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    Dynamical system is a system which changes (evolves) over the time, and its state at any time is uniquely determined by the initial state. If the time changes continuously, a dynamical system can often be described by means of an ordinary differential equations system that relate the rate of its state change (i.e., the time derivative) to the system state itself. The changes in the state of such system can be uniform, as well as more complex - periodic and even chaotic

 

   Examples of dynamical systems: 

 

 

     An important method for the dynamical systems studying is a numerical solution of the equations describing the system and futher analysis of the resulting dependencies. The best representation for the solution is graphical - as a graph of the system states depending on time or in the form of curves depicting the evolution of a particular system in the space of its possible states («phase space»), such curves are called «phase trajectories». For this analysis is designed DEREK - application for Windows operating systems. 

    What is the difference of DEREK from other programs for the study of dynamical systems (some of which are listed on the «Links» page)? This program is really SIMPLE and truly VISUAL. If you can make a model of a dynamical system as a set of differential equations, but do not want to go into details of numerical methods to solve it, and wrestle with the formation of a graphical representation of the results - DEREK may be very useful for you. 

 

   DEREK allows you to:

  • create the description of the system as a set of differential equations, initial conditions and parameters of the equations; 

  • find errors in the description of the system, just pointing out the place and nature of the error; 

  • automatically choose the parameters of the numerical method for solving the system. If something goes wrong in the numerical solution (it sometimes happens), DEREK does not issue obscure messages (such as «Runtime error 203 at address 049F») and does not stop running with a plaintive squeak, but simply and clearly explains what you need to correct in order to find a solution (of course, you can make DEREK die, but you have to work hard). The number of parameters to control the numerical solution of the system is small, and their meaning is quite transparent, so it's difficult to be confused with them

  • build visual graphic representation of solutions and of any dependent on solution variables. DEREK can itself scale a field for plots to best display them, but admits that the layout can be performed manually;

  • explore the three-dimensional image of the phase trajectory. DEREK can rotate a three-dimensional phase trajectory, as well as zoom it in and out. 

 

   DEREK also includes some special methods for the analysis of dynamical systems: 

  • construction of a «family» of solutions that depend on one or two parameters, or on a set of different initial conditions. For each value of the parameter graph is plotted, where you can watch for the changes in the behavior of the system (for example, when the parameter varies, the motion from uniform can become periodic or chaotic - such change of the system behavior is called «bifurcation»

  • building a «Poincaré sections» - the set of points of the system trajectory intersection with a given plane. Poincare sections allow to better understand the nature of the motion 

  • calculation of «Lyapunov exponents» - sets of numbers that characterize the behavior of the system at infinite evolution and do not depend on the initial conditions (except, perhaps, some exceptions). DEREK is also able to build a graph of the Lyapunov exponents depending on the system parameter.

 

    In addition, DEREK contains a large (you can say «great») number of examples of real dynamic systems from mechanics, physics, chemistry, biology, and (well, how one can avoid it in the modern theory of dynamical systems!) - meteorology. Many examples are provided with links to resources on the Internet, which include detailed descriptions of the systems, the history of their research, and more. 

 

   The unique feature of DEREK is its ability to deal with a collection of dynamical systems known to it as a database. It allows you to search systems by specifying sets of complex criteria. Thus, you can get answers to questions like the following: «select the systems of  at least  third order and, moreover, non-autonomous» or «select the systems that have been modified after March 1, 2015 and their right-hand sides are polynomials of not more than second order in dynamical system variables» or even «select the systems which are described by the Sturm–Liouville equation».

 

   Samples of images that DEREK is able to create can be seen on the «Examples» and «Experiments» pages. On the «How it works» an investigation of a system from entering its description to solutions analysis is shown. On «Attractive attractors» page  the complex and beautiful images of dynamical systems attractors  can be found. Finally, on «Reconstruction» page user can learn about another aspect of the dynamical systems study - the constructing of differential equations based on experimental data.

  Dynamical systems research methods can be applied to studying the spread of epidemics, the example is on the «Epidemic dynamics» page, based on the COVID-19 pandemic case.

 

   If you prefer viewing pictures to reading boring text - the «Gallery »page is for you ...

Dynamical systems exploring-gallery
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