1 1 0 .5 -.5 1 .5 .5 -.5 -.5 1 .5 .5 -.5 -.5 1 .5 .5 -.5 -.5 1 .5 % bowl
0 .5 -.5 -.5 .5 -.5 .5 .5 -.5 .5 -.5 -.5 .5 -.5 .5 .5 -.5 .5 -.5 -.5 .5 -.5 .5 .5]; % ball
% Build the assembly
set(gcf,'color','black');
daspect(newplot,[1 1 1]);
xform=@(R)makehgtform('axisrotate',[0 1 0],R,'zrotate',pi/2,'yrotate',pi,'translate',[2 0 0]);
P=hgtransform('Parent',gca,'Matrix',makehgtform('xrotate',pi*.5,'zrotate',pi*-.8));
for i = 1:numblock
P = hgtransform('Parent',P,'Matrix',xform(shapes(end,i)*pi));
patch('Parent',P, 'Vertices', v, 'Faces', f, 'FaceColor',clr(i,:),'EdgeColor','none');
patch('Parent',P, 'Vertices', v*.75, 'Faces', f(end,:), 'FaceColor','none',...
'EdgeColor','w','LineWidth',2);
end
view([10 60]);
axis tight vis3d off
camlight
% Setup vectors for animation
h=findobj(gca,'type','hgtransform')'; h=h(2:end);
r=shapes(end,:)*pi;
steps=100;
% Animate between different shapes
for si = 1:size(shapes,1)
sh = shapes(si,:)*pi;
diff = (sh-r)/steps;
% Animate to a new shape
for s=1:steps
arrayfun(@(tx)set(h(tx),'Matrix',xform(r(tx)+diff(tx)*s)),1:numblock);
view([s*360/steps 20]); drawnow();
end
r=sh;
for s=1:steps; view([s*360/steps 20]); drawnow(); end % finish rotate
end
I asked my question in the general forum and a few minutes later it was deleted. Perhaps this is a better place?
Rather than using my German regional forum (as I do not speak German), I want to ask questions in an international English-speaking forum. Presumably there should be an international English forum for everyone around the world, as English is the first or second language of everyone who has gone to school. Where is it?
And what do you do for Valentine's Day?
which technical support should I contact/ask for the published Simscape example?
I was looking into the possibility of making a spin-to-win prize wheel in MATLAB. I was looking around, and if someone has made one before they haven't shared. A labeled colored spinning wheel, that would slow down and stop (or I would take just stopping) at a random spot each time. I would love any tips or links to helpful resources!
Greetings to all MATLAB users,
Although the MATLAB Flipbook contest has concluded, the pursuit of ‘learning while having fun’ continues. I would like to take this opportunity to highlight some recent insightful technical articles from a standout contest participant – Zhaoxu Liu / slandarer.
Zhaoxu has contributed eight informative articles to both the Tips & Tricks and Fun channels in our new Discussions area. His articles offer practical advice on topics such as customizing legends, constructing chord charts, and adding color to axes. Additionally, he has shared engaging content, like using MATLAB to create an interactive dragon that follows your mouse cursor, a nod to the upcoming Year of the Dragon in 2024!
I invite you to explore these articles for both enjoyment and education, and I hope you'll find new techniques to incorporate into your work.
Our community is full of individuals skilled in MATLAB, and we're always eager to learn from one another. Who would you like to see featured next? Or perhaps you have some tips & tricks of your own to contribute. Remember, sharing knowledge is a collaborative effort, as Confucius wisely stated, 'When I walk along with two others, they may serve me as my teachers.'
Let's maintain our commitment to a continuous learning journey. This could be the perfect warm-up for the upcoming 2024 contest.
