Olga Kuznetsova PhD student at the Department of Mathematical Information Technology, University of Jyväskylä, Member of FICS (Finnish Graduate School in Computational Sciences) Phone: +7 921 4023104 Address: P.O. Box 35 (Agora), FI-40014 University of Jyväskylä, Finland E-mail: [email protected],

Date of birth: 03.05.1987 High education: St. Petersburg St. University, Faculty of Mathematics and Mechanics. Speciality: Applied mathematics and computer science. Master Degree (diploma with honour), average grade: 4.8. Foreign languages: English – excellent, German – with dictionary. Science advisors of PhD study: Professor Pekka Neittaanmäki (University of Jyväskylä) and Professor Gennady Leonov (St. Petersburg State University). PhD topic: Investigation of limit cycles of dynamical systems. Publications: 10 works, including international conference papers and mathematical magazine publications. The full list of publications can be found on the site http://sites.google.com/site/okuznetsovamath/home/publicationslist.pdf. ******************************************************************************

Briefly about PhD study topic

”Investigation of limit cycles in dynamical systems” 1. HISTORY OF THE PROBLEM. Considering the history of the problem, it is necessary to say first about a distinguished German mathematician David Hilbert, who stated the famous problem of estimating the number of limit cycles in 2-dimensional polynomial systems dy dx = -α y+f(x,y), = α x+g(x,y), α > 0. dt dt Great Russian mathematician, Academician Kolmogorov, examined the cycles in quadratic systems. In [1] V.I. Arnold writes that Andrey Kolmogorov had distributed a

few hundreds of quadratic systems among the students of Mechanics and Mathematics Faculty of Moscow State University. Each student had to find the number of limit cycles of his/her system. The result of this experiment was absolutely unexpected: not a single system had a limit cycle! It means that the probability of existence of limit cycles in system is apparently small and analytical methods of the search are demanded. In the history of investigation of quadratic dynamical systems, the following scientists have to be mentioned: -N. Bautin - Three small limit cycles for quadratic system are found [2]; - I. Petrovskii & E. Landis - It is stated that quadratic systems have no more than 3 cycles [3]; - S. Shi - A counter-example: the quadratic system with 4 cycles [4] is found; -Yu. Ilyashenko - It is proved that the number of limit cycles of quadratic (and any polynomial) system is finite [5]. 2. METHODS FOR THE SOLUTION OF PROBLEM. Limit cycles can be divided into two groups: large limit cycles (which can be obtained with the help of computer modeling) and small cycles (which can be obtained in purely analytical way). In the classical works of Poincare [6] and Lyapunov [7] there was developed the method of Lyapunov quantities (also called the Poincare-Lyapunov constants or focus values), which determine a behavior of system in the neighborhood of the boundary of stability domain. The method can also be used [2] for the search of small limit cycles. Let the point (0, h) (h is small enough) be the initial data of a solution of the system (see above) and let T(h) be the time at which the trajectory (x, y) meets first the upper semiaxis y. It is proved that y(T(h)) can be represented as series in h. So the sign of y(T(h))-h = L 0 h+L1h 3 +L 2 h 5 +... is defined by the sign of the first nonzero coefficient L , that coincides with the k-th Lyapunov quantity. k

In this figure the technique of constructing small cycles is illustrated. If we have a system with L 0 =0 and L1 >0, then by small disturbances of coefficients of system we can obtain L0 <0 , L1 >0 , and | L0 |<<| L1|. It means the winding of trajectory with very small initial data h1 and the unwinding of trajectory with h2 ( h1 <
conditions of existence of large limit cycles. These two methods together with the method of Lyapunov quantities are actively used in my work. 3. USING THE COMPUTER. 1.) For investigation of small limit cycles the Lyapunov quantities are usually obtained. They can be computed as the symbolic expressions in the terms of coefficients of system. Note that the expressions for the first and second Lyapunov quantities were obtained in 40-50s of the 20th century by N. Bautin [2] and N. Serebryakova [10], respectively. The expressions for Lyapunov quantities of higher degrees are lengthy and cannot be obtained without help of powerful computer. For the computation of Lyapunov quantities, the algorithms for computation software packages (MatLab in my case) are created. The study of symbolic computation of Lyapunov quantities is a difficult and important problem [2, 10]. In the figure is shown the expression for the fifth Lyapunov quantity for Lienard system. 2.) Kolmogorov's experiment permits one to suggest that for system the probability to have a limit cycle is, evidently, small. But, in fact, even for systems with cycles, visualization of large limit cycles is a difficult problem because we need to find proper initial data, options of system, and time. The use of computation software packages makes the visualization more simple but creating effective algorithms for visualization is still of importance. The three large limit cycles obtained by me for quