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Hello, everyone!
Over the past few weeks, our community has been buzzing with activity, showcasing the incredible depth of knowledge, creativity, and innovation that makes this forum such a vibrant place. Today, we're excited to highlight some of the noteworthy contributions that have sparked discussions, offered insights, and shared knowledge across various topics. Let's dive in!
Interesting Questions
Fatima Majeed brings us a thought-provoking mathematical challenge, delving into inequalities and the realms beyond (e^e). If you're up for a mathematical journey, this question is a must-see!
lil brain tackles a practical problem many of us have faced: efficiently segmenting a CSV file based on specific criteria. This post is not only a query but a learning opportunity for anyone dealing with similar data manipulation challenges.
Popular Discussions
Discover a simple yet effective trick for digit manipulation from goc3. This tip is especially handy for those frequenting Cody challenges or anyone interested in enhancing their number handling skills in MATLAB.
Chen Lin shares an exciting update about the 'Run Code' feature in the Discussions area, highlighting how our community can now directly execute and share code snippets within discussions. This feature marks a significant enhancement in how we interact and solve problems together.
From the Blogs
A Deep Dive into EEG Analysis for Predicting Neurological Outcomes By Tanya Kuruvilla
Connell D`Souza, alongside Team Swarthbeat, explores the cutting-edge application of EEG analysis in predicting neurological outcomes post-cardiac arrest. This blog post offers an in-depth look into the challenges and methodologies of modern medical data analysis.
Mihir Acharya discusses the pivotal role of MATLAB and Simulink in the future of robotics simulation. Through an engaging conversation with industry analyst George Chowdhury, this post sheds light on overcoming simulation challenges and the exciting possibilities that lie ahead.
We encourage everyone to explore these contributions further and engage with the authors and the community. Your participation is what fuels this community's continual growth and innovation.
Here's to many more discussions, discoveries, and breakthroughs together!
We're thrilled to share an exciting update with our community: the 'Run Code' feature is now available in the Discussions area!
Simply insert your code into the editor and press the green triangle button to run it. Your code will execute using the latest MATLAB R24a version, and it supports most common toolboxes. Moreover, this innovative feature allows for the running of attached files, further enhancing its utility and flexibility.
The ‘run code’ feature was first introduced in MATLAB Answers. Encouraged by the positive feedback and at the request of our community members, we are now expanding the availability of this feature to more areas within our community.
As always, your feedback is crucial to us, so please don't hesitate to share your thoughts and experiences by leaving a comment.
The study of the dynamics of the discrete Klein - Gordon equation (DKG) with friction is given by the equation :
In the above equation, W describes the potential function:
to which every coupled unit 
adheres. In Eq. (1), the variable $
$ is the unknown displacement of the oscillator occupying the n-th position of the lattice, and 
is the discretization parameter. We denote by h the distance between the oscillators of the lattice. The chain (DKG) contains linear damping with a damping coefficient 
, while
is the coefficient of the nonlinear cubic term.
adheres. In Eq. (1), the variable $$ is the unknown displacement of the oscillator occupying the n-th position of the lattice, and is the discretization parameter. We denote by h the distance between the oscillators of the lattice. The chain (DKG) contains linear damping with a damping coefficient , whileis the coefficient of the nonlinear cubic term.For the DKG chain (1), we will consider the problem of initial-boundary values, with initial conditions
and Dirichlet boundary conditions at the boundary points 
and 
, that is,
and , that is,
Therefore, when necessary, we will use the short notation 
for the one-dimensional discrete Laplacian
for the one-dimensional discrete Laplacian
Now we want to investigate numerically the dynamics of the system (1)-(2)-(3). Our first aim is to conduct a numerical study of the property of Dynamic Stability of the system, which directly depends on the existence and linear stability of the branches of equilibrium points.
