What is Dynamical systems: Definition and 73 Discussions
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.
At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.
The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.
Hi everybody.
I like to model dynamical systems, but over the last few years, I've been busy implementing simulations, without actually deriving their equations of motions. I'm thus here to check with members whether some systems for which I wrote the equations of motions are actually corrects.
Hi everyone,
(this is my first post so be gentle)
I am currently getting my masters is mechanical engineering, was admitted to aero Ph.D. programs as Vtech, MSU, and Cinci last year but decided to get masters locally and apply to "better" schools (UofM) for next cycle with a better resume and...
I' m reading wiki article about Solitons and have some some troubles to understand the meaning of the following:
Question: In context of systems of differential equations, what means precisely "integrability of the equations"?
Is there any good intuition how to think about it? Has it some...
Hi!
Suppose that someone had solved an old but open problem in the great area of mathematics and physics, for instance, dynamical systtems, algebraic geometry and differential equations. Based on your broad experience, what are the best scientific journals to submit such a discovery?
In...
Hi,
I was attempting a question on the dynamical systems topic of Poincare maps, and was struggling to understand a certain part of it.
Knowledge from prior parts of the questions:
There was a system which we converted to polar coordinates to get: (## a ## is an arbitrary real constant)...
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...
Homework Statement:: Can someone explain the finite number of equilibria outcome of the Poincaré-Bendixson Theorem?
Relevant Equations:: Poincaré-Bendixson Theorem
[Mentor Note -- General question moved from the schoolwork forums to the technical math forums]
Hi,
I was reading notes in...
In this video we show how we can exploit the linearity in the symbol of Liouville operators to define an inner product on dynamical systems that give rise to densely defined Liouville operators over a reproducing kernel Hilbert space (RKHS).
Koopman Generators, Liouville Operators, and Transfer Functions! We talk about the role of operators in Dynamical Systems theory. This requires a discussion of Hilbert spaces, Densely Defined Operators, and Occupation Kernels.
Summary:: What is the best way to get into the Aerospace Engineering Industry?
Hi everyone,
I'm new to the physics forums. My name is Andrew, I'm going to be in my undergrad Senior year in mechanical engineering this coming fall. I've recently been looking into PhD programs and I've been...
Background in teaching (mathematics and physics)
Signed up here as an alternative to physics.stackexchange.
Pursuing inspiration for complicated dynamical systems.
Actively working on cutting edge numerical methods and educational (free) software
Glad to answer questions in my field.
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...
In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist...
Homework Statement
Both m1 and m2 (m1=2m2) masses can slide without friction over parallel and rigid bars that are placed at a distance d from each other. A spring with elastic constant k and with zero natural length connects both masses. The system is placed on a table. The system is released...
This thread is a shoot-off from this thread.
Assuming some relation between human language and logical reasoning, how would this relate, let's say, to the arrival and evolution of logic in human and animal psychology?
I would presume that some logic, for example classical logic, can be more or...
A few years ago I read two pretty groundbreaking linguistic papers from the 90s arguing that natural languages are networks which can be conceptualized from the perspective of nonlinear dynamical systems theory, with a lexicon being a state space and grammatical rules being attractors in that...
hi everyone, I'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...
I need your advice on understanding a proof of a lemma from a book I am reading.
I asked my question in overflow:
https://mathoverflow.net/questions/299408/lemma-4-5-1-on-page-77-in-the-book-averaging-methods-in-nonlinear-dynamical-syst
Does anyone understand the proof and can answer my...
It is NOT about the heat equation. I'm asking about a dynamical system or equations set to describe fire evolution, with given fuctions of air, material and enviorment change.
Homework Statement
Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is how does the system of equations 1 which I produced myself differ from the set of equations 2. Below I have...
Hi (Sleepy),
I suspect this is trivial, but I couldn't find any info onlin.
Consider the folowing map: $\phi_{n+1} = f(\phi_n ; \Theta, a) = (\phi_n + \Theta + a \sin \phi_n) \mod 2\pi$.
I need to check if is invertible: $\phi_n = f^{-1} (\phi_{n+1}; \Theta, a)$ when a = 1/2 or 3/2...
I am in the process of writing a lecture out for my Graduate modeling class I teach. I normally don't write lectures out in LaTex or use PDF's because I write on the dry erase board, but if anyone is interested I wouldn't mind spending the time to type out some notes on the topic.
The topic...
On an exam we just took, we were asked to find the dimension of a set using the box counting technique. So choose an epsilon, and cover your object in boxes of side length epsilon, and count the minimum number of boxes required to cover the object. Then use a smaller epsilon and and count the...
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...
Hi all!
I am taking an online course on aerial robotics and am currently on the topic of linearizing a 2-D quadrotor dynamic model. See slide (link below):
The equations under "linearized dynamics" are derived using the equilibrium hover configuration (e.g. y = y0, z = z0, Φ0 = 0, u1,0 =...
Homework Statement
Consider a quantum mechanical system with three states. At each step a particular particle transitions from one state to a different state.
Empirical data show that if the particle is in State 1, then it is 7 times more likely to go to State 2 at the next step than to State...
Homework Statement
In any given year a person may or may not get the flu. Past records show that if a person has the flu one year then (due to a build up of antibodies) there is a 85% chance that they will not get the flu in the following year. If they don't have a flu in a given year then...
How to change the damping ratio of dynamical system if it is say for example robotic arm? Can it be changed directly or one need to change natural freqvancy? by which it will automatically change.
