This equation will change how you see the world
18:39
The logistic map connects fluid convection, neuron firing, the Mandelbrot set and so much more. Fasthosts Techie Test competition is now closed! Learn more about Fasthosts here: Code for interactives is available below...
Animations, coding, interactives in this video by Jonny Hyman ????
Try the code yourself:
References:
James Gleick, Chaos
Steven Strogatz, Nonlinear Dynamics and Chaos
May, R. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).
Robert Shaw, The Dripping Faucet as a Model Chaotic System
Crevier DW, Meister M. Synchronous period-doubling in flicker vision of salamander and man.
J Neurophysiol. 1998 Apr;79(4):1869-78.
Bing Jia, Huaguang Gu, Li Li, Xiaoyan Zhao. Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns Cogn Neurodyn (2012) 6:89–106 DOI 10.1007/s11571-011-9184-7
A Garfinkel, ML Spano, WL Ditto, JN Weiss. Controlling cardiac chaos
Science 28 Aug 1992: Vol. 257, Issue 5074, pp. 1230-1235 DOI: 10.1126/science.1519060
R. M. May, D. M. G. Wishart, J. Bray and R. L. Smith Chaos and the Dynamics of Biological Populations
Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 413, No. 1844, Dynamical Chaos (Sep. 8, 1987), pp. 27-44
Chialvo, D., Gilmour Jr, R. & Jalife, J. Low dimensional chaos in cardiac tissue. Nature 343, 653–657 (1990).
Xujun Ye, Kenshi Sakai. A new modified resource budget model for nonlinear dynamics in citrus production. Chaos, Solitons and Fractals 87 (2016) 51–60
Libchaber, A. & Laroche, C. & Fauve, Stephan. (1982). Period doubling cascade in mercury, a quantitative measurement. 43. 10.1051/jphyslet:01982004307021100.
Special thanks to Patreon Supporters:
Alfred Wallace, Arjun Chakroborty, Bryan Baker, DALE HORNE, Donal Botkin, halyoav, James Knight, Jasper Xin, Joar Wandborg, Lee Redden, Lyvann Ferrusca, Michael Krugman, Pindex, Ron Neal, Sam Lutfi, Tige Thorman, Vincent
Special thanks to:
Henry Reich for feedback on earlier versions of this video
Raquel Nuno for enduring many earlier iterations (including parts she filmed that were replaced)
Dianna Cowern for title suggestions and saying earlier versions weren't good
Heather Zinn Brooks for feedback on an earlier version.
Music from:
What We Discovered A Sound Foundation 1 Seaweed Colored Spirals 4
Busy World Children of Mystery
The Feigenbaum Constant - Numberphile
18:55
Binge on learning at The Great Courses Plus:
The Feigenbaum Constant and Logistic Map - featuring Ben Sparks.
Catch a more in-depth interview with Ben on our Numberphile Podcast:
Ben Sparks:
Random numbers:
Mandelbrot Set:
Logistic Map graph:
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI):
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
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Whats so special about the Mandelbrot Set? - Numberphile
16:53
Featuring Ben Sparks discussing the Mandelbrot Set (and Julia Sets). Catch a more in-depth interview with Ben on our Numberphile Podcast:
More links & stuff in full description below ↓↓↓
More videos with Ben:
Ben Sparks website:
And on Twitter:
Holly Krieger has done a few Mandelbrot videos on Numberphile...
The Mandelbrot Set:
63 and -7/4 are special:
Pi and the Mandelbrot Set:
Fibonacci Numbers hidden in the Mandelbrot Set:
Filled Julia Set:
Ben was using Geogebra software:
Files from this video:
He mentioned the book Chaos by James Gleick:
Golden Spiral T-Shirt available in the Numberphile Teespring store:
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI):
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
And support from Math For America -
NUMBERPHILE
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Chaos: The Science of the Butterfly Effect
12:51
Chaos theory means deterministic systems can be unpredictable. Thanks to LastPass for sponsoring this video. Click here to start using LastPass:
Animations by Prof. Robert Ghrist:
Want to know more about chaos theory and non-linear dynamical systems? Check out:
Butterfly footage courtesy of Phil Torres and The Jungle Diaries:
Solar system, 3-body and printout animations by Jonny Hyman
Some animations made with Universe Sandbox:
Special thanks to Prof. Mason Porter at UCLA who I interviewed for this video.
