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.

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  1. djulzz1982

    Modelling Guy Needing Refresher

    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.
  2. R

    Programs Universities for Research in Dynamical Systems of Fluids

    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...
  3. T

    A Integrability of Systems of Differential Equations

    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...
  4. V9999

    Open problems and suggestions of great mathematical journals

    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...
  5. M

    Dynamical Systems: how to find equation for Poincare map?

    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)...
  6. 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...
  7. M

    Dynamic Systems: Poincaré-Bendixson Theorem finite # of equilibria

    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...
  8. Dynamical System Inner Products and Projections with Liouville Operators (Koopman Generators)

    Dynamical System Inner Products and Projections with Liouville Operators (Koopman Generators)

    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).
  9. Operators for (Nonlinear) Dynamical Systems

    Operators for (Nonlinear) Dynamical Systems

    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.
  10. AJSayad

    Finding Your Path to the Aerospace Engineering Industry

    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...
  11. E

    Dynamical systems and cutting edge numerical methods

    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.
  12. V

    How Did a Physics Graduate Become a Researcher in Italy?

    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...
  13. S

    I Interpretations of phase space in Dynamical Systems Theory

    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...
  14. O

    What is the speed of each particle?

    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...
  15. Auto-Didact

    How Does Logic Evolve in Human and Animal Psychology?

    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...
  16. Auto-Didact

    Lingusitics How Have Dynamical Systems Approaches Influenced Contemporary Linguistics?

    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...
  17. Martin T

    Vladimir I. Arnold ODE'S book, about action group

    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...
  18. A

    MHB Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

    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...
  19. T

    I Equation for Fire Evolution Dynamics

    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.
  20. J6204

    Determing the differences between two sets of differential eqs

    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...
  21. Joppy

    MHB Discrete dynamical systems - Invertible maps

    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...
  22. D

    MHB Gathering interest in Dynamical Systems (Bifurcation, Stable Manifolds, Hamiltonian Systems) f

    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...
  23. T

    I Dimension using box counting technique

    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...
  24. 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...
  25. Michael Wu

    Linearizing a 2-D quadrotor dynamic model

    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 =...
  26. R

    Finding transition matrix, no % probability given

    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...
  27. R

    Trying to find long run behaviour of a dynamical system

    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...
  28. U

    Can You Change the Damping Ratio of a Second Order Linear System Easily?

    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.
  29. P

    Find the limit cycle for this dynamical system

    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...
  30. P

    Classify the fixed points of this dynamical system

    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...
  31. J

    Dynamical systems and thermodynamics

    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.
  32. U

    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...
  33. U

    Period of Limit Cycle: Find B for Hopf Bifurcation

    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...
  34. U

    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...
  35. laramman2

    Applied What Are the Best Textbooks for Self-Studying Nonlinear Dynamics?

    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.
  36. A

    Classical Looking for rigorous text on dynamical systems

    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
  37. S

    Toplogy book for study of dynamical systems

    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...
  38. N

    MHB Recommend books for dynamical systems?

    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...
  39. S

    Topological Conjugation between two dynamical systems

    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)...
  40. S

    Topological Conjugation between two dynamical systems

    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...
  41. Z

    Riemann Hypothesis for dynamical systems

    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...
  42. L

    Want a book/notes that cover this syllabus (dynamical systems).

    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...
  43. D

    MHB Bifurcations of dynamical systems

    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 -...
  44. S

    MHB Dynamical Systems and Markov Chains

    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 \(\rho\).
  45. B

    Benchmark problems for dynamical systems

    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"...
  46. T

    Ideas for a thesis on dynamical systems?

    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)...
  47. D

    Books on orbital transfers, satellite control theory, dynamical systems etc

    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...
  48. Rasalhague

    What is the meaning of the shorthand notation used in dynamical systems?

    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 =...
  49. A

    Nonlinear Orthonormal Bases for Dynamical Systems in 3D

    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...
  50. O

    What is the Kronecker index in dynamical systems?

    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.