Keynote lecture 1: nonlinear explicit ordinary differential equations: are they really nonlinear?
Metin Demіralp · International Conference on Mathematical and Computational Methods in Science and Engineering · 2011
We (myself, my son, my colleagues in my group and some other collaborators from certain universities) were concerning with construction of a fictituos probabilistic dynamics behind the ordinary differential equations. This was based on the fact that some dynamical systems like quantum dynamics or Liouville dynamics are governed by partial differential equations. However, it is also possible to construct an infinite degree of freedom system over the expectations or expected values of some operators such that these expectations are connected through some ordinary differential equations in infinite number since the operators appearing in the expectations are related to each other recursively through commutators without stopping. At the beginning even these expectation ODEs could have nonlinearities. After intense studies in our group we could be able to show that these ODEs can be structured in linear format by using certain appropriate basis operators. What we have observed was the fact that the lower number of ODEs were resulting in mostly nonlinear structures unless very specific characters are existing in the system under consideration. However infinite number of equations were capable of bringing linearity. This was a one way research. From probabilistic equations to ODEs. Later we focused on the reverse action. To find an appropriate probabilistic structure like partial differential equation became the target of our efforts. Four years ago, my son Emre Demiralp, joined with a group in University Michigan at Ann Arbor to work on neuroscientific problems. By his pushes on me we started to deal with the mathematical structure behind the brain activities. Since these problems were having too many parameters and necessitating huge number of data, the models might need causal and probabilistic aspects. However all systems were at most considered as dynamical systems which are governed by ODEs. If there would be a probabilistic structure it would be behind these mathematical objects. Our efforts intensified and focused on this issue. The last year our project group took some important steps and possibility of birth of a quantum mechanical background system seemed to be appearing on the horizon. This encouraged us to continue and eventually we could be able to develop a probabilistic structure not like quantum or Liouville mechanics but something new. There appeared an evolution operator instead of the Hamiltonian. A first order partial differential equation could have been constructed. Its solution was possessing wave function properties and so on. When this is done what we have seen was an infinite linear ordinary differential equation set with constant coefficient matrix. This was valid for all explicit ODEs, linear or nonlinear. Then we focused on to reveal this feature without using probabilistic toolsand considerations. Today we are at point that all these are accomplished. There is no nonlinear explicit ODEs but the nonlinear representation of infinitely linear structures. In other words, nonlinearity is the folded form of linearity. Many of well known ODEs are now under the attack of our group. Our major studies are directed to this area.