I have long wanted to make a video about chaos, ever since reading James Gleick's fantastic book, Chaos. I hope this video gives an idea of phase space - a picture of dynamical systems in which each point completely represents the state of the system. For a pendulum, phase space is only 2-dimensional and you can get orbits (in the case of an undamped pendulum) or an inward spiral (in the case of a pendulum with friction). For the Lorenz equations we need three dimensions to show the phase space. The attractor you find for these equations is said to be strange and chaotic because there is no loop, only infinite curves that never intersect. This explains why the motion is so unpredictable - two different initial conditions that are very close together can end up arbitrarily far apart.
Music from The Longest Rest A Sound Foundation Seaweed
The Bayesian Trap
10:37
Bayes' theorem explained with examples and implications for life.
Check out Audible:
Support Veritasium on Patreon:
I didn't say it explicitly in the video, but in my view the Bayesian trap is interpreting events that happen repeatedly as events that happen inevitably. They may be inevitable OR they may simply be the outcome of a series of steps, which likely depend on our behaviour. Yet our expectation of a certain outcome often leads us to behave just as we always have which only ensures that outcome. To escape the Bayesian trap, we must be willing to experiment.
Special thanks to Patreon supporters:
Tony Fadell, Jeff Straathof, Donal Botkin, Zach Mueller, Ron Neal, Nathan Hansen, Saeed Alghamdi
Useful references:
The Signal and the Noise, Nate Silver
The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy, by Sharon Bertsch McGrayne
Bayes' theorem or rule (there are many different versions of the same concept) has fascinated me for a long time due to its uses both in mathematics and statistics, and to solve real world problems. Bayesian inference has been used to crack the Enigma Code and to filter spam email. Bayes has also been used to locate the wreckage from plane crashes deep beneath the sea.
Music from Flourishing Views 3
Maps That Will Change The Way You See The World
11:09
From politics to pop culture to history, maps can teach us exactly what people think and thought. Prepare to be amazed by these top 10 maps that will change how you see the world.
Subscribe for more! ► ◄
Stay updated ► ◄
For copyright queries or general inquiries please get in touch: hello@beamazed.com
Credit:
The Quadratic Formula that will change your life
10:46
In this video I present an unbelievably elegant method to factor out quadratic functions, which gives a nice geometric insight into polynomials. After watching this, you will never use the quadratic formula ever again! This method is pioneered by Prof. Po-Shen Loh at Carnegie Mellon University and has been invented by the Babylonians, but somehow forgotten with time. Enjoy this wonderful adventure into the wonders of mathematics.
Po-Shen Loh's webpage:
Subscribe to my channel: youtube.com/drpeyam
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These are the asteroids to worry about
20:06
Stephen Hawking thought an asteroid impact posed the greatest threat to life on Earth. Thanks to Kiwico for sponsoring this video. For 50% off your first month of any crate, go to
For other potential world ending catastrophes, check out Domain of Science:
Special thanks to:
Prof. Dave Jewitt from UCLA Earth, Planetary, and Space Sciences
Prof. Mark Boslough from Sandia National Labs
Scott Manley:
Ryan Wyatt at Morrison Planetarium
Prof. Amy Mainzer
Alexandr Ivanov for the opening shot of Chelyabinsk Meteor
Maps of Asteroid Impacts —
Time passing animation from Universe Sandbox -
Opposition Effect —
Belskaya, I. N., & Shevchenko, V. G. (2000). Opposition effect of asteroids. Icarus, 147(1), 94-105.
Potentially Hazardous Asteroids —
Perna, D., Barucci, M. A., & Fulchignoni, M. (2013). The near-Earth objects and their potential threat to our planet. The Astronomy and Astrophysics Review, 21(1), 65.
Survey of Potentially Hazardous Asteroids —
Population Vulnerability —
Rumpf, C. M., Lewis, H. G., & Atkinson, P. M. (2017). Population vulnerability models for asteroid impact risk assessment. Meteoritics & Planetary Science, 52(6), 1082-1102.
Size distribution of NEOs —
Trilling, D. E., Valdes, F., Allen, L., James, D., Fuentes, C., Herrera, D., ... & Rajagopal, J. (2017). The size distribution of near-earth objects larger than 10 m. The Astronomical Journal, 154(4), 170.
2020 NEOWISE Data Release —
National Research Council Report—
Board, S. S., & National Research Council. (2010). Defending planet earth: Near-Earth-Object surveys and hazard mitigation strategies. National Academies Press.
Tug Boat —
Schweickart, R. L., Lu, E. T., Hut, P., & Chapman, C. R. (2003). The asteroid tugboat. Scientific American, 289(5), 54-61.
Gravity Tractor 1 —
Lu, E. T., & Love, S. G. (2005). Gravitational tractor for towing asteroids. Nature, 438(7065), 177-178.