For the discussion of numerical results, it is also important to emphasize the role of the parameter 
. By changing the time variable 
, we rewrite Eq. (1) in the form
. By changing the time variable , we rewrite Eq. (1) in the form
. We consider spatially extended initial conditions of the form:
where 
is the distance of the grid and 
is the amplitude of the initial condition We also assume zero initial velocity:
the following graphs for 
and 
and % Parameters
L = 200; % Length of the system
K = 99; % Number of spatial points
j = 2; % Mode number
omega_d = 1; % Characteristic frequency
beta = 1; % Nonlinearity parameter
delta = 0.05; % Damping coefficient
% Spatial grid
h = L / (K + 1);
n = linspace(-L/2, L/2, K+2); % Spatial points
N = length(n);
omegaDScaled = h * omega_d;
deltaScaled = h * delta;
% Time parameters
dt = 1; % Time step
tmax = 3000; % Maximum time
tspan = 0:dt:tmax; % Time vector
% Values of amplitude 'a' to iterate over
a_values = [2, 1.95, 1.9, 1.85, 1.82]; % Modify this array as needed
% Differential equation solver function
function dYdt = odefun(~, Y, N, h, omegaDScaled, deltaScaled, beta)
U = Y(1:N);
Udot = Y(N+1:end);
Uddot = zeros(size(U));
% Laplacian (discrete second derivative)
for k = 2:N-1
Uddot(k) = (U(k+1) - 2 * U(k) + U(k-1)) ;
end
% System of equations
dUdt = Udot;
dUdotdt = Uddot - deltaScaled * Udot + omegaDScaled^2 * (U - beta * U.^3);
% Pack derivatives
dYdt = [dUdt; dUdotdt];
end
% Create a figure for subplots
figure;
% Initial plot
a_init = 2; % Example initial amplitude for the initial condition plot
U0_init = a_init * sin((j * pi * h * n) / L); % Initial displacement
U0_init(1) = 0; % Boundary condition at n = 0
U0_init(end) = 0; % Boundary condition at n = K+1
subplot(3, 2, 1);
plot(n, U0_init, 'r.-', 'LineWidth', 1.5, 'MarkerSize', 10); % Line and marker plot
xlabel('$x_n$', 'Interpreter', 'latex');
ylabel('$U_n$', 'Interpreter', 'latex');
title('$t=0$', 'Interpreter', 'latex');
set(gca, 'FontSize', 12, 'FontName', 'Times');
xlim([-L/2 L/2]);
ylim([-3 3]);
grid on;
% Loop through each value of 'a' and generate the plot
for i = 1:length(a_values)
a = a_values(i);
% Initial conditions
U0 = a * sin((j * pi * h * n) / L); % Initial displacement
U0(1) = 0; % Boundary condition at n = 0
U0(end) = 0; % Boundary condition at n = K+1
Udot0 = zeros(size(U0)); % Initial velocity
% Pack initial conditions
Y0 = [U0, Udot0];
% Solve ODE
opts = odeset('RelTol', 1e-5, 'AbsTol', 1e-6);
[t, Y] = ode45(@(t, Y) odefun(t, Y, N, h, omegaDScaled, deltaScaled, beta), tspan, Y0, opts);
% Extract solutions
U = Y(:, 1:N);
Udot = Y(:, N+1:end);
% Plot final displacement profile
subplot(3, 2, i+1);
plot(n, U(end,:), 'b.-', 'LineWidth', 1.5, 'MarkerSize', 10); % Line and marker plot
xlabel('$x_n$', 'Interpreter', 'latex');
ylabel('$U_n$', 'Interpreter', 'latex');
title(['$t=3000$, $a=', num2str(a), '$'], 'Interpreter', 'latex');
set(gca, 'FontSize', 12, 'FontName', 'Times');
xlim([-L/2 L/2]);
ylim([-2 2]);
grid on;
end
% Adjust layout
set(gcf, 'Position', [100, 100, 1200, 900]); % Adjust figure size as needed
Dynamics for the initial condition , 
, for 
, for different amplitude values. By reducing the amplitude values, we observe the convergence to equilibrium points of different branches from 
and the appearance of values 
for which the solution converges to a non-linear equilibrium point 
Parameters: 
, for , for different amplitude values. By reducing the amplitude values, we observe the convergence to equilibrium points of different branches from and the appearance of values for which the solution converges to a non-linear equilibrium point Parameters: 
Detection of a stability threshold 
: For 
, the initial condition , 
, converges to a non-linear equilibrium point
.
