Chaos Definition and 188 Threads

Suppose we are talking about a purely classical phenomena (OK, nothing is purely classical, but suppose we consider quantum effects as insignificant, that is, we ignore them). In this context, I came across someone talking about "a particle in chaotic continuous motion as the particle is...

Dear all, I've never really understood how exactly Liouville's theorem about time-conservation of phase space volume can be reconciled with the second law of thermodynamics. Recently I came across this popular article, http://www.necsi.edu/projects/baranger/cce.pdf "Chaos, Complexity, and...

Chaos could appear in the system if there is some nonlinearity. My question is how to explain that there is no chaos in constant acceletating system s(t)=v_0t+\frac{at^2}{2} when equation is nonlinear? Why is important only that difference and differential equation be nonlinear. It confusing me.

Hi, I was curious to know what the perception of quantum chaos is. It seems a bit on the fringe, but at the same time it seems interesting. An adviser I'm considering working for works on both solid state physics applications and does some theoretical work in quantum chaos, so I'm wondering if...

The universe is random or chaotic?

I work with an electromagnetic molecule trap, and I'd like to determine which orbits are chaotic. To this end, I intend to study the evolution of a perturbation on a trajectory with time. I'd like to compute something called the fast lyapunov indicator for various trajectories y(t), where I...

Suppose you want to solve the Navier-Stokes equation for an incompressible turbulent flow. This is, in principle, possible. However, for higher Reynolds number, this isn't really feasible since the smallest eddy sizes are really small. First question 1) Why would this prevent you from...

Can someone please suggest me a good reference for studying the duffing oscillator?

I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

Homework Statement I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by ## \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = fcos(\omega t)## Lyapunov exponent is ## \lambda ## in the equation ## \delta x(t) = \delta x_0 e^{\lambda t} ## The attempt at a...

Hi. As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular...

Here is the table of contents of Nonlinear Dynamics and Chaos (by Strogatz) Overview Flows on the Line Bifurcations Flows on the Circle Linear Systems Phase Plane Limit Cycles Bifurcations Revisited Lorenz Equations One-Dimensional Maps Fractals Strange Attractors Last quarter, there was a...

What are the topics which are been perceived by researchers around the World on the topic of Quantum Chaos? I know my questions is not specific, but I do not have any idea to ask about anything specific.

Take the Lorenz equations x'=σ(y-x) y'=rx-y-xz z'=xy-bz with σ=10, b=8/3 and r=28 as a typical example of chaos (I am using primes to indicate total time derivatives in this post). A basic property of a chaotic system (where the flow in phase space is a strange attractor) is that if you pick...

Hi, We normally use a simple symmetry argument to show that the probability of each outcome of a throw of a fair, cube-shaped die is 1/6. However, is it possible to actually model the physics of the throw and show that the probabilities are 1/6? Since this is classical physics, the outcome can...

I want to investigate the phenomenon of Chaos in the form of how its driving amplitude affects _____, in a driven, damped pendulum, using a computer simulation given. Initially I was looking at 'degree of chaos' for the dependent variable - to measure this I wanted to use the Lyapunov...

Homework Statement Given the lorentz system for ##\sigma=10, b = \frac{8}{3}, r = 28##, and ##x(t)## from the first lorentz system, show that we can solve for y(t) and z(t) for the modified lorentz system by finding ##\dot E##.[/B] Homework EquationsThe Attempt at a Solution I have found...

Rugby balls and American footballs are prolate spheroids. As such, their bounce patterns seem sporadic - they tend to bounce to different heights and in different directions even when they appear to hit the ground with a constant angle, speed, and spin. Does this behaviour relate to chaos...

What textbooks would you recommend for self studying Nonlinear Dynamics? I am a undergraduate junior who will be doing research on nonlinearity of spiking neurons. I have taken courses on ODE, vector calculus, probability, statistics, and linear algebra.

So, I was playing around with a couple of voltage multiplier circuits a few months ago, and while optimizing one design, I came up with a pretty neat (not to brag) way of converting a sine wave to a square wave by using transformers in a completely different way than normally. A little while...

I am solving a nonlinear ODE in the form of Newton's Second Law. In the equation, there is a Heaviside Theta Function of the function which I am solving (##\theta (x(t)##). Since it is quite troublesome to have both the left side of the ODE and the imput of the ODE to contain function of unknown...

Homework Statement (a): Show the lagrangian derivative in phase space (b)i: Show how the phase space evolves over time and how they converge (b)ii: Find the fixed points and stability and sketch phase diagram (c)i: Find fixed points and stability (c)ii: Show stable limit cycles exist for T>ga...

Found this question while think of determine&random. If a system if very complex, it may looks like random. Even GUT is found, it is still impossible to tell what a determined system will be after a long period because of Heisenberg's uncertainty principle. An error in initial conditions, even...

Hello all, For an undergraduate essay, I am studying the development of quantum chaos in a 1D spin 1/2 chain (my main source paper can be found here:http://scitation.aip.org/content/aapt/journal/ajp/80/3/10.1119/1.3671068). One of the main tools used to distinguish chaotic from non chaotic...

Homework Statement The figure below shows the path of a particle governed by the Lorenz equations with r = 28, σ = 10, b = 8/3. The x'es and boxes show points where the path crosses the plane z = r − 2σ > 0. (a) Which indicator shows a decreasing z and which shows an increasing z? (b) Show...