Laser Ablation —
Thiry, N., & Vasile, M. (2014). Recent advances in laser ablation modelling for asteroid deflection methods. SPIE Optical Engineering+ Applications, 922608-922608.
Yarakovsky Effect —
DART Mission —
Nuclear 1 —
Ahrens, T. J., & Harris, A. W. (1992). Deflection and fragmentation of near-Earth asteroids. Nature, 360(6403), 429-433.
Nuclear 2 —
Bradley, P. A., Plesko, C. S., Clement, R. R., Conlon, L. M., Weaver, R. P., Guzik, J. A., ... & Huebner, W. F. (2010, January). Challenges of deflecting an asteroid or comet nucleus with a nuclear burst. In AIP Conference Proceedings (Vol. 1208, No. 1, pp. 430-437). American Institute of Physics.
Researched and Written by Petr Lebedev, Jonny Hyman and Derek Muller
3D animations, VFX, SFX, Audio Mixing by Jonny Hyman
2D animation by Ivàn Tello
Intro animation by Nicolas Pratt
With Filming by Raquel Nuno
Music from Stellar Dance Orbit That Notebook What We Discovered Out of Poppies Handwriting
Images and video supplied by Getty Images
The Map of Doom | Apocalypses Ranked
21:32
This follows my journey to find and rank all of the biggest threats to humanity. Grab yourself the Map of Doom poster here:
Link to Veritasium’s Asteroids video video:
This year was the first experience we’ve had of a global disaster affecting every single person on Earth. And also how unprepared society was to deal with it, despite plenty of people giving warnings that this was going to happen at some stage.
But in the midst of all the doom I started to wonder, what other things could threaten humanity, that we are not thinking about? So I made the Map of Doom to list all the threats to humanity in one place.
But just finding them all is not enough. I wanted to find a way of comparing the risks of all of these disaster scenarios. So this video follows my attempt at doing that, and I think I’ve hit on a great way to visually compare all these dangers, which I haven’t seen anyone do before, so hopefully by the end of the video you’ll have a better idea about what the biggest threats are, and how they compare to this pandemic.
#existentialrisk #climatechange #DomainOfScience
--- References ----
Thanks to Kurtis Baute for climate change info:
[1] 10 ways the world is most likely to end, explained by scientists
[2] Fifth of countries at risk of ecosystem collapse, analysis finds
[3] Global Catastrophic Risk
[4] Human Extinction
[5] Global Challenges Foundation (pdf)
[6] Deadliest Earthquake
[7] If all the plankton died
[8] Gamma ray burst
[9] Asteroid Impacts Death Risk
[10] West coast US earthquake probabilities
[11] How many people die of earthquakes per year?
[12] Most deadly Earthquakes in history
[13] Asteroid Impact Frequency and Risks
[14] Supervolcanoes
[15] Average frequency of supervolcanoes
[16] Climate change deaths per year
[17] Deaths due to climate change
[18] Global Carbon Emissions Per Year
[19] Estimating Future Death Tolls
[20] Frequency of Pandemics
[21] Worst pandemics in history
[22] Climate Tipping Points
[23] The richest 1% carbon emissions
[24] People killed by asteroids
[25] The London Beer Flood
--- Posters ----
DFTBA Store:
RedBubble Store:
I have also made posters available for educational use which you can find here:
-- Some Awesome People ---
And many thanks to my $10 supporters on Patreon, you are awesome!
Theodore Chu
Petr Murmak
Sebastian
Eric Epstein
Alex Polo
Kevin Delaney
Reggie Fourmyle
Mark Pickenheim
Join the gang and help support me produce free and high quality science content:
--- My Science Books ----
I also write science books for kids called Professor Astro Cat. You can see them all here:
--- Follow me around the internet ---
--- Credits ---
Music, art, and everything else by Dominic Walliman
Additional music: Higher Kiss by TrakTribe
Sound effects obtained from
Veritasium: this equation will change how you see the world, a John Gribbin plagiarism?
4:24
If you are going to use the same examples as someone else word for word in your video, you might want to give some credit.
FORSEN REACTS to This equation will change how you see the world
21:29
Original Video :
FORSEN TWITCH :
You own content in this video and would like it removed? please contact me at:
g4minside@gmail.com
Simulating the Logistic Map in Matlab
16:29
This video shows how simple it is to simulate discrete-time dynamical systems, such as the Logistic Map, in Matlab.
Esta ecuación cambiará tu modo de ver el mundo
17:23
La aplicación logística conecta la convección de fluidos, la activación de neuronas, el conjunto de Mandelbrot y mucho más. Bienvenidos al Caos. ???? SUSCRÍBETE para ver todos nuestros videos:
Una vez que descubras esta ecuación nada volverá a ser igual. Entonces ¿ya conoces la constante de Feigenbaum?