Many of the examples in the MATLAB documentation are extremely high quality articles, often worthy of attention in their own right. Time to start celebrating them! Today's is how to increase Image Resolution using deep learning
https://uk.mathworks.com/help/deeplearning/ug/single-image-super-resolution-using-deep-learning.html
Struct is an easy way to combine different types of variants. But now MATLAB supports classes well, and I think class is always a better alternative than struct. I can't find a single scenario that struct is necessary. There are many shortcomings using structs in a project, e.g. uncontrollable field names, unexamined values, etc. What's your opinion?
I am confused, is the matlab answer better or Julia’s?
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Enjoy yourself and have fun! We're committed to fostering a supportive and educational environment. Dive into discussions, share your expertise, and grow your knowledge. We're excited to see what you'll contribute to the community!
I recently discovered a 2-minute video that introduces MatGPT, and I believe it's a resource worth sharing. The creator highlights MatGPT's impressive capabilities by demonstrating how it tackles the classic Travelling Salesman Problem.
With more than 13,000 downloads on File Exchange, MatGPT is gaining traction among users. I strongly recommend taking it for a spin to experience its potential firsthand.
Have you ever used Live Tasks in MATLAB? MathWorks development team would like to get some feedback on your experience – what did you like and not like. Especially, if you know about it but don’t use it frequently, we would like to understand why?
Please tell us what you think by submitting your response to this form https://forms.office.com/r/ui1EGqAFDx
how accurate are the answers of the AI Playground regarding information that are not specifiyed in the documentation?
We're thrilled to announce the roll-out of some new features that are going to supercharge your Playground experience! Here's what's new:
Copy/Download code from the script area
You can now effortlessly Copy/Download code from the script area with just a single click. Copy code or Download your script directly as .m files and keep your work organized and portable.We hope this will allow you to effortlessly transfer your work from Playground to MATLAB Desktop/Online.
Run Code directly from the Chat panel
Execute code snippets from the chat section with a single click. This new affordance means saving a step since you no longer have to insert code and then hit run from the toolstrip to execute instead just hit run in the chat panel to see the output immediately in the script area
Enhanced visual Experience
Customize your Playground workspace by expanding or collapsing the chat and script sections. Focus on what matters most to you, whether it's AI chat or working on your script.
We hope you will love these updates. Try them out and let us know your feedback.
MathWorks just released three new courses on Coursera liseted below. If you work with image or video data and are wanting to incorporate deep learning techniques into your workflow, this is a great opporutnity. The course creators monitor the discussion forums, so you can ask questions and get feedback on your work. Below are links to the three courses and a quick description of a project you'll complete in each.
- Introduction to Computer Vision for Deep Learning. You'll train a classifier to classify images of people signing the American Sign Language alphabet.
- Deep Learning for Object Detection. Move from just classification to finding object locations. You'll train a model to find different types of parking available on the MathWorks campus.
- Advanced Deep Learning Techniques for Computer Vision. You'll train anomaly detection models for medical images and use AI-assisted labeling auto label images.
Can anyone provide insight into the intended difference between Discussions and Answers and what should be posted where?
Just scrolling through Discussions, I saw postst that seem more suitable Answers?
What exactly does Discussions bring to the table that wasn't already brought by Answers?
Maybe this question is more suitable for a Discussion ....
When I want to understand a problem, I'll often use different sources. I'll read different textbooks, blog posts, research papers and ask the same question to different people. The differences in the solutions are almost always illuminating.
I feel the same way about AIs. Sometimes, I don't want to ask *THE* AI...I want to ask a bunch of them. They'll have different strengths and weaknesses..different personalities if you want to think of it that way.
I've been playing with the AI chat arena and there really is a lot of difference between the answers returned by different models. https://lmarena.ai/?arena
I think it would be great if the MATLAB Chat playgroundwere to allow the user to change which AI they were talking with.
What does everyone else think?
how can i use this AI?
Hello Community!
We are working on a new translation experience for the MathWorks website and products. The goal is to make it easy for people to see content in the best language for them.
Step 1 is learning from those of you who use another language instead of, or in addition to English. If this sounds like you, we'd love your response to a brief survey.
Feel free to comment here as well. Thanks in advance!
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