What is Chaos: Definition and 199 Discussions

The Symbol of Chaos originates from Michael Moorcock's Eternal Champion stories and its dichotomy of Law and Chaos. In them, the Symbol of Chaos comprises eight arrows in a radial pattern. In contrast, the symbol of Law is a single upright arrow. It is also called the Arms of Chaos, the Arrows of Chaos, the Chaos Star, the Chaos Cross, the Star of Discord, the Chaosphere (when depicted as a three-dimensional sphere), or the Symbol of Eight.
Alternative symbols of chaos (owing nothing to Moorcock) include the Sacred Chao and the Five Fingered Hand of Eris of Discordianism.

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  1. V9999

    I Open problems in nonlinear dynamics and Chaos

    What are the remaining open problems and challenges of nonlinear dynamics and chaos?
  2. A

    I Does the Hamilton-Jacobi equation exist for chaotic systems?

    Given a Hamiltonian ##H(\mathbf{q},\mathbf{p})##, in the time-independent Hamilton-Jacobi approach we look for a canonical transformation ##(\mathbf{q},\mathbf{p})\rightarrow(\mathbf{Q},\mathbf{P})## such that the new Hamiltonian is one of the new momenta...
  3. A

    Classical Introductory books to Hamiltonian chaos

    I'm looking for books (or any other reference) to start studying the emergence of Hamiltonian chaos and KAM theory. You know, something that doesn't require a Ph.D in math to understand but is comprehensive enough to give a good understanding of the topic. Added bonus if it has a discussion on...
  4. C

    A N-Body Simulations of Chaos in the Orbits of Trappist System Planets

    As a retired physics professor with a long experience in complex simulation software for high energy physics experiments (e.g. the LHC) I revisited last July the n-body planetary simulations which I taught in an undergraduate physics course during the Spring 2017. It was then that the...
  5. codebpr

    A Can a black hole horizon act as a source of Chaos?

    I was going through this paper where on page 5 they argue that in the given Poincare section: I am a bit confused by this statement. How does the given saddle point correspond to the black hole horizon and is it necessary that it acts as a source of chaos? Any explanation would be truly...
  6. P

    I Does chaos amplify inherent quantum level randomness?

    Does chaos amplify inherent quantum level randomness/uncertainty to macroscopic level?
  7. G

    I Chaos theory has no fine edge -- does it fluctuate?

    Does nature err slightly beyond order into chaos in the context of chaos theory? And I'd like to proffer the idea that the point at which order tips into chaos actually fluctuates. Nothing in nature is absolutely perfect therefore do natural errors at the point in which order tips into chaos...
  8. M

    Dynamical Systems - Chaos: Stability condition for a 2-cycle system

    Hi, (This question is part of the same example as a previous post of mine, but I have a question about a different part of it) I was looking at a question from an exam for a course I am self-teaching. There is a sub-question which asks us to find the values of a parameter for which the 2-cycle...
  9. StenEdeback

    I Good introductory book for chaos theory?

    Hi, I have undergraduate level knowledge about mathematics, quantum physics, and general theory of relativity. Now I am curious about chaos theory, and I would be grateful for suggestions of good introductory books to chaos theory. They may be both introductory and a bit more advanced.Sten Edebäck
  10. MathematicalPhysicist

    A Quantum Chaos: Expert Discussion and Q&A

    I might have a few questions about this topic in the future. So I am just asking if there are any experts on this topic here? Thanks in advance! BTW I asked also in physics stackexchange why they don't have a tag of QC, but you know how that site is...
  11. F

    A Problem calculating arbitrary Polynomial Chaos polynomials using SAMBA

    Hello everyone. I have recently read the following article (which title is SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate...
  12. HansBu

    Periodic and Chaotic Solutions to Chen System/Attractors

    Here is the Chen System I am given the initial condition (t=0) that a particle lies on the xyz-plane at a point (-10,0,35). I was notified that if I plugged in a=40, b=5, and c=30, the trajectory of the particle will be chaotic. On the other hand, if I retained the values of a and c, and...
  13. S

    I Models and theories of laws of physics emerging from chaos?

    Some physicists (like John A Wheeler, Holger B Nielsen or Ilya Prigogine) have proposed that all the laws of physics (including the most fundamental ones) emerged from a primordial chaos (for example, in the case of Wheeler, he proposed that laws of physics emerged from an initial random and...
  14. O

    Problem in solving differential equation

    Hello everyone! I was studying chaotic systems and therefore made some computer simulations in python. I simulated the driven damped anhatmonic oscillator. The problem I am facing is with solving the differential equation for t=0s-200s. I used numpy.linspace(0,200,timesteps) for generate a time...
  15. thaiqi

    I Can we imagine that an electron moves on an orbit of chaos?