Al mirar el diagrama de bifurcación, puedes notar que se parece a un fractal.
Las características a gran escala parecen repetirse en escalas cada vez más pequeñas. Y, efectivamente, si nos acercamos, veremos que, de hecho, es un fractal.
Podría decirse que el fractal más famoso es el conjunto de Mandelbrot.
Es realmente la ecuación más bella y por eso hará que cambies tu forma de ver el mundo.
Video Original en Inglés de este video Esta ecuación cambiará tu modo de ver el mundo: This equation will change how you see the world
Animaciones, codificación, e interactivos en este video de Jonny Hyman ????
Prueba el código tú mismo:
Referencias:
James Gleick, Chaos
Steven Strogatz, Dinámica No Lineal y Caos
May, R. Modelos matemáticos simples con una dinámica muy complicada. Nature 261, 459-467 (1976).
Robert Shaw, El grifo que gotea como un modelo de sistema caótico
Crevier DW, Meister M. Periodo de duplicación sincrónico en la visión parpadeante de la salamandra y el hombre.
J Neurophysiol. 1998 Apr;79(4):1869-78.
Bing Jia, Huaguang Gu, Li Li, Xiaoyan Zhao. Dinámica de la bifurcación del período doble al caos en los patrones de disparo neural espontáneo Cogn Neurodyn (2012) 6:89-106 DOI 10.1007/s11571-011-9184-7
Un Garfinkel, ML Spano, WL Ditto, JN Weiss. Controlando el caos cardíaco
Ciencia 28 de agosto de 1992: Vol. 257, Número 5074, pp. 1230-1235 DOI: 10.1126/science.1519060
R. M. May, D. M. G. Wishart, J. Bray y R. L. Smith El caos y la dinámica de las poblaciones biológicas
Fuente: Actas de la Sociedad Real de Londres. Serie A, Ciencias Matemáticas y Físicas, Vol. 413, No. 1844, Caos Dinámico (8 de septiembre de 1987), pp. 27-44
Chialvo, D., Gilmour Jr, R. y Jalife, J. Caos de baja dimensión en el tejido cardíaco. Nature 343, 653-657 (1990).
Xujun Ye, Kenshi Sakai. Un nuevo modelo modificado de presupuesto de recursos para la dinámica no lineal en la producción de cítricos. Caos, solitones y fractales 87 (2016) 51-60
Libchaber, A. & Laroche, C. & Fauve, Stephan. (1982). Periodo de duplicación de cascada en el mercurio, una medida cuantitativa.
Esta ecuación cambiará tu modo de ver el mundo
Un agradecimiento especial a:
Henry Reych por sus comentarios sobre las versiones anteriores de este vídeo
Raquel Nuno por soportar muchas iteraciones anteriores (incluyendo partes que ella filmó y que fueron reemplazadas)
Dianna Cowern por sugerencias de títulos y diciendo que las versiones anteriores no eran buenas
Heather Zinn Brooks por los comentarios sobre una versión anterior.
Música de:
What We Discovered A Sound Foundation 1 Seaweed Colored Spirals 4
Busy World Children of Mystery”
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Servicios de doblaje:
Twitter: @Unilingo_HQ
Traductora: Paula Salomone
Voz de doblaje: Pato Lago, Diego Rivas,
Ingeniero de sonido: Gastón Adriel Álvarez
Edición y post-producción de video: Juan Caille Tornquist
------------------------------------------------------------------------
#veritasiumenespañol
What are Logistic Maps
10:40
Logistic Maps interactive exploration:
Logistic Maps on Wikipedia:
What Is The Coastline Paradox?
2:16
How long is the coastline of Australia? One estimate is that it's about 12,500 km long. However the CIA world factbook puts the figure at more than double this, at over 25,700 km. How can there exist such different estimates for the same length of coastline? Well this is called the coastline paradox. Your estimate of how long the coastline is depends on the length of your measuring stick - the shorter the measuring stick the more detail you can capture and therefore the longer the coastline will be.
The Logistic Map - Fractal Zoom
2:25
This video is a little demonstration that the Logistic Map equation is in the same family as The Mandelbrot Set. The Logistic Equation & Map come up a lot of places, such as biology where it can model the growth of animal populations. It also has many other uses. The Logistic Map, is the discrete version of the Logistic Equation (it is the recurrence relation.) If you are interested in this equation, how it appears in real life, and how it relates to the Mandelbrot, please see this great video by Veritasium:
This equation will change how you see the world
Equation: z = c*z*(1-z). [The initial value of z is 0.5].
Thank-you to my supporters on Patreon.
