What is Nonlinear dynamics: Definition and 34 Discussions
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.
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...
Hi, my name is Vini.
I am Graduated in Physics. I worked as a visiting researcher at the Institute of Complex Systems (ISC) at the National Research Council (Consiglio Nazionale Delle Ricerche) in Florence, Italy. My research interests are differential geometry, statistical mechanics, and...
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...
This thread is a direct shoot-off of this post from the thread Atiyah's arithmetic physics.
Manasson V. 2008, Are Particles Self-Organized Systems?
The author convincingly demonstrates that practically everything known about particle physics, including the SM itself, can be derived from first...
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...
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...
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.
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...
In my rough understanding Molecular Dynamics using Classical Newtonian mechanics is a 6N dimensional non linear system. 6N dimension because you have 3 position vectors and 3 momentum vectors for each N particles. Nonlinearity because of the terms in force fields. In principle this system can...
What would be the best book for me if I want to learn nonlinear dynamics ? I have my basics clear in linear differential equations, linear system theory, integral transforms and random process if they suffice as prerequisites.
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...
Hi all,
I'm a ME graduate student concentrating in fluids and I'm trying to decide between two electives. The two I'm looking at are Nonlinear Control Systems or Advanced Heat Transfer. Nonlinear controls looks interesting, but I'm not sure how it could tie in with fluids...as opposed to heat...
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...
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.
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...
Hello all,
My question is really simple. I really like working on problems that involve Nonlinear Dynamics and Chaos, and I also really enjoy fields of Theoretical Physics that probe the nature of reality (quantum mechanics, high energy & elementary particle physics, string theory, etc.) I...
Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$
for $n\geq 1$.
However, $\langle\sin^6\rangle = \frac{5}{16}\neq\frac{15}{48}$.
What is the deal here?
Show that all vector fields on the line are gradient systems.
This is exercise 7.2.4 in the book "Nonlinear Dynamics and Chaos" by Steven H.Strogatz
Thanks very much!
Practice Problems to "Nonlinear Dynamics..." Strogatz Book
Hi,
I'm taking a dynamics course which uses the "nonlinear dynamics and chaos" book by Strogatz. I get a half-descent understanding from the lectures of the prof and the book's explanations of things, but \ is\ there\ a\ good\...
Homework Statement
Consider the system x' + x = F(t), where F is smooth, T-periodic function. IS it true that the system necessarily has a stable T-periodic solution x(t)? If so, Prove it; If not, find an F that provides a counterexample.
Homework Equations
x' + x = F(t)
The...
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...
Homework Statement
Find the fixed points and classify them using linear analysis. Then sketch the nullclines, the vector field, and a plausible phase portrait.
dx/dt = x(x-y), dy/dt = y(2x-y)
Homework Equations
The Attempt at a Solution
f1(x,y) = x(x-y)
x-nullcline: x(x-y) = 0 \Rightarrow...
Hi everyone,
I am studying a 2-D nonlinear dynamics system, with two key parameters. But I have trouble when I want to locate where the homoclinic bifurcation occurs in the parameter space. Can anyone give me some ideas or reference readings? Thx
Homework Statement
2.1.5(A mechanical Analog)
a) Find a mechanical system approximately governed by dx/dt=sinx
b)Using your physical intuition explain why it becomes obvious that x*=0 is an unstable fixed point and x*=pi is a stable fixed point.
(note*this is exactly how it appears in...
First off, though I've been reading through these forums for a while now, this is my first post here, so let me briefly introduce myself.
I'm finishing up my third year as an undergraduate in mathematics. Next year, I want to apply to grad school in math, specifically, I'd like to study...
Hi,
I am a physics student who just finished his bachelor thesis in nonlinear dynamics. I am about to start the physics master programme at my home university (in Germany). Within a year, I will have to start writing my master thesis, so I'm trying to figure out in which domain I want to work...
What is meant by "waveform". Working in strogatz nonlinear dynamics, global bifurcati
Homework Statement
Consider the system r' = r(1-r^2), O' = m - sin(O) for m slightly greater than 2. Let x = rcos(O) and y = rsin(O). Sketch the waveforms of x(t) and y(t). (These are typical of what one...
I'm interested in focusing on nonlinear dynamics or mathematical physics for my PhD and was wondering if anyone could tell me what US universities have strong departments in these topics. I've heard that Cornell is good for dynamics and chaos but haven't heard much about other colleges.
Thanks.
I recently came across Nonlinear Dynamics and Chaos by Strogatz and I'm recommending it to all my Physics/Applied Math friends. This is a great introductory book on the subject and you don't need any more Math than is taught in a basic Diff. Eq. course.
I love this book 'cause
a) Strogatz...