# What is Chaos theory: Definition and 77 Discussions

Chaos theory is a branch of mathematics focusing on the study of chaos — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a hurricane in China.Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic. This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, pandemic crisis management, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.

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1. ### A Does the Maximum Lyapunov exponent depend on the eigenvalues?

I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
2. ### 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...
3. ### Robert Jansen (new member)

I posted my comment because I am an amateur (in the nonpejorative sense of the word) student of chaos theory. Straightforward applications of the subject have led to 2 utility patents. Presently interested in the mathematics of self-organizing nonlinear dynamic systems, which is obviously an...
4. ### 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...
5. ### 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
6. ### 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...
7. ### A Unstable sets embedded in a chaotic attractor

I am having a hard time understanding the discussion of chaotic sets on invariant manifolds as given in Chaos in Dynamical Systems by Edward Ott. If the invariant manifold of a particular system contains a chaotic attractor ##A##, then the transverse Lyapunov exponent ##h## will generally...
8. ### A Path between fixed points in a logistic map

Hello, I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, ##f(x) = 4\lambda x(1-x)##. Let me then compare 1,2 and 4 iterations of this map on fixed points. I assume that ##\lambda## is large enough such that two period doublings...
9. ### 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...
10. ### I How to check chaotic system using Lyapunov

Greetings! Hey, can anyone help me? I need an explanation how can Lyapunov help me to check the system weather it is chaotic or not. Let say I have this equation Rossler System Eq.(1) So how can you tell that the system have chaotic behavior or not? Does it depends on parameters? or from...
11. ### 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...
12. ### I Can Euler Integration Simplify Chaotic Systems?

Hey, I have this chaotic system. It is a modified Hamiltonian Chaotic System and it is based on Henon-Heiles chaotic system. So I have this functions (as shown below). I want to know how can I make it as a discrete function. Like, how can I know the value for x dot and y dot. 1. Prefer to know...
13. ### 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...
14. ### 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)...
15. S

### Can subharmonics in a system be also termed as bifurcation?

I think that the existence of subharmonics is also bifurcation.Is that true
16. 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...
17. ### 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...
18. 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?
19. 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.
20. S

### I What is the difference between phase space and state-space?

In state space, the coordinates are the state variables of a system.So,each point in state space represents a specific value of state variables.Thus,state space representation represents the changes in a dynamical system. The state variables are the minimum number of variables which uniquely...
21. 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.
22. ### 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...
23. 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.
24. 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?
25. ### I Calculating Hamiltonian matrix elements in a chaotic system

The system in which I tried to calculate the Hamiltonian matrix was a particle in a stadium (Billiard stadium). And I used the principle where we take a rectangle around the stadium in which the parts outside the stadium have a very high potential V0. We know the wave function of a rectangular...
26. ### Writing a Script - Chaos Theory

Hey everyone, So I'm writing the script for a Youtube show I'm making, and it's pretty heavy on the science/physics side in terms of exposition. So before i jump into what i want to ask let me just briefly explain the story. A Time traveler and his time traveling machine go back 30 years in...
27. ### Defining chaos: expansion entropy

Hunt & Ott 2015, Defining Chaos NB: For a more introductory version, phys.org ran a piece on this article two summers ago This paper was published as a review of the concept of chaos in the journal Chaos for the 25th anniversary of that journal. The abstract is extended with a clearer...
28. ### Chaos and the Lorenz Equations

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...
29. ### Chaos (Non-Linear Dynamics) Driven Damped Pendulum

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...
30. ### 2D Phase portrait - Black hole?

Homework Statement Trajectories around a black hole can be described by ## \frac{d^2u}{d\theta^2} + u = \alpha \epsilon u^2 ##, where ##u = \frac{1}{r}## and ##\theta## is azimuthal angle. (a) By using ##v = \frac{du}{d\theta}##, reduce system to 2D and find fixed points and their stability...
31. ### Lorentz Chaos - The 'Butterfly Effect'

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...
32. ### Chaos Theory and the Prolate Spheroid

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...
33. ### Can't tell if I made a chaotic circuit or if I broke the sim

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...
34. ### Quantum Chaos, Level spacing distr. in integrable system

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...
35. ### Help with research in Chaos theory

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...
36. ### Why are fractals and chaos theory synonymous?

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...
37. ### What Causes Chaos? Can We Predict Its Patterns?

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.
38. ### Exploring Chaos Theory & Statistics: How Do They Intersect?

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...
39. ### Applying Chaos Theory to history

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...
40. ### Chaos Theory and Schrodinger's Cat: A Quirky Connection

Can we categorize the Schrodinger's cat as the chaotic system?:cool:
41. ### Chaos theory vs catastrophe theory

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...
42. ### Schools Studying Chaos Theory & Dynamic Systems: A Guide for Postgraduates

Hi everyone, it's been a while since I visited here. However, I now find myself in need of help as to how to best go about studying something related to chaos theory and dynamic systems in postgraduate school. I'm currently in my third year of physics and my first question would be what...
43. ### Minimal model in Chaos theory

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...
44. ### Prerequisites for Chaos Theory

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...
45. ### Understanding Chaos Theory: Unpredictability in Deterministic Systems

What is chaos theory? I know this is a broad subject, so feel free to direct me to books or links.
46. ### Where are nonlinear dynamics and chaos theory used?

Basically where are nonlinear dynamics and chaos theory used in the real world? Like if someone studies it what type of areas might they find it being useful for? The only example I can seem to think of is stuff like weather/fluids/air resistance/physics. What might be some other more...
47. ### Interested in Chaos Theory, Complex Systems, Nonlinear Systems

As the thread title says I'm interested in Chaos Theory, Complex Systems, and Nonlinear Systems. If I can help it, I'd like to study these at graduate level. My question is what kind and how much mathematics I'm supposed to know if I'm to study these?
48. ### Books on Chaos theory for an engineering graduate?

I'll be self-studying about chaos theory. I am looking for a comprehensive book on chaos theory which also touches upon the mathematical aspects involved in it from scratch .I'm no math graduate.. I hold an undergraduate degree in mechanical engineering.So, my knowledge in math is limited to...
49. ### What to include in a report on Chaos Theory

I have to conduct a physics research report on Chaos Theory. From background reading i understand it to be a very maths based topic so my question is how do i relate chaos theory to physics? What sort of sub topics should i be researching? Any responses greatly appreciated!
50. ### Chaos Theory: Periodic Points

Homework Statement How do I find periodic points of a given function? I'm looking at discrete cases only (iterations of the function). Homework Equations A point is defined to be a periodic point of period n if f^n(x)=x, where f^n(x) is defined recursively as f(f^n-1(x)). [If this...