Homework Statement
Consider the dynamical system:
$$\dot{r}=-ar^4+ar^3+r^6-r^5+r^2-r~;~~\dot{\theta}=1$$
Find all fixed points and limit cycles for:
a) ##~~a=2##
b)##~~a<2##
c)##~~2<a<2\sqrt{2}##
Homework Equations
Not applicable.
The Attempt at a Solution
For all three values/ranges...
Homework Statement
$$\dot{x_1}=x_2-x_2^3,~~~~~~\dot{x_2}=-x_1-3x_2^2+x_1^2x_2+x_2$$
I need help in determining the type and stability of the fixed points in this system.
Homework Equations
The Jordan Normal Form[/B]
Let A be a 2x2 matrix, then there exists a real and non singular matrix M...
Is there a mathematical way to show that a dissipative (or general?) dynamical system obeys the laws of thermodynamics?
I am looking for books/sites/references, thanks for any reply.
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...
Homework Statement
I'm given this system:
\dot x = Ax^2 y + 1 - (B+1)x
\dot y = Bx - Ax^2 y
(a) Find the value of B when hopf bifurcation occurs.
(b) Estimate the period of the limit cycle in terms of ##A## and ##B##.Homework EquationsThe Attempt at a Solution
I have found fixed point to be...
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...
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.
Hi,
I'm looking for a modern rigorous text on (Hamiltonian) dynamical systems, perhaps with emphasis on perturbation theory. It should be in the same vein is Poincare's "methodes nouvelles", but modern.Thanks
need a good book on topology and metric spaces! I'm an undergrad taking a course on non-linear dynamical systems, just realising my pre-knowledge is slightly under the requirement since I have not taken any course on topology. I only know basic real analysis and some complex analysis.
so any...
I am finding some topics a bit obscurely explained.
I have attached the curriculum/study guide we are using, and was hoping someone could suggest books which cover these bases.
Particularly finding week 8 a bit difficult at the moment, especially stability (surely just more robust revision and...
Find a topological conjugation between g(x) and T(x) where g and T are mappings (both tent maps [graphically speaking])
g:[-1, 1] → [-1,1]
g(x) = 1-2|x|
T:[0,1] → [0, 1]
T(x) = 2x when x ≤ 1/2 and 2(1-x) when x ≥ 1/2
h ° T = g ° h (homeomorphism)
h:[0, 1] → [-1, 1]
h(x) = cos(∏x)...
Homework Statement
Find a topological conjugation between g(x) and T(x) where g and T are mappings (both tent maps [graphically speaking])
Homework Equationsg:[-1, 1] → [-1,1]
g(x) = 1-2|x|
T:[0,1] → [0, 1]
T(x) = 2x when x ≤ 1/2 and 2(1-x) when x ≥ 1/2
h ° T = g ° h (homeomorphism)The...
what are teh differential equations associated to Riemann Hypothesis in this article ??
http://jp4.journaldephysique.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/jp4/abs/1998/06/jp4199808PR625/jp4199808PR625.html
where could i find the article for free ? , have...
I am looking for a book that covers these topics at a self-contained level for self-study (ie: a book designed for a short course on the subject or lecture notes):
Things in bold are of most interest to me. I notice there are a lot of pure math books on this subject but I'm looking for...
Something doesn't seem right in regards to my analysis of the Jacobian. What about when $a=0$ at the second fixed point?
\begin{alignat*}{3}
x' & = & y - ax\\
y' & = & -y + \frac{x}{1 + x}
\end{alignat*}
First, we need to determine the fixed points in the system. So let
\begin{alignat*}{3}
y -...
Prove that if \(P\) is a stochastic matrix whose entries are all greater than or equal to \(\rho\), then the entries of \(P^{2}\) are greater than or equal to \(\rho\).
Homework Statement
Not really homework, but I couldn't find a better place to ask. I found an interesting paper and I'd like to test out the ideas it contains, since the code is available for free. The paper is
"Evolving Fractal Gene Regulatory Networks for Robot Control".
While the "test"...
Hi there,
I'm looking for a topic for my physics master thesis. The topic is free, and I'm really interested in (parallel) computing, as well as dynamical systems and "chaos". However, I'm kinda short on specific ideas, so this is where I'm looking for help. Current ideas for the project:
1)...
Hey folks wondering if anyone knows some good books on the following subjects...
Orbital transfers, specifically to Lagrange points with information on stable/unstable manifolds.
Control theory of satellites, new to control theory so perhaps I should be looking at a general control theory...
I have some questions about what I think is a fairly standard and common short-hand notation used in physics.
Today I watched lecture 2 in the nptelhrd series Classical Physics by Prof. V. Balakrishnan. In it, he models a kind of system called a simple harmonic oscillator, I think using TC =...
I propose 27 scalar functions {fn : n = 1,2,…,27}, 9 delay-coupled 3D unit vectors {eij : i,j = 1,2,3} of general periodic nature, 7 ortho-normal bases {Zk : k = 0,1,2,…,6} each having three 3D unit vectors, 4 tensors {Y3,Y4,Y5,Y6} of order 2 and a tensor E of order 3. Z3-Z6 and E are defined...
Hello all,
I was wondering what is the Kronecker index in relation to dynamical systems. For instance a sample question would be find the Kronecker indices ind(0,x-F(x)) and ind(inf,x-F(x)) for the dynamical system {(dx/dt)=g(x,y),(dy/dt)=h(x,y)}. Thanks in advance.