Dynamical Systems

Homework Help & Tutoring

We offer an array of different online Dynamical Systems tutors, all of whom are advanced in their fields and highly qualified to instruct you.
Dynamical Systems
Send your subject help request Submit your homework problem, or a general tutoring request.
Get quotes from qualified tutors Receive a response from one of our tutors as soon as possible, sometimes within minutes!
Collaborate with your tutor online Work together with your tutor to answer your question within minutes!
Dynamical Systems Tutors Available Now
25 tutors available
Leo
Leo
(Leo)
Doctor of Philosophy (PhD)
Hi! I have been a professor in New York and taught in a math department and in an applied math department.
4.9/5(6,268+ sessions)
2 hours avg response
PhysMath18
Israel
(PhysMath18)
Doctor of Philosophy (PhD)
Offering help to undergraduate, graduate, and postgraduate students in subjects involving Math, Physics, and Computation.
5/5(3,792+ sessions)
2 minutes avg response
statistics_help_online
Yohay
(statistics_help_online)
Master of Business Administration (M.B.A.)
A Statistician (BA) and an MBA Graduate with a DECADE of experience in 1-on-1 lessons, ‎homework assistance, Data analyses and much more.
4.9/5(1,587+ sessions)
3 minutes avg response
midnitc
Midhun
(midnitc)
Doctor of Philosophy (PhD)
10+ yrs experience in MATLAB,MAPLE,Mathematica, Robotics, Mechanics,Dynamics,Image processing,Python,C,C++,Computer Vision
4.6/5(1,113+ sessions)
1 hour avg response
SMHR022
Syed
(SMHR022)
Doctor of Science
PhD in Mathematics, 10+ years experience in teaching and mentoring Mathematics at High School to University level Math subjects.
4.9/5(675+ sessions)
16 minutes avg response
Henry_Stanley
Hranislav
(Henry_Stanley)
Master of Science (M.S.)
A PhD student in the Department of Functional Analysis and Research Assistant at home University. Also 5+ years of teaching experience.
5/5(298+ sessions)
26 minutes avg response
See 25 More Tutors
See what our students are saying
Describe your homework help.
FAQ Frequently Asked Questions
Can you help me with my homework in less than 24 hours?
Can you help me with my exam/quiz/test?
How much will it cost?
What kind of payments do you accept?

Dynamical Systems

Dynamical systems(1,9,10) as a field of study have been around since the time of Newton due to their great importance in the sciences. Only in rare instances can such systems be solved algebraically, with linear (time independent) systems and some Hamiltonian systems as exceptions. Usually we need computers to find the solution.

A dynamical system is usually presented in one of two ways:

  1. As a set of differential equations, for example Newton's laws for the motion of planets, population dynamics, fluid mechanics, etc. With a given initial condition, such a differential equation has typically one solution which varies with time. The solution then depends on the initial condition, time, and relevant parameters (for example the masses of the various planets).
  1. As a transformation which models the change of an initial condition after one unit of time. Iterating such transformations then produces the solution at a discrete set of times in the future.

The questions to be answered using dynamical systems are:  What is the behavior of the solution as time tends to infinity? Does the solution converge to a constant (equilibrium/steady state) solution, a periodic one, or a chaotic one, and what do these outcomes depend on? How stable are the dynamics? What is turbulence? Can we define some average behavior?

A typical class in dynamical systems will devote itself to a few of these topics:

  1. Basic definitions. Notions of stability and instability. Becoming familiar with the types of phenomena encountered.
  2. Bifurcation Theory(2), which addresses the question of how changes in parameters affect the behavior of solutions. This topic tends to have a heavily algebraic/analytic flavor. The student will learn about many different kinds of bifurcations. An important example is the Hopf bifurcation(11), in which a stable steady state solution becomes unstable, while a stable periodic solution emerges.
  3. Hamiltonian systems(3,12). These are of great interest in physics. Such systems have at least one conserved quantity (energy). A similar class of transformations is one that preserves an additional quantity, such as area, volume, or a symplectic form. Such dynamical systems occur frequently and have special properties.
  4. Systems with sensitive dependence on initial conditions and chaotic systems(4,13). This builds on a class of very interesting examples of hyperbolic dynamical systems that can be fully analyzed using tools such as symbolic dynamics that help elucidate their behavior. An important classical example has been Smale's Horseshoe.
  5. Ergodic theory(5,14). The historical background for this emerges when one tries to relate the movement of microscopic atoms in a gas to macroscopic thermodynamic quantities, such as pressure and temperature. In modern mathematical language, one learns here about invariant measures, averages, ergodic theorems (Birkhoff, Von Neumann) and its important generalization: Oscledec's multiplicative ergodic theorem. Lyapunov exponents are defined. Consider two nearby points and follow these as time evolves. Does their distance increase or decrease at an exponential rate? If so, then these points lie along unstable or stable directions of the system, respectively.
  6. Perturbation theory(6,15). When we consider the motion of planets, the sun has by far the greatest mass, and so the motion of individual planets is mostly dominated by just the sun. The corresponding solutions are the Keppler solutions which give rise to elliptical planetary orbits. In reality of course the motion of a single planet is determined not just by the sun, but also by the other planets which have small mass relative to the sun, and perhaps large distances to the planet of interest. Therefore, the deviations of planetary motion from a Keppler orbit can be analyzed in terms of power series that involve the masses and distances of the other planets. This was the beginning of perturbation theory. More modern and far more sophisticated methods are also used, for example the Kolmogorov-Arnold-Moser (KAM) theory(7,16).
  7. Special systems. Some classes in dynamical systems concentrate on very specific examples, such as holomorphic (analytic) transformations of the complex plane. Already for quadratic transformations z->z^2+c this leads to wonderful graphics and theory, with concepts such as Julia sets, Mandelbrot sets(8,17), and fractals.

References:

  1. https://www.springer.com/mathematics/dynamical+systems/journal/10884
  2. https://www.math.colostate.edu/~shipman/47/volume3b2011/M640_MunozAlicea.pdf
  3. https://www.unige.ch/~hairer/poly_geoint/week1.pdf
  4. http://www.colby.edu/mathstats/wp-content/uploads/sites/81/2017/08/2017-Manning-Thesis.pdf
  5. http://www.staff.science.uu.nl/~kraai101/lecturenotes2009.pdf
  6. https://www.sciencedirect.com/topics/chemistry/perturbation-theory
  7. http://math.bu.edu/people/cew/preprints/introkam.pdf
  8. https://plus.maths.org/content/what-mandelbrot-set
  9. https://ocw.mit.edu/courses/mechanical-engineering/2-032-dynamics-fall-2004/
  10. https://en.wikipedia.org/wiki/Dynamical_system
  11. https://en.wikipedia.org/wiki/Hopf_bifurcation
  12. https://en.wikipedia.org/wiki/Hamiltonian_system
  13. https://en.wikipedia.org/wiki/Chaos_theory
  14. https://en.wikipedia.org/wiki/Ergodic_theory
  15. https://en.wikipedia.org/wiki/Perturbation_theory
  16. https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem
  17. https://en.wikipedia.org/wiki/Mandelbrot_set

 

To fulfill our tutoring mission of online education, our college homework help and online tutoring centers are standing by 24/7, ready to assist college students who need homework help with all aspects of dynamical systems. Our mathematics tutors can help with all your projects, large or small, and we challenge you to find better online dynamical systems tutoring anywhere.

Read More

College Dynamical Systems Homework Help

Since we have tutors in all Dynamical Systems related topics, we can provide a range of different services. Our online Dynamical Systems tutors will:

  • Provide specific insight for homework assignments.
  • Review broad conceptual ideas and chapters.
  • Simplify complex topics into digestible pieces of information.
  • Answer any Dynamical Systems related questions.
  • Tailor instruction to fit your style of learning.

With these capabilities, our college Dynamical Systems tutors will give you the tools you need to gain a comprehensive knowledge of Dynamical Systems you can use in future courses.

24HourAnswers Online Dynamical Systems Tutors

Our tutors are just as dedicated to your success in class as you are, so they are available around the clock to assist you with questions, homework, exam preparation and any Dynamical Systems related assignments you need extra help completing.

In addition to gaining access to highly qualified tutors, you'll also strengthen your confidence level in the classroom when you work with us. This newfound confidence will allow you to apply your Dynamical Systems knowledge in future courses and keep your education progressing smoothly.

Because our college Dynamical Systems tutors are fully remote, seeking their help is easy. Rather than spend valuable time trying to find a local Dynamical Systems tutor you can trust, just call on our tutors whenever you need them without any conflicting schedules getting in the way.

Start Working With Our College Dynamical Systems Tutors
To fulfill our tutoring mission of online education, our college homework help and online tutoring centers are standing by 24/7, ready to assist college students who need homework help with all aspects of Dynamical Systems.