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ADS : Vol 4 : Chapter 4.2 : The Logistic Map
9:28
The logistic map is a great example of how one transitions from simple to chaotic dynamics. In this case, there's a curious pattern of period doubling bifurcations.
Lecture - 9 The Logistic Map and Period doubling
55:25
Lecture Series on Chaos, Fractals and Dynamical Systems by Prof.S.Banerjee,Department of Electrical Engineering, IIT Kharagpur. For more details on NPTEL visit
The Logistic Equation
13:27
MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course:
Instructor: Gilbert Strang
When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state.
License: Creative Commons BY-NC-SA
More information at
More courses at
Sonification of the Logistic Map, LMap to sinewave FM
5:26
More experiments with the module can be found here:
This experiment was inspired by Veritasium's recent video on the logistic function:
I sample the logistic map and use the result to modulate a sinewave oscillator. I also mess about a bit with the setup as it goes on.
I found that a module with this function exists for VCV rack, it is made by m80
I made the high res spectrogram using After Effects, to a quite high spatial and temporal (120px per second) resolution visualisation
Covid 19: Flattening of the curve, chaos, and fractals
8:13
This video is a follow-up of the previous video on the exponential aspect of a virus propagation ( In the present video, I present the use of the logistic map equation to represent the flattening of the curve of a virus propagation. I then present the chaotic and fractal aspects of the logistic map equation and link it to the famous fractal image known as the Mandelbrot set using LabVIEW VIs.
The link to the code used in the video is:
A version of the code that includes only the bifurcation diagrams is available at: The code also includes a compiled executable. The executable can only be run on Windows and it requires the LabVIEW 2016 run-time engine that can be downloaded for free from:
Licenses for images:
The climat change image is from kai Stachowiak under public domain license CC0 1.0 Universal (CC0 1.0)
The top left lemming image is from Ansgar Walk under Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) license (
The bottom right lemming image is from Argus Fin under a public domain license (
Logistic map zoom
2:57
Zoom into the logistic map bifurcation diagram from which the Feigenbaum constant, δ = 4.669 201…, is calculated indicating the rate at which branches of the tree split (How to calculate .. is at ).
The graph is drawn by repeating the simple calculation: a * x * (1 - x)
which gives a new value of x. More than 1000 successive values of x are plotted vertically, against a set value of a, plotted horizontally.
The Feigenbaum constant, δ, is a universal constant (= 4.669 201 609 102 990 671 853 203 821 578) and is deeply connected to many natural processes that display chaotic dynamics.
Further information at
The Sound of the Logistic Map
1:12
In case you ever wondered what the logistic map sounded like!
The logistic map is an excellent example of bifurcation and the period-doubling route to chaos, as well as the concept of sensitive dependence on initial conditions (SDIC). It reveals how even the simplest non-linear system can create great complexity. Used as a basic population model, the logistic equation is just the product of linear growth, X, and linear decay, (1 - X), scaled by a driving factor, alpha. The map is the iteration X(i+1) = alpha * X(i) * (1 - X(i)). For small alpha (which controls the rate of birth and death), the population reaches a stable, steady state value - i.e. birth rate and death rate have attained equilibrium. Eventually as alpha is increased the population suddenly starts to oscillate between two stable values, regardless of the initial number. As alpha is increased further still, each stable value itself bifurcates (splits into two, period-doubling) to give overall four stable values, then eight, sixteen etc. until there are infinite - chaos! This happens at an ever increasing rate, which turns out to be governed by a newly discovered universal constant known as the Feigenbaum constant, d = 4.6692.... In words, the next period doubling happens over a distance in alpha a factor of 4.6692 sooner than the previous period doubling. One implication of this is that once period-doubling starts, it exponentially accelerates to chaos (in the driving parameter space). Such behaviour has also been observed in a range of physical systems, for example convection cells in a closed volume, transition from laminar to turbulent flow, dripping faucets....
It is very easy to hear the difference between the regions of chaos (fuzzy in appearance) and the ordered regions which can re-appear momentarily, despite the system having already entered a chaotic state. Within the chaotic regions there exists a sensitivity to initial conditions. What this means is that if you changed the initial value of the system by even the tiniest amount, for the same number of iterations you will get a very different result! This is NOT due to randomness, but rather the complexity of the system. This chaos is actually deterministic, meaning that it can be exactly repeated / computed at any time given only the rules of the system and the initial conditions. It is important to consider this in reality, just because something appears too complex to describe does not mean it is random. There may be a surprisingly simple set of rules behind it all, and it just seems complex because the rules have played out long enough and involved a large enough number of entities. Whether this applies to the entire Universe or not is a question many deep thinkers have pondered on since the dawn of time, as the consequences are profound when it comes to concepts such as free will and the existence of God....
