We usually use differential equations, ordinary and partial, difference and delayed. But, could dynamics be captured using discrete mathematics structures, or combinatorics?
Hello Sonam,
It depends on your model, what you actually want to model, and what your outputs you would like to have. But, yes, you can do that, I am referring to the use of discrete mathematics to model dynamical systems. I think for control you can also use it.
I am doing discrete event simulations (that are built on the idea of discrete mathematics) for very dynamic systems. Until now, I did not have any problem. It is rue that sometimes you really need to think about the formulas of the simulation in order to encompass true dynamism, but that is part of the fun and the challenge.
Regards,
Alex
I remember a talk of Wolf Kohn and Zelda Zabinsky at the ISMP 2009 in Chicago on exactly this topic. You can download their research article here: http://www.mdpi.com/1099-4300/15/9/3592
It is quite common to turn continuous (dynamic) data set into discrete, but turning discrete data set into continuous requires more tricks. In data test we do it quite often.
Assume a data set [1,2,3,4,5] as a possible choice of answer on a continuous successive integer scale; each is a quantitative value. Its dynamism may be evidence by giving 50 surveys to 50 people; the answer to the survey may be all over the place but with the confine of [1,2,3,4,5]. This is a dynamic system. It could be treated with continuous probability tools: PDF and CDF. This same data set may also be turned into a discrete set by using the mean as a reference point, say if equal to or greater than mean = yes and less than mean = no. Now the 50 surveys are divided into two distinct groups. Discrete probability may be used. What was once CDF now becomes MDF.
http://mathworld.wolfram.com/DiscreteMathematics.html
There are also approaches to combine System Dynamics (i.e., differential equations) and Optimization (mixed-integer nonlinear programming), see the attached publications.
Technical Report A Global Approach to the Control of an Industry Structure Sy...
Technical Report Modern Nonlinear Optimization Techniques for an Optimal Cont...
Prof. Armin: Thank you loads. I am lucky to have gotten such motivation at an young age.
Hello Sonam,
Perhaps it will be useful my (with co-author) paper : The generalized entropy in the generalized topological spaces, Topology and its Applications 159 (2012) 1734–1742.
If you look applications of tools provided by discrete mathematics in dynamical systems then this book can be good place to start:
Lind, Douglas; Marcus, Brian. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. xvi+495 pp. ISBN: 0-521-55124-2; 0-521-55900-6
It provides a very smooth introduction into symbolic dynamics from point of view of discrete structures (graphs, formal languages, Perron-Frobenius theory, etc.) and then reveal connections with topology, mixing, topological entropy, etc.
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