What is dynamic systems: Definition and 17 Discussions

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in 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. Ugnius

    I need sources to learn about dynamic systems

    Hi! I have exam in couple of weeks , and now am looking for sources to learn about dynamic systems , chaotic systems and etc. My main goal is to learn characteristics of such systems , learn about special points in dynamic plane. For the most part we used to create dynamic system simulations...
  2. M

    Dynamic Systems: Question about Isoclines of Systems

    Hi, I was doing some practice problems online for dynamical systems and came across the following question about isoclines. It left me with 2 questions that I hoping to get some insight to. Question: 1. What are isoclines? (I have tried doing an internet search, but the results don't help me)...
  3. 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...
  4. S

    Applied Master Slave schemes in dynamic systems

    Does anyone know a good book on master-slave schemes in dynamic systems? I'm on a dynamics book covering oscillators and it kind of glances over the topic and expects me to know it. I've done previous basic dynamics books and haven't seen it in them and checking on the internet I have found very...
  5. Feodalherren

    Dynamic systems: two rotating disks interconnected by damper

    Homework Statement Homework EquationsThe Attempt at a Solution So I'm assuming my solution is just as valid as the one in the book, if not, can somebody explain to me why? I prefer my solution since they say that phi is a known function.
  6. V

    Typical examples of second order nonlinear dynamic systems

    Hi guys, after hours of searching internet I couldn't find much real-life examples of second order nonlinear dynamic systems (only tons of tons of equation and system theory... got totally frustrated). They will serve as a base process for modeling controllers. So far I found propeller pendulum...
  7. B

    Is a Self-Powered Vibration Generator Possible?

    I'm fascinated by the prospect of them, but I can't seem to find much information/progress online aside from like small sensors which aren't quite the same thing. Has there been anything developed that is a self powered vibration generator? As in, it vibrates and then is powered by its own...
  8. G

    Dynamic Systems - DE with Unit Impulse

    Homework Statement y(t) = e^{a(t-t_{0})} y(t_{0}) + \int_{t_{0}}^{t} e^{a(t-\tau)}b \dot{u} (\tau) d\tau u(t) = \delta(t) = \frac{1}{2} c^{3} t^{2} e^{-ct} where c >> |a|, t_0 = 0, and y(0) = 0 Find y(t) and represent the unit impulse, delta, in the solution. The remaining terms should not...
  9. F

    Power Law Derivation for dynamic systems

    Just a quick question. Let A and B be two points. Electrical work is defined as the amount of energy it takes to move an amount of charge Q through a potential difference VB-VA (for our purposes here, we will assume that the voltage values are measured with respect to an Earth ground) and is...
  10. T

    [Dynamic Systems] Computing the orbit of a number

    Homework Statement Choose an even number in the interval [100,199] and compute its orbit under the proper divisor function. Homework Equations Proper Divisor Function = σ(n) = Sum of all divisors of n, excluding n The Attempt at a Solution I am unsure what it means by "compute the...
  11. R

    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...
  12. K

    How Can I Self-Study Dynamic Systems?

    Hi guys, I'm a pure math master degree. Interested in studying dynamic systems, I want to learn more about this subject without any academic course.(just self study) How can I start it? Which books , articles and subjects?
  13. K

    Dynamic Systems: Calculating Boom Crane's Transfer Motion

    There is boom crane with a bucket attached at the end. The angle of the boom "theta" is 60 degree. The weight of the bucket with a man in it is 200Kg. The mass of the boom is acting at centre of the boom (length of the boom is 10m) is 600 Kg. There is a cylinder attached to lift the boom 1m...
  14. S

    Mark's Struggle to Solve Dynamic Systems Qn

    [SOLVED] dynamic systems I Have Just Received This Question = A Car Of Mass 1.2 Tonnes Rolls Down A Hill Which Is 600m Long And Is Inclined At An Angle Of 9 Degrees To The Horizontal. Ignoring The Effects Of Air Resistance And Friction And Assuming The Car To Start From Rest At The Top Of The...
  15. R

    Strategies for Programming Feedback Control Systems

    Hi, I'm developing a feedback control system which explicitly comprises a model of the dynamic plant (inc. actuators). Any thoughts on programming methods I can use to describe the dynamic system which should provide valid results for a given input (step, ramp, impulse) in both time and...
  16. A

    How can I solve a problem involving dynamic systems in physics?

    I have really big prob., please. anyone who know how to do this i'll be very pleased. My Homewok is little problem.We work Dynamic systems and then we test that system on computer ( on Mathlab).I attached you my prob. you and I'll translete you upper text: Two equal bars AiBi length...
  17. quantumdude

    Modeling and Analysis of Dynamic Systems

    This lonely little Forum hasn't seen much traffic, and I suspect it's because most of you don't know exactly what the field of Systems Engineering actually entails. Given the importance of "the systems approach" to modern engineering, I think a tutorial thread is in order. The discussion...