Coded in LabVIEW (still images and audio rendering). Video put together using Hitfilm Express.
This particular output used a sampling rate of 5760 Hz and 192 samples per alpha (to give 30 fps matching with the video frame-rate). Alpha was incremented such that it represented one pixel at 1080p resolution. This led to a decent length video with a frequency not unbearably high or inaudibly low!
Mandelbrot Sound seizure warning
1:41
Built with JMP software, following contour 24 to generate audio samples.
Thanks Keshikan for the font!
Project 1: Logistic Map | Lecture 11 | Numerical Methods for Engineers
16:38
Getting ready to do a numerical calculation of the logistic map. Let's first learn a little theory.
Join me on Coursera:
Lecture notes at
Subscribe to my channel:
Bailout CCR Series #3 | Mark Powell, Edoardo Pavia and Paul Toomer
1:23:41
Our final video of the 3 part Bailout rebreather series is here! Part 3 will focus on how training agencies and instructors plan to handle the training of dual rebreathers.
This time Joe is hosting the following guests:
???? Mark Powell - TDI Instructor Trainer and author of Deco for Divers
???? Edoardo Pavia - IANTD Instructor Trainer and TDI Cavern Instructor at dive center Sea Dweller Divers and our brand ambassador
???? Paul Vincent Toomer - Co-owner at Dive RAID International and our brand ambassador
More info:
#divesoft #rebreather #liberty #ccrliberty
Logistic Map Orbits
4:01
Logistic map, part 1: period doubling route to chaos
17:18
The logistic map is a simple discrete model of population growth with very complicated dynamics. It depends on a growth rate parameter r. We consider the dynamics at various values of the parameter and find that there’s a branch of stable fixed points which bifurcates into stable attractor cycles of period 2, 4, 8, 16, .... The period-doubling cascade. The bifurcation diagram shows chaos intermingled with periodic windows.
► Next, the bifurcation diagram and self-similarity
► Additional background
Introduction to mappings
Logistic equation (1D ODE)
Lorenz map on strange attractor
Lorenz equations introduction
Definitions of chaos and attractor
Lyapunov exponents to quantify chaos
► Robert May's 1976 article introducing the logistic map (PDF)
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe
► Follow me on Twitter
► Course lecture notes (PDF)
► Advanced lecture on maps from another course of mine
References:
Steven Strogatz, Nonlinear Dynamics and Chaos, Chapter 10: One-Dimensional Maps
period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Poincare map largest Liapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator Duffing oscillator nonlinear oscillators nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation Liapunov gradient systems Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices topology Verhulst Oscillators Synchrony Torus friends on track roller racer dynamics on torus Lorenz equations chaotic strange attractor convection chaos chaotic
#NonlinearDynamics #DynamicalSystems #PopulationGrowth #LogisticMap #PeriodDoubling #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #LyapunovExponent #Lyapunov #Liapunov #Oscillators #Synchrony #Torus #Bifurcation #Hopf #HopfBifurcation #NonlinearOscillators #AveragingTheory #LimitCycle #Oscillations #nullclines #RelaxationOscillations #VanDerPol #VanDerPolOscillator #LimitCycles #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics #Poincare #mathematicians #maths #mathsmemes #math4life #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian
Logistic map, part 2: bifurcation diagram and self-similarity
15:19
The logistic map has an iconic bifurcation diagram, showing chaotic attractors intermingled with periodic windows, the largest being the period-3 window. We numerically explore trajectories via cobweb diagrams and properties of the bifurcation diagram, like self-similarity, and hint at how it arises.
► Next, the bifurcation that leads to period doubling
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► Logistic map introduction
► Additional background
Introduction to mappings
Logistic equation (1D ODE)
Lorenz map on strange attractor
Lorenz equations introduction
Definitions of chaos and attractor
Lyapunov exponents to quantify chaos
► Robert May's 1976 article introducing the logistic map (PDF)
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe
► Follow me on Twitter
► Course lecture notes (PDF)
► Advanced lecture on maps from another of my courses
References:
Steven Strogatz, Nonlinear Dynamics and Chaos, Chapter 10: One-Dimensional Maps
period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Poincare map largest Liapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator Duffing oscillator nonlinear oscillators nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation Liapunov gradient systems Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices topology Verhulst Oscillators Synchrony Torus friends on track roller racer dynamics on torus Lorenz equations chaotic strange attractor convection chaos chaotic
#NonlinearDynamics #DynamicalSystems #Bifurcation #LogisticMap #PeriodDoubling #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #LyapunovExponent #Lyapunov #Liapunov #Oscillators #Synchrony #Torus #Hopf #HopfBifurcation #NonlinearOscillators #AveragingTheory #LimitCycle #Oscillations #nullclines #RelaxationOscillations #VanDerPol #VanDerPolOscillator #LimitCycles #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics #Poincare #mathematicians #maths #mathsmemes #math4life #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian
The Logistic Map
10:18
Lecture of Physics with Andrés Aragoneses (Eastern Washington University) on iterative maps and the logistic map.