: For , the initial condition , , converges to a non-linear equilibrium point.Characteristics for 
, with corresponding norm 
where the dynamics appear in the first image of the third row, we observe convergence to a non-linear equilibrium point of branch 
This has the same norm and the same energy as the previous case but the final state has a completely different profile. This result suggests secondary bifurcations have occurred in branch 
, with corresponding norm where the dynamics appear in the first image of the third row, we observe convergence to a non-linear equilibrium point of branch This has the same norm and the same energy as the previous case but the final state has a completely different profile. This result suggests secondary bifurcations have occurred in branch By further reducing the amplitude, distinct values of 
are discerned: 1.9, 1.85, 1.81 for which the initial condition 
with norms 
respectively, converges to a non-linear equilibrium point of branch 
This equilibrium point has norm 
and energy 
. The behavior of this equilibrium is illustrated in the third row and in the first image of the third row of Figure 1, and also in the first image of the third row of Figure 2. For all the values between the aforementioned a, the initial condition 
converges to geometrically different non-linear states of branch 
as shown in the second image of the first row and the first image of the second row of Figure 2, for amplitudes 
and 
respectively.
are discerned: 1.9, 1.85, 1.81 for which the initial condition with norms respectively, converges to a non-linear equilibrium point of branch This equilibrium point has norm and energy . The behavior of this equilibrium is illustrated in the third row and in the first image of the third row of Figure 1, and also in the first image of the third row of Figure 2. For all the values between the aforementioned a, the initial condition converges to geometrically different non-linear states of branch as shown in the second image of the first row and the first image of the second row of Figure 2, for amplitudes and respectively. Refference:
Many MATLAB enthusiasts come Cody to sharpen their skills, face new challenges, and engage in friendly competition. We firmly believe that learning from peers is one of the most effective ways to grow.
With this in mind, the Cody team is thrilled to unveil a new feature aimed at enriching your learning journey: the Cody Discussion Channel. This space is designed for sharing expertise, acquiring new skills, and fostering connections within our community.
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Together, let's create an environment where every member feels empowered to share, learn, and connect.
Check out this episode about PIVLab: https://www.buzzsprout.com/2107763/15106425
Join the conversation with William Thielicke, the developer of PIVlab, as he shares insights into the world of particle image velocimetery (PIV) and its applications. Discover how PIV accurately measures fluid velocities, non invasively revolutionising research across the industries. Delve into the development journey of PI lab, including collaborations, key features and future advancements for aerodynamic studies, explore the advanced hardware setups camera technologies, and educational prospects offered by PIVlab, for enhanced fluid velocity measurements. If you are interested in the hardware he speaks of check out the company: Optolution.
Let's talk about probability theory in Matlab.
Conditions of the problem - how many more letters do I need to write to the sales department to get an answer?
To get closer to the problem, I need to buy a license under a contract. Maybe sometimes there are responsible employees sitting here who will give me an answer.
Thank you
Hello MATLAB Community!
We've had an exciting few weeks filled with insightful discussions, innovative tools, and engaging blog posts from our vibrant community. Here's a highlight of some noteworthy contributions that have sparked interest and inspired us all. Let's dive in!
Interesting Questions
Cindyawati explores the intriguing concept of interrupting continuous data in differential equations to study the effects of drug interventions in disease models. A thought-provoking question that bridges mathematics and medical research.
Pedro delves into the application of Linear Quadratic Regulator (LQR) for error dynamics and setpoint tracking, offering insights into control systems and their real-world implications.
Popular Discussions
Chen Lin shares an engaging interview with Zhaoxu Liu, shedding light on the creative processes behind some of the most innovative MATLAB contest entries of 2023. A must-read for anyone looking for inspiration!
Zhaoxu Liu, also known as slanderer, updates the community with the latest version of the MATLAB Plot Cheat Sheet. This resource is invaluable for anyone looking to enhance their data visualization skills.