    Bohr's hydrogen atom model is outdated facing Schrodinger's wave equation. Now that wave mechanics doesn't use a concept of orbit for the electron in hydrogen atom. But can we suppose the electron is still circling around the atom core, not necessarily in circles or ellipses, but in chaos like a...
  16. S

    B Non-linear Behavior, Chaos to Order

    I often hear about non-linear or chaos having order, and have difficulty grasping the concept. For example, you have weather disorder, how can it bring order? or you have Covid chaos, how can it bring order? Can you give some examples of the concept? What is fact, myth, and misconception...
  17. Isaac0427

    I Exploring Randomness and Chaos in Classical & Quantum Physics

    Note: This question involves both classical and quantum physics, so I didn't know where to put it. I'll start with the coin flip: People often compare electron spins to a coin flip, citing that coin flips are random. I am wondering if that is a true analogy, or just another faulty analogy...
  18. O

    I The Trapping Region of the Lorenz equations

    I was dealing with nonlinear systems of differential equations like the Lorenz equations (https://en.wikipedia.org/wiki/Lorenz_system). Now there is a trapping region of this system defined by the ellipsoid ρx^2+σy^2+σ(z-2ρ)^2<R. I wondered how this region is found and I found out that a...
  19. maxstronge

    Programs Need some insight into switching from social science to physics

    Hello. First of all, I'm really glad to have stumbled across this community. You guys seem amazing, and I can't wait to keep lurking (and hopefully, one day, participating). I'm in a strange situation and I could really use the guidance of some people with experience. If the explanation is...
  20. M

    I Connection between General Relativity & Chaos Theory?

    I am very new to such ideas but was wondering if there is any connection to what I am asking. Taking two events, let's say at opposite ends of the globe. Would even A, only have a potential on event B, if light could travel between these event in the given time frame of these event occurring...
  21. frostysh

    Exploring Chaos & Randomness in Deterministic Dynamics

    For myself is horrible to imagine that indeterministic randomness can appear in deterministic, absolutely known and visible dynamics. I have briefly look on the mathematics behind it and understand almost nothing... Well I have a very basic knowledge in mathematics. There is so-called "Rannou's...
  22. E

    Discontinuities in a Poincare map for a double pendulum

    I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong. The condition for these sections is the standard ##\theta_1 = 0## and ##\dot{\theta}_1 > 0##. Looking at one of the maps, we see that most of the...
  23. m4r35n357

    Damped & Driven Pendulums (in _pure_ Python)

    This is another application of using Taylor recurrences (open access) to solve ODEs to arbitrarily high order (e.g. 10th order in the example invocation). It illustrates use of trigonometric recurrences, rather than the product recurrences in my earlier Lorenz ODE posts. Enjoy! #!/usr/bin/env...
  24. O

    I How to get a Chaos Sequence from this Equation?

    Hi, I have this equation about Rossler system, describe as Eq. 1. Given that the chaotic behavior of the system for parameter values a=b=0.2 and c=5.7. How can I calculate the chaotic sequence for this equation below. The equation also referred from...
  25. E

    A Differential Equation to Difference Equation

    Hi all, I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions: 1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that? 2) Consider Lorenz system...
  26. L

    Impact of chaos on a system's entropy

    From a heuristic standpoint it makes sense that when a system goes from being periodic to chaotic, the occupied volume of the phase space increases (while not violating liouville theorem). Since the volume of phase space is proportional if not equal to the entropy, shouldn’t entropy always...
  27. L

    A Does the unforced quartic oscillator behave chaotically?

    I thought that quartic oscillator is chaotic, but here http://www.scholarpedia.org/article/Duffing_oscillator it seems this is only when there is driving force. It also says that for unforced quartic oscillator we can find equilibria and then determine if equilibria are stable or unstable...
  28. osilmag

    I Exploring Chaos Theory Constants: Beta, Prandtl, and Rayleigh

    One of the constants in chaos theory is symbolically labeled as beta, however I haven't found an official definition. The other constants Prandtl and Rayleigh deal with viscosity and diffusivity so they must be appropriate for the specific situation. Is beta simply a constant that can be changed...
  29. D

    Velocity correlations and molecular chaos

    I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first...
  30. Benhur

    I Deterministic chaos in linear systems

    I was reading a paper about deterministic chaos, where the author had commented that non-linear systems can be affected by this. My question is whether some linear systems can also have chaotic behavior. It is possible? Thank you
  31. Jezza

    I Physical examples of different types of bifurcation

    I have to sit an exam on non linear dynamical systems in a couple of weeks. Something that's been asked in the past is to name physical examples of different types of bifurcation. I've consulted Strogatz's book and the internet to try and find some, but I can't seem to find many (or even any)...
  32. Zafa Pi

    I Chaos like phenomena on a simple metric space?

