“There is nothing more practical than a good theory.” James C. Maxwell

“. . . le souci du beau nous conduit aux mêmes choix que celui de l’utile.” Henri Poincaré

In Theory of Ordinary Differential Equations (ODEs) and Nonlinear Dynamics:

• Laid down the foundations of Theory of Nonlinear Non-Autonomous Dynamic Systems of Generic Configuration using the Henri Poincare’s strategy of the investigation of ODEs and manifolds (https://getinfo.de/en/search/id/BLSE%3ARN026870380/). The term of "generic configuration" means that the extended phase spaces of the systems have nor equilibrium points neither all other kinds of degeneracies or their motions take place in the domains located so "far away" from them that the method of linearization does not work.*
• Proposed a solution to the famous mathematical problem of stability, which since its formulation by A. M. Lyapunov in 1892, was considered practically insurmountable: The General Method of Construction of Lyapunov Functions (http://arxiv.org/abs/1403.5761). 

In Control Theory:

• Developed Method of Differential-Topological Reduction of Dimensions of Dynamic Systems with Control in the Neighborhood of Attracting Sets Intrinsically Existing in Extended Phase Spaces and Principle of Quasi-Equivalency between Optimal Controls of Original Full-Order and Corresponding Reduced Systems in Identical Optimization Problems (https://getinfo.de/en/search/id/tema%3ATEMAI93115552273/A-method-of-differential-topological-reductions/). 
• Created Henri-Poincaré-Strategy-Based Backstepping Method originally named as "A method of goal-oriented formation of the local topological structure of co-dimension one foliations for dynamic systems with control" (https://getinfo.de/en/search/id/tema%3ATEMAI93116204273/A-method-of-goal-oriented-formation-of-the-local/). 

The full list of the prime publications can be found at The German National Library of Science and Technology (http://www.tib-hannover.de/en/). 

So, why might you need to be familiar with the above-mentioned results of the researches?

These theoretical discoveries enable us to see the world in new, absolutely amazing light where the inexplicable metamorphoses into the explicable, the impossible turns into the possible and the difficult becomes the easy. It has become the reality because the proposed research is the tetrad with Differential-Topological Structure of Nonlinear Non-Autonomous Dynamic Systems of Generic Configuration as a cornerstone and its three pillars of Stability, Bifurcation and Control. The key role of Differential-Topological Structure lies in giving the researchers, who have to deal with mathematical models, the general mathematical approach to the investigation and control of the intrinsically nonlinear and unsteady physical phenomena very often happening in the form of transient or oscillating processes. The main concepts used in the above-mentioned researches are topological, namely the concepts of foliations, fiber bundles and coverings. The central idea is formulated in the form of some special classification of the isomorphisms of stratified manifolds. Thus it is also fundamentally topological. It is very innovative but if you are unacquainted with Topology this fact should not discourage you. Almost all its concepts can be explained with clear and simple geometrical illustrations.

How powerful and effective is the proposed approach based on Poincare’s ideas and the above-mentioned topological concepts? By the way, Henri Poincaré is the father-founder of Topology. The answer is that it is extremely powerful and effective! In aerospace aerodynamics it ensures the desired control performance of hypersonic flight maneuvering. In biology and medicine the approach can help us destroy the division cycles of malignant cells. In physics the one can suppress the plasma instabilities. In chemistry and atmospheric dynamics using it we are able to control the processes of non-equilibrium thermodynamics as, for example, the Belousov–Zhabotinsky reaction. This list of examples can be continued. 

The author is available for giving the short course of lectures aimed to form the mainstay for the deep understanding of Nonlinear Non-Autonomous Dynamic Systems of Generic Configuration and its Applications to Mathematical Modeling and Simulation in Modern Science and Technology. If it sounds interesting to you please do not hesitate to contact me using the following email: [email protected] or Linked-In and ResearchGate communication resources. 

* For a long time it is the method of linearization that has remained the favorite tool for the investigation of nonlinear dynamic systems and especially their most popular elements, namely equilibrium points and limit cycles. They are also called critical elements. Why has so much attention been paid to the critical elements? This is a very pertinent question since it is well known that except for the trivial cases and some special ones the critical elements form countable or countable infinite as maximum sets of phase spaces in contrast to their rest non-critical elements (we call them regular ones) that form uncountable sets. This means that the cardinality of the regular elements unfathomably larger than the cardinality of the critical elements. Translating the previous statement into layperson language one can say that if you pick any element of an arbitrarily given phase space at random it will always be a regular element. The answer was given by Henri Poincaré. Despite the fact that the critical elements are extremely rare birds in phase spaces but at least locally (or in their neighborhood) they completely define the behavior of the systems. The classic strategy of the qualitative investigation of dynamic systems sprang just from here and it is as follows. First, you need to find all the critical elements of a given phase space. Second, you have to analyze their properties. Third, having compiled and collated all the obtained information you should try to paint the whole picture of the phase space of the corresponding dynamic system. The main tool on these three phases is the method of linearization. Whereas it has usually produced excellent results for the equilibrium points, however its success for the limit cycles has been mostly limited to planar cases. 

Let our system

                                                           dx/dt = f(t, x)

be defined for all t from some time interval T and have a n-dimensional phase vector x. We can find its equilibrium points by solving the functional vector-equation

                                                           f(t, x) = 0

for x. In order that the system has at least one equilibrium point it is necessary and sufficient that the above-mentioned functional equation would have a constant vector as a solution. That is

                                                           f(t, a) = 0, 

where all the components of x = a are the same real numbers for all t from T.  It is obvious that there are immeasurably more chances that some of the components of the solution of a will be  dependent of t. But this means that the system have no equilibrium points.

Regarding limit cycles the status quo here as follows. There are the well developed methods for their investigation only in two cases. The methods of the first one works for the general form of the function of f(•) but it should be independent of t and the dimension of x should equal 2. For n > 2 and even with f(•) depending on t the Floquet theory and its variations with additional mathematical  tools have been designed to work. But the Floquet theory is a part of the method of linearization.  For n > 2 and with f(•) depending on t  it represents some theoretical interest without doubts but no more because in practice it is from a little to no use.

Now imagine the following situation.  Your system has a 6-dimensional phase vector of x. The first thing you have to check is the existence of equilibrium points for your system. Blam!-Blam!-Blam! The first alarm goes off.  Among the solutions of the functional vector-equation  

                                                            f(t, x) = 0 

for x there is not a single constant vector a. This means that the system does not have equilibrium points. Surely, this so-called find disappoints you but it is not an end. You turn your attention to limit cycles.  Maybe they exist in the phase space of the system?  But the dimension of the phase space, you have to work in, is 6. This fact just annihilates all your hopes to get a slightest foothold in your efforts to investigate the system. It is clear why. Just to find (we do not talk about its analysis) a closed 1-dimensional curve in the 6-dimensional space when f(t, x) is of a general nonlinear form without any regular tool for this task looks incomparably more difficult than to seek a tiny needle in a huge haystack being blindfolded.

Thus you have got in some kind of an impasse and it seems to you that there is no way-out of it. Luckily, not everything looks terribly desperate because many years ago your most humble servant found himself in a similar forlorn situation. However, he managed to take over it having created the special mathematical tools that helped him and will help you become an episodic winner in the quiz of Plan, Genesis and Evolution with Mother Nature as a hostess.