From File Exchange
Giorgio introduces a toolbox for frequency estimation, making it simpler for users to import signals directly from the MATLAB workspace. A significant contribution for signal processing enthusiasts.
From the Blogs
Cleve Moler revisits a classic program for predicting future trends based on census data, offering a fascinating glimpse into the evolution of computational forecasting.
Boost Your App Design Efficiency – Effortless Component Swapping & Labeling in App Designer by Adam Danz
With contributions from Dinesh Kavalakuntla, Adam presents an insightful guide on improving app design workflows in MATLAB App Designer, focusing on component swapping and labeling.
We're incredibly proud of the diverse and innovative contributions our community members make every day. Each post, discussion, and tool not only enriches our knowledge but also inspires others to explore and create. Let's continue to support and learn from each other as we advance in our MATLAB journey.
Happy Coding!
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other (comment below)
3%
117 票
In the MATLAB description of the algorithm for Lyapunov exponents, I believe there is ambiguity and misuse.
The lambda(i) in the reference literature signifies the Lyapunov exponent of the entire phase space data after expanding by i time steps, but in the calculation formula provided in the MATLAB help documentation, Y_(i+K) represents the data point at the i-th point in the reconstructed data Y after K steps, and this calculation formula also does not match the calculation code given by MATLAB. I believe there should be some misguidance and misunderstanding here.
According to the symbol regulations in the algorithm description and the MATLAB code, I think the correct formula might be y(i) = 1/dt * 1/N * sum_j( log( ||Y_(j+i) - Y_(j*+i)|| ) )
Are you local to Boston?
Shape the Future of MATLAB: Join MathWorks' UX Night In-Person!
When: June 25th, 6 to 8 PM
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🚀 This Opportunity Is Not to Be Missed:
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- Voice Your Expertise: Share your insights and suggestions directly with MathWorks developers.
- Learn, Discover, and Grow: Expand your MATLAB knowledge and skills through firsthand experience with unreleased features.
- Network Over Dinner: Enjoy a complimentary dinner with fellow MATLAB enthusiasts and the MathWorks team. It's a perfect opportunity to connect, share experiences, and network after work.
- Earn Rewards: Participants will not only contribute to the advancement of MATLAB but will also be compensated for their time. Plus, enjoy special MathWorks swag as a token of our appreciation!
👉 Reserve Your Spot Now: Space is limited for these after-hours sessions. If you're passionate about MATLAB and eager to contribute to its development, we'd love to hear from you.
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2784 票
A high school student called for help with this physics problem:
- Car A moves with constant velocity v.
- Car B starts to move when Car A passes through the point P.
- Car B undergoes...
- uniform acc. motion from P to Q.
- uniform velocity motion from Q to R.
- uniform acc. motion from R to S.
- Car A and B pass through the point R simultaneously.
- Car A and B arrive at the point S simultaneously.
Q1. When car A passes the point Q, which is moving faster?
Q2. Solve the time duration for car B to move from P to Q using L and v.
Q3. Magnitude of acc. of car B from P to Q, and from R to S: which is bigger?
Well, it can be solved with a series of tedious equations. But... how about this?
Code below:
%% get images and prepare stuffs
figure(WindowStyle="docked"),
ax1 = subplot(2,1,1);
hold on, box on
ax1.XTick = [];
ax1.YTick = [];
A = plot(0, 1, 'ro', MarkerSize=10, MarkerFaceColor='r');
B = plot(0, 0, 'bo', MarkerSize=10, MarkerFaceColor='b');
[carA, ~, alphaA] = imread('https://cdn.pixabay.com/photo/2013/07/12/11/58/car-145008_960_720.png');
[carB, ~, alphaB] = imread('https://cdn.pixabay.com/photo/2014/04/03/10/54/car-311712_960_720.png');
carA = imrotate(imresize(carA, 0.1), -90);
carB = imrotate(imresize(carB, 0.1), 180);
alphaA = imrotate(imresize(alphaA, 0.1), -90);
alphaB = imrotate(imresize(alphaB, 0.1), 180);
carA = imagesc(carA, AlphaData=alphaA, XData=[-0.1, 0.1], YData=[0.9, 1.1]);
carB = imagesc(carB, AlphaData=alphaB, XData=[-0.1, 0.1], YData=[-0.1, 0.1]);
txtA = text(0, 0.85, 'A', FontSize=12);
txtB = text(0, 0.17, 'B', FontSize=12);
yline(1, 'r--')
yline(0, 'b--')
xline(1, 'k--')
xline(2, 'k--')
text(1, -0.2, 'Q', FontSize=20, HorizontalAlignment='center')
text(2, -0.2, 'R', FontSize=20, HorizontalAlignment='center')
% legend('A', 'B') % this make the animation slow. why?