Introduction to Complexity: Logistic Map
11:11
These are videos from the Introduction to Complexity course hosted on Complexity Explorer. You will learn about the tools used by scientists to understand complex systems. The topics you'll learn about include dynamics, chaos, fractals, information theory, self-organization, agent-based modeling, and networks. You’ll also get a sense of how these topics fit together to help explain how complexity arises and evolves in nature, society, and technology.
This course was developed by professor Melanie Mitchell, and is based on her book Complexity: A Guided Tour.
Chaos #2 - Logistic Map
33:52
Chaos, Strange Attractor, Feigenbaum Constant, Cobweb Map, Bifurcation, Feigenbaum diagram,
Logistic Map
9:58
To understand the iterations and nature of simple Quadratic equation in Logistic Map .
MSN 514 - Lecture 14: Logistic map
50:06
1D maps, Logistic map, Period doubling, Chaos, Liapunov exponent, Rössler attractor, Feigenbaum constant, Universality, Renormalization
MAE5790-19 One dimensional maps
1:14:35
Logistic map: a simple mathematical model with very complicated dynamics. Influential article by Robert May. Numerical results: Fixed points. Cycles of period 2, 4, 8, 16, .... The period-doubling route to chaos. An icon of chaos: The orbit diagram. Chaos intermingled with periodic windows. Period-3 window. Analytical results: Fixed points and their stability. Flip bifurcation (eigenvalue = --1) at period doubling. Period-2 points and their stability.
Reading: Strogatz, Nonlinear Dynamics and Chaos, Sections 10.0-10.2.
Fractal Music - Image Sonifications - Mandelbrot Set
2:59
Click here if you want to subscribe:
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Sonification of the Mandelbrot set fractal. Composed by Gustavo Díaz-Jerez.
Procedure:
- X axis of image mapped to time, in seconds.
- Y axis of image mapped to frequency (27.5 - 4163Hz, continuous, exponential scale, using sinusoids).
- Brightness of image mapped to dynamic range. Black (0,0,0) = silence (-INF dB). White (255,255,255) = Max (0 dB)
The right side shows a spectrogram and a bar diagram of the sound. The bottom shows the wave form.
Postprocessing: bass frequencies boost.
Notice that this is not somehow inspired or based on the image. It IS how the image translates to sound for the given paramenters.
More information and sheet music available at
Introduction to Complexity: Logistic Map Quiz Solution
43
These are videos from the Introduction to Complexity course hosted on Complexity Explorer. You will learn about the tools used by scientists to understand complex systems. The topics you'll learn about include dynamics, chaos, fractals, information theory, self-organization, agent-based modeling, and networks. You’ll also get a sense of how these topics fit together to help explain how complexity arises and evolves in nature, society, and technology.
This course was developed by professor Melanie Mitchell, and is based on her book Complexity: A Guided Tour.
Nonlinear Dynamics: Feigenbaum and Universality
5:57
These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer.org) taught by Prof. Liz Bradley. These videos provide a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language (C, Java, Matlab, R, ...).
After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.
In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.
MAE5790-20 Universal aspects of period doubling
1:11:56
Exploring the logistic map and period doubling with online applets. Interactive cobweb diagrams. Interactive orbit diagram. Zooming in to see the periodic windows. Self-similar fractal structure: each periodic window contains miniature copies of the whole orbit diagram. Smooth curves running the orbit diagram: supertracks. How are periodic windows born? Example: Birth of period three. Tangent bifurcation. The mechanism underlying the fractal structure.
Introduction to Mitchell Feigenbaum's work on universality, what he found, and why it matters. Testable predictions about periodic doubling in physical and chemical systems. Sine map vs. logistic map. The universal scaling constants alpha and delta.
Reading: Strogatz, Nonlinear Dynamics and Chaos, Sections 10.2¬--10.4, 10.6.
Logistic Differential Equation
10:52
Solving Logistic Differential Equation,
Cover up for partial fractions (why and how it works):
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How Learning Ten Equations Can Improve Your Life - David Sumpter
54:09
Mathematics has a lot going for it, but David Sumpter argues that it can not only provide you with endless YouTube recommendations, and even make you rich, but it can make you a better person.
