Abstract:

We are delivering theories of two types of chaos, which have been developed in our recent papers. The first one is the chaos on labels, and the second one is the unpredictable dynamics. The cornerstone of the newly introduced unpredictable dynamics is the unpredictable point, which is a specification of the Poisson stable point. Let us recall that the Poisson stable point is known as the ultimate type of oscillation in the Poincare-Birkhoff theory. Thus, our suggestions prolong the line of recurrent motions to a new frontier such that the research joins with chaos investigation. It is important to emphasize that Devaney's chaos relies entirely on the collective behavior of orbits, such as sensitivity, dense cycles, and transitivity. Oppositely, the Poincare chaos is based on the concept of unpredictability, which is focused uniquely on a single orbit motion. Moreover, by simulating a solution, we can conclude that not only is chaos present, but also that motion is unpredictable. This is, from one side, principally new for chaos study and, from another side, leads us back to the roots of classical dynamics, which considers a single stable motion as the main object of discussion. What we are doing is like that of a single electron idea in physics. In our opinion, the novel motion will make the research of chaos more beneficial inside of the theoretical study as well as for applications. The first discoveries have already been made in our papers, where we have provided the perception of the random processes through the two types of chaos invented for our research.

Dynamics on labels introduced in our papers is a universal way to analyze chaos in fractals, neural networks, and random processes. Theoretically, in the basis of the dynamics on labels, the abstract similarity dynamics lays, which happens in abstract similarity sets through the abstract similarity map, and can be exemplified by the symbolic dynamics, but most formally, as it relates to symbolic dynamics as integration to summation in calculus. That is, symbolic dynamics is a special case of abstract similarity dynamics when the labeling for the set of symbol sequences is maximally simple. Our proposals also surpass the symbolic dynamics in a methodological sense. To study a motion by symbolic dynamics, one must project it on the Bernoulli shift along a sequence of symbols. We avoid a search for projection's characteristics and solve a problem completely inside of the state space. We are confident that the new method can be utilized as a universal method for research on the most sophisticated dynamics as well as a way of chaos initiation in many applications. The domain structured chaos (chaos on labels) has already been applied for the extension of chaos in hyperbolic sets, chaotic attractors, fractals, finite-dimensional cubes, stochastic dynamics, probability, differential equations, and neural networks.

This talk is tailored for those who work with the theory of functions, differential equations, geometry, and the history of mathematics. We will avoid diving into technical details, focusing instead on the broader implications and applications of our research.

Bio:

Dr. Marat Akhmet is a mathematics professor at Middle East Technical University, Ankara, Türkiye. He received his Bachelor's degree in Mathematics from Aktobe State University, Kazakhstan, and his Doctorate in Differential Equations and Mathematical Physics from Kyiv State University, Ukraine.

Dr. Akhmet has been awarded the Science Prize of TUBITAK (Türkiye, 2015) for his achievements in scientific research and the Prof. C.S. Hsu Award (UC Berkeley, 2021) for outstanding results in nonlinear analysis investigations. Has been invited as a Plenary speaker for 17 international conferences in Turkey, Kazakhstan, the USA, France, Italy, and the UK, and has organized more than 20 international and national scientific meetings. Dr. Akhmet is passionate about scientific research as well as his role as a teacher, and he has continually been one of the recipients of the 'Best Performance Award' at Middle East Technical University over the 20 years of his work there. Dr. Akhmet has mentored 19 Ph.D. students over his career, who now work in research centers and universities in Türkiye, Kazakhstan, USA, Saudi Arabia, and Libya.

He solved the Second Peskin conjecture for integrate-and-fire biological oscillators by a skillful formalization of the firing connection to achieve synchronization.

Dr. Akhmet made significant contributions to chaos theory, suggested a method of chaos replication, extended the types of recurrent motions with a concept of unpredictable points, and developed domain-structured dynamics as a universal approach in this research area.

In papers and books by the applicant, fractals are related to Dynamics not only as the results of iterations but also as their states and orbits. Thus, the beauty of fractals can now also be extended along the time axis.

He has contributed to the fundamentals of the theory of discontinuous, almost periodic functions and the solutions of impulsive systems. He has also introduced and developed the B-equivalence method for systems with variable impulse moments. The most sophisticated results were developed for bifurcation problems, discontinuous dynamics with grazing phenomena, and singularity.

Dr. Akhmet has achieved significant results for differential equations with piecewise constant argument. The classical results of the operator theory are now used for this research.

It is important that many of the applicant's results have been realized in research on chaos and synchronization in areas of science and industry such as neuroscience, semiconductor gas discharge models, economics, medicine, and biology.