xlim([0, 3])
ylim([-.3, 1.3])
%% axes2: plots velocity graph
ax2 = subplot(2,1,2);
box on, hold on
xlabel('t'), ylabel('v')
vA = plot(0, 1, 'r.-');
vB = plot(0, 0, 'b.-');
xline(1, 'k--')
xline(2, 'k--')
xlim([0, 3])
ylim([-.3, 1.8])
p1 = patch([0, 0, 0, 0], [0, 1, 1, 0], [248, 209, 188]/255, ...
EdgeColor = 'none', ...
FaceAlpha = 0.3);
%% solution
v = 1; % car A moves with constant speed.
L = 1; % distances of P-Q, Q-R, R-S
% acc. of car B for three intervals
a(1) = 9*v^2/8/L;
a(2) = 0;
a(3) = -1;
t_BatQ = sqrt(2*L/a(1)); % time when car B arrives at Q
v_B2 = a(1) * t_BatQ; % speed of car B between Q-R
%% patches for velocity graph
p2 = patch([t_BatQ, t_BatQ, t_BatQ, t_BatQ], [1, 1, v_B2, v_B2], ...
[248, 209, 188]/255, ...
EdgeColor = 'none', ...
FaceAlpha = 0.3);
p3 = patch([2, 2, 2, 2], [1, v_B2, v_B2, 1], [194, 234, 179]/255, ...
EdgeColor = 'none', ...
FaceAlpha = 0.3);
%% animation
tt = linspace(0, 3, 2000);
for t = tt
A.XData = v * t;
vA.XData = [vA.XData, t];
vA.YData = [vA.YData, 1];
if t < t_BatQ
B.XData = 1/2 * a(1) * t^2;
vB.XData = [vB.XData, t];
vB.YData = [vB.YData, a(1) * t];
p1.XData = [0, t, t, 0];
p1.YData = [0, vB.YData(end), 1, 1];
elseif t >= t_BatQ && t < 2
B.XData = L + (t - t_BatQ) * v_B2;
vB.XData = [vB.XData, t];
vB.YData = [vB.YData, v_B2];
p2.XData = [t_BatQ, t, t, t_BatQ];
p2.YData = [1, 1, vB.YData(end), vB.YData(end)];
else
B.XData = 2*L + v_B2 * (t - 2) + 1/2 * a(3) * (t-2)^2;
vB.XData = [vB.XData, t];
vB.YData = [vB.YData, v_B2 + a(3) * (t - 2)];
p3.XData = [2, t, t, 2];
p3.YData = [1, 1, vB.YData(end), v_B2];
end
txtA.Position(1) = A.XData(end);
txtB.Position(1) = B.XData(end);
carA.XData = A.XData(end) + [-.1, .1];
carB.XData = B.XData(end) + [-.1, .1];
drawnow
end
Are you a Simulink user eager to learn how to create apps with App Designer? Or an App Designer enthusiast looking to dive into Simulink?
Don't miss today's article on the Graphics and App Building Blog by @Robert Philbrick! Discover how to build Simulink Apps with App Designer, streamlining control of your simulations!
Northern lights captured from this weekend at MathWorks campus ✨
Did you get a chance to see lights and take some photos?
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2455 票
Hi to all.
I'm trying to learn a bit about trading with cryptovalues. At the moment I'm using Freqtrade (in dry-run mode of course) for automatic trading. The tool is written in python and it allows to create custom strategies in python classes and then run them.