Our latest Oxford Mathematics Public Lecture.
Oxford Mathematics Public Lectures are generously supported by XTX Markets
This Simple Equation Will CHANGE the Way You See the World!
2:35
E + R = O. As the title says, this equation may look simple, but it could change your life. It's all about how we react to what happens to us in order to take control of our lives.
#ShortSweetSat
14. Bugs, The Logistic Map
42:58
Intro to nonlinear dynamics via a discrete mapping.
This is just one of 61 Lectures covering a full one-year Course in Computational Physics previously taught by Rubin Landau at Oregon State University. (The number refers to the chapter and section in reference 1 below.)
There are video modules (Flash), slides, codes, quizzes, and all the lectures at:
The YouTube playlist with all videos is at:
References:
1. Computational Physics, 3rd Ed, Problem Solving with Python, Rubin H Landau, Manuel J Paez & Cristian Bordeianu (deceased), © Wiley 2015.
2. A Survey of Computational Physics with Java, Rubin H Landau, Manuel J Paez & Cristian Bordeianu (deceased), © Princeton University Press, 2008.
Iterates of the Logistic Map in 3D
16
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.
The logistic map is defined by the iteration xn+1=rxn(1-xn). See the sequence of values of xn reached from a grid of initial values. For small r, the behavior is simple. But as r increases, the system exhibits a series of bifurcations, eventually showing complex behavior.
By Stephen Wolfram
Nonlinear Dynamics: Matlab Logistic Map Code
3:10
These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer.org) taught by Prof. Liz Bradley. These videos provide a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language (C, Java, Matlab, R, ...).
After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.
In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.
models for population growth - logistic growth, logistic differential equations, & carrying capacity
14:25
in this video i cover logistic growth, logistic differential equations, & how to identify carrying capacity. thanks for watching! @mathwithyosh
Logistic Bifurcation Animation
32
This video shows trajectories of the Logistic Map for parameter values of r ranging from 0 to 4. As the trajectory moves, the stable equilibrium values are plotted on the bifurcation diagram to the right. From this, we can see how the famous period doubling bifurcation is traced out, which eventually leads to chaotic behavior.
Logistic map, part 3: bifurcation point analysis | bottlenecks in maps, intermittency chaos
20:35
The logistic map bifurcation diagram can be analytically explained. We calculate the value of first few bifurcation points, where the non-zero fixed point emerges and stable cycles of period 2 and 4 emerge via a period-doubling bifurcation (or flip bifurcation). We see a map version of fixed point ghosts and bottlenecks, regions of high residence time, related to the intermittency route to chaos.
► Next, the universality of features in the logistic map
► Logistic map
Introduction
Bifurcation diagram
► Additional background
Introduction to mappings
Logistic equation (1D ODE)
Lorenz map on strange attractor
Lorenz equations introduction
Definitions of chaos and attractor
► Ghosts and bottlenecks
In 1D differential equations
In 2D differential equations
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe
► Follow me on Twitter
► Course lecture notes (PDF)
► Advanced lecture on maps from another of my courses
► Robert May's 1976 article introducing the logistic map (PDF)
References:
Steven Strogatz, Nonlinear Dynamics and Chaos, Chapter 10: One-Dimensional Maps
intermittent period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Poincare map largest Liapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator Duffing oscillator nonlinear oscillators nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation Liapunov gradient systems Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices topology Verhulst Oscillators Synchrony Torus friends on track roller racer dynamics on torus Lorenz equations chaotic strange attractor convection chaos chaotic
#NonlinearDynamics #DynamicalSystems #Bifurcation #LogisticMap #PeriodDoubling #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #LyapunovExponent #Lyapunov #Liapunov #Oscillators #Synchrony #Torus #Hopf #HopfBifurcation #NonlinearOscillators #AveragingTheory #LimitCycle #Oscillations #nullclines #RelaxationOscillations #VanDerPol #VanDerPolOscillator #LimitCycles #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics #Poincare #mathematicians #maths #mathsmemes #math4life #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian
Logistic Map - algorithms to investigate the logistic equation and logistic map
13:38
Two algorithms are presented here
The first can be used to investigate the logistic equation (x_(n+1)=r x_n(1-xn) for different fixed values of r
The second is to generate the logistic map where long term values for x are plotted as a function of r.
Both algorithms describe how to generate x,y data that can be plotted with Excel or Gnuplot
ADS : Vol 4 : Chapter 5.2 : The Henon Map
7:01
Here's a classic -- the Henon map! It's chaotic! What's it good for?
Hey, it's chaotic!