So I get the gist of Chaos Theory, and I read through another thread on this forum stating that Chaos does exist, which quotes Chaos Theory. Does this not seem to be a tad counter intuitive ? Chaos theory simply states not knowing the starting attributes of a mathematical system that you can't...

Hello I'm an undergraduate who is currently doing research in Chaos theroy. So far I've built a double pendulum and simulated it on my computer using mathematica. I'm going to use a tracker software to track the empirical motion of the pendulum and try to match it with the my theoretical...

Numbers generated by a pseudo-random process will eventually repeat, which is how PR processes can be differentiated from a truly "random" process. Some chaotic processes follow closed trajectories in state space; i.e., their attractors are periodic, so number sequences generated by those...

I'm doing a presentation in a few weeks on fractals and chaos theory. To me, their link is more intuitive than mathematically/physically sound, and I'm really struggling to put the link into words. I've tried googling it, but no where seems to give a satisfactory explanation of the link...

Can we consider +inf as an attractor for a map for which trajectories emanating from any point in the state space tends to +inf.

I was reading this post on livescience web site It's saying that the #1 unsolved mystery in physics is that we are not sure if there is order in chaos I was confused since I thought that chaos only exist due to the lake of infinitely precise calculations, but if there is an all seeing eye...

Well here's my question: what does really "create" chaos?jump between attractions?Can one sit and produce a function which will determine the chaos? P.S my question migh seem a little stupid just because I'm still trying to get a general sense of everything. Thanks.

I have a list of chaos numbers Average each 10 results as display below Sum each 10 results as display below Average the 30 results as displayed below here’s my question: I believe that I can see as the average goes up we have larger sums So I think I can predict something from the last...

Testing for chaos in data I have data for 3 variables ,each with respect to the discrete time values. How do I check for the existence of chaos for this discrete 3D system?(I don't have the analytic eqs.,just the data.) MY IDEAS ON CHECKING FOR CHAOS FROM DATA:(which of these are feasible...

Good afternoon, I've come here to hopefully resolve an issue that cropped up in a debate with a friend. We were discussing weather patterns and chaos theory was brought up. My understanding of chaos theory is that it is a way of explaining behavior of certain deterministic systems aka...

Let's say you wanted to determine what day in a certain amount of time had been the most influential in our lives today. I theorized that whatever the time period, the first day in that time period would automatically be the most influential day. I though this because as you go farther back in...

Homework Statement a) Find a mechanical system that is approximately governed by \dot{x}=sin(x) b) Using your physical intuition, explain why it now becomes obvious that x*=0 is an unstable fixed point and x*=\pi is stable. Homework Equations \dot{x}=sin(x) (?) The Attempt at a Solution...

Hi all, I'm trying to self-learn about chaos, fractal, or anything that correspondence to random analysis (maybe with some material from statistical physics). Anyone know what the best textbook for these fields?

Homework Statement Show that there is a certain ellipsoidal region E of the form rx2 + σy2 + σ(z-2r)2 ≤ C such that all trajectories of the Lorenz equations eventually enter E and stay in there forever. Homework Equations Lorenz Equations: \dot{x} = \sigma (y - x) \dot{y} = rx -...

I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet the requirements for either in my last year and all of them align better with my interests...

Can we categorize the Schrodinger's cat as the chaotic system?:cool:

Hal Haggard's ILQGS talk should be quite interesting. The plan is to have the slides PDF uploaded sometime Monday at http://bohr.physics.berkeley.edu/hal/pubs/Talks/ILQGS2013/haggard021213.pdf It's potentially helpful to let people look at the slides a day in advance of the Tuesday 12 February...

Hello everyone, I would like to know if Bohm Mechanics (and weak measurement) could be useful in understanding Quantum Chaos theories? I'm currently in search of a PhD in Foundation of QM and I would like to know if doing a project that combine the trajectories coming from Bohm Mechanics with...

I am taking a course in non-linear dynamics and I read that Lorenz systems exhibit 'chaotic behaviour' and the spruce-budworm non-linear D.E follows the criteria of 'catastrophe theory'.Is there a difference between these 2 theories?If yes,does this mean that small changes in the spruce-budworm...

I am going to be simulating damped driven oscillators for my project and I was wondering if someone could give me a definitive answer on the matter. I know MATLAB is more of a 'tool' than a language so I'm thinking the maths behind damped driven oscillators might be easier to implement into...

I am reading a book on chaos theory by Robert M.May.There is a reference to a 'minimal model of a ecosystem' through which the author describes hysteresis and bistable states. Q. What does a minimal model mean in mathematical terms,&also intuitively?Is it a concept of topology referring to...

I've taken an interest in chaos theory of late, but only from a casual standpoint. What are the math requirements for truly understanding chaos theory? What about physics? Any other knowledge required? I will earn my BS in computer science and minor in mathematics in spring 2013. I've...

For systems under an energy constraint (a conserved Hamiltonian that is a function of the q's and p's in generalized coordinates), is it possible for it to behave chaotically (chaos by its dynamical definition), or will it always have some equilibrium it tends towards?

What is chaos theory? I know this is a broad subject, so feel free to direct me to books or links.

Does " Chaos " really exist ? Does " Chaos " really exist, i think not, there is an understanding of every situation, no matter how small, and a Digital view of this, is by way no means a true view, this is the only reason " Chaos " exists, because "Digital Fails to understand and gives false...