I've written some strategy just to learn how to do, but now I'd like to create some interesting algorithm. I've a matlab license, and I'd like to know what are suggested tollboxes for following work:
- Create a criptocurrency strategy algorythm (for buying and selling some crypto like BTC, ETH etc).
- Backtesting the strategy with historical data (I've a bunch of json files with different timeframes, downloaded with freqtrade from binance).
- Optimize the strategy given some parameters (they can be numeric, like ROI, some kind of enumeration, like "selltype" and so on).
- Convert the strategy algorithm in python, so I can use it with Freqtrade without worrying of manually copying formulas and parameters that's error prone.
- I'd like to write both classic algorithm and some deep neural one, that try to find best strategy with little neural network (they should run on my pc with 32gb of ram and a 3080RTX if it can be gpu accelerated).
What do you suggest?
Dear MATLAB contest enthusiasts,
I believe many of you have been captivated by the innovative entries from Zhaoxu Liu / slanderer, in the 2023 MATLAB Flipbook Mini Hack contest.
Ever wondered about the person behind these creative entries? What drives a MATLAB user to such levels of skill? And what inspired his participation in the contest? We were just as curious as you are!
We were delighted to catch up with him and learn more about his use of MATLAB. The interview has recently been published in MathWorks Blogs. For an in-depth look into his insights and experiences, be sure to read our latest blog post: Community Q&A – Zhaoxu Liu.
But the conversation doesn't end here! Who would you like to see featured in our next interview? Drop their name in the comments section below and let us know who we should reach out to next!
Hey MATLAB Community! 🌟
In the vibrant landscape of our online community, the past few weeks have been particularly exciting. We've seen a plethora of contributions that not only enrich our collective knowledge but also foster a spirit of collaboration and innovation. Here are some of the noteworthy contributions from our members.
Interesting Questions
Victor encountered a puzzling error while trying to publish his script to PDF. His post sparked a helpful discussion on troubleshooting this issue, proving invaluable for anyone facing similar challenges.
Devendra's inquiry into interpolating and smoothing NDVI time series using MATLAB has opened up a dialogue on various techniques to manage noisy data, benefiting researchers and enthusiasts in the field of remote sensing.
Popular Discussions
Adam Danz's AMA session has been a treasure trove of insights into the workings behind the MATLAB Answers forum, offering a unique perspective from a staff contributor's viewpoint.
The User Following feature marks a significant enhancement in how community members can stay connected with the contributions of their peers, fostering a more interconnected MATLAB Central.
From File Exchange
Robert Haaring's submission is a standout contribution, providing a sophisticated model for CO2 electrolysis, a topic of great relevance to researchers in environmental technology and chemical engineering.
From the Blogs
Verification and Validation for AI: From model implementation to requirements validation by Sivylla Paraskevopoulou
Sivylla's comprehensive post delves into the critical stages of AI model development, from implementation to validation, offering invaluable guidance for professionals navigating the complexities of AI verification.
In this engaging Q&A, Ned Gulley introduces us to Zhaoxu Liu, a remarkable community member whose innovative contributions and active engagement have left a significant impact on the MATLAB community.
Each of these contributions highlights the diverse and rich expertise within our community. From solving complex technical issues to introducing new features and sharing in-depth knowledge on specialized topics, our members continue to make MATLAB Central a vibrant and invaluable resource.
Let's continue to support, inspire, and learn from one another
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4070 票
The study of the dynamics of the discrete Klein - Gordon equation (DKG) with friction is given by the equation :
above equation, W describes the potential function :
The objective of this simulation is to model the dynamics of a segment of DNA under thermal fluctuations with fixed boundaries using a modified discrete Klein-Gordon equation. The model incorporates elasticity, nonlinearity, and damping to provide insights into the mechanical behavior of DNA under various conditions.
% Parameters
numBases = 200; % Number of base pairs, representing a segment of DNA
kappa = 0.1; % Elasticity constant
omegaD = 0.2; % Frequency term
beta = 0.05; % Nonlinearity coefficient
delta = 0.01; % Damping coefficient
- Position: Random initial perturbations between 0.01 and 0.02 to simulate the thermal fluctuations at the start.