    Let M = {p, x1, x2, x3, ...} be a metric space with no isolated points. f: M → M is continuous with f(xn) = xn+1, and f(p) = p. We say f separates if ∃ δ > 0, ∋ for any y and z there is some n with |fn(y) - fn(z)| > δ, where fn+1(y) = f(fn(y)). QUESTION: Does f separate?
  33. Giulio Prisco

    I Deterministic chaos and randomness

    The term "deterministic chaos" emphasizes that chaotic behavior is random in-practice but deterministic in-principle. But does this even make sense in a finite universe where computation is physical? There ain't no such thing as a Turing machine with an infinite tape. If computing the behavior...
  34. A. Neumaier

    I Classical chaos and quantum mechanics

    It is the resolution given by my thermal interpretation, and this resolution is valid (independent of the thermal interpretation) even without being accepted. Bell assumes that measurement outcomes follow strictly and with infinite precision Born's rule for a von Neumann measurement. But the...
  35. wolram

    B Can the Solar System's Chaos Lead to Earth's Sudden Demise?

    Can anyone predict where the planets will be in 1 million years time, could the Earth be kicked out of orbit a far quicker way to die than waiting for the sun to go nova?
  36. S

    Difference between bifurcation and chaos

    Chaos is when the waveforms become aperiodic. I think bifurcation is the phenomenon inclusive of chaos and in addition, it is also termed for situations in which the waveforms become n-periodic.Does bifurcation include period-n phenomenon as well as chaos? From period-n it means that still the...
  37. T

    Programs PHD in Computer Science with a concentration in Chaos Theory

    Hi everyone, I am working on a second bachelor's degree in Computer Science, and am hoping to enter a Phd program in fall of 2019. Recently, I have taken an interest in Chaos Theory and was wondering if it is possible to do research in the field in the Computer Science department, or if it is...
  38. T

    I How to identify if a system exhibits chaos?

    How to tell if a non linear equation exhibits chaos? Sorry, I am a beginner on this topic. And my library doesn't have book on this topic. I only read about this from John R Taylor's Mechanics book. I am looking for further resources. TIA
  39. MathematicalPhysicist

    A Chaos, quantum gravity, theory of everything....

    How is chaos incorporated into quantum gravity theories, or in theories that incorporate all the known 4 interactions? I don't believe I've seen a thread where chaos theory is discussed in relation to superstring theories or LQG. I've seen some papers and dissertations on quantum chaos and...
  40. S

    Why is chaos more studied in dc-dc converters compared to other circuits?

    Why is chaos only more studied in dc-dc converters and not in other nonlinear circuits, such as, rectifiers?
  41. S

    What is 'phase space in chaos theory and nonlinear dynamics?

    The term 'phase space' is often used in the study of nonlinear dynamics.What is it.
  42. S

    Is Chaos Predictable or Random?

    Chaos is deterministic behavior.Why is chaos deterministic.Why chaos is not random. Chaos is sensitive dependence on initial conditions,a slight change in initial condition can give rise to totally different trajectories.
  43. Sunny Singh

    I Chaos vs purely exponentially growing systems

    I have just started reading chaos from the MIT OpenCourseWare and the following passage has confused me. "The sensitivity to initial conditions is important to chaos but does not itself differentiate from simple exponential growth, so the aperiodic behavior is also important. In the definition...
  44. S

    What is the best software for the analysis of chaos in electric circuits?

    What is the best software for the analysis of chaos in electric circuits.
  45. S

    I Does chaos exist in circuits with linear elements?

    I have heard that chaos exists in all dynamical systems.Does this mean that chaos exists in circuits with linear elements too?Which software is best for analyzing chaos in electric circuits?
  46. anorlunda

    Chaos in flocks & schools of animals

    A documentary on chaos talked about the macro behavior of flocks of birds and schools of fish. They appear organized, but without central command. The documentary said that this is an example of complex behavior arising from simple rules plus chaos. It compared them to the Mandelbrot set...
  47. Sasho Andonov

    A Lyapunov coefficient for function f(x) = 1/(1+x)

    May I use function f(x) = 1/(1+x) to investigate chaos? I am trying to understand chaos using this function, but things are not going as I expected... Could you please advice me how I can calculate Lyapunov coefficient for this function?
  48. Sasho Andonov

    I like to learn, but I do not understand....

    I would like to attend this forum just to ask some things regarding theory of chaos... I found a lot of books, but still no one of them provided good information to understand chaos... I hope it will not be bothering to all of you...(?)
  49. MarcoJV

    I Please share your chaotic systems....

    Hi. I am an undergrad student taking a course on dynamical systems. Our final assignment is to find and study a dynamic system (not necesarily mechanic, but chaotic, natually). I was wondering if there is experenced people in this community that could help me find an interesting system or...
  50. haushofer

    A Chaos: difference vs differential equation

    Dear all, I have a question concerning chaos. As you may well know, the logistic mapping $$x_{n+1} = rx_n (1-x_n) $$ exhibits chaos, depending on the value of r. This logistic mapping is a reparametrized version of the difference equation $$x_{n+1} = x_n + k x_n (1 - \frac{x_n}{M}) $$...