- Velocity: All bases start from rest, assuming no initial movement except for the thermal perturbations.
% Random initial perturbations to simulate thermal fluctuations
initialPositions = 0.01 + (0.02-0.01).*rand(numBases,1);
initialVelocities = zeros(numBases,1); % Assuming initial rest state
The simulation uses fixed ends to model the DNA segment being anchored at both ends, which is typical in experimental setups for studying DNA mechanics. The equations of motion for each base are derived from a modified discrete Klein-Gordon equation with the inclusion of damping:
% Define the differential equations
dt = 0.05; % Time step
tmax = 50; % Maximum time
tspan = 0:dt:tmax; % Time vector
x = zeros(numBases, length(tspan)); % Displacement matrix
x(:,1) = initialPositions; % Initial positions
% Velocity-Verlet algorithm for numerical integration
for i = 2:length(tspan)
% Compute acceleration for internal bases
acceleration = zeros(numBases,1);
for n = 2:numBases-1
acceleration(n) = kappa * (x(n+1, i-1) - 2 * x(n, i-1) + x(n-1, i-1)) ...
- delta * initialVelocities(n) - omegaD^2 * (x(n, i-1) - beta * x(n, i-1)^3);
end
% positions for internal bases
x(2:numBases-1, i) = x(2:numBases-1, i-1) + dt * initialVelocities(2:numBases-1) ...
+ 0.5 * dt^2 * acceleration(2:numBases-1);
% velocities using new accelerations
newAcceleration = zeros(numBases,1);
for n = 2:numBases-1
newAcceleration(n) = kappa * (x(n+1, i) - 2 * x(n, i) + x(n-1, i)) ...
- delta * initialVelocities(n) - omegaD^2 * (x(n, i) - beta * x(n, i)^3);
end
initialVelocities(2:numBases-1) = initialVelocities(2:numBases-1) + 0.5 * dt * (acceleration(2:numBases-1) + newAcceleration(2:numBases-1));
end
% Visualization of displacement over time for each base pair
figure;
hold on;
for n = 2:numBases-1
plot(tspan, x(n, :));
end
xlabel('Time');
ylabel('Displacement');
legend(arrayfun(@(n) ['Base ' num2str(n)], 2:numBases-1, 'UniformOutput', false));
title('Displacement of DNA Bases Over Time');
hold off;
The results are visualized using a plot that shows the displacements of each base over time . Key observations from the simulation include :
- Wave Propagation: The initial perturbations lead to wave-like dynamics along the segment, with visible propagation and reflection at the boundaries.
- Damping Effects: The inclusion of damping leads to a gradual reduction in the amplitude of the oscillations, indicating energy dissipation over time.
- Nonlinear Behavior: The nonlinear term influences the response, potentially stabilizing the system against large displacements or leading to complex dynamic patterns.
% 3D plot for displacement
figure;
[X, T] = meshgrid(1:numBases, tspan);
surf(X', T', x);
xlabel('Base Pair');
ylabel('Time');
zlabel('Displacement');
title('3D View of DNA Base Displacements');
colormap('jet');
shading interp;
colorbar; % Adds a color bar to indicate displacement magnitude
% Snapshot visualization at a specific time
snapshotTime = 40; % Desired time for the snapshot
[~, snapshotIndex] = min(abs(tspan - snapshotTime)); % Find closest index
snapshotSolution = x(:, snapshotIndex); % Extract displacement at the snapshot time
% Plotting the snapshot
figure;
stem(1:numBases, snapshotSolution, 'filled'); % Discrete plot using stem
title(sprintf('DNA Model Displacement at t = %d seconds', snapshotTime));
xlabel('Base Pair Index');
ylabel('Displacement');
% Time vector for detailed sampling
tDetailed = 0:0.5:50; % Detailed time steps
% 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(tspan, x(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');
colorbar; % Adds a color bar to indicate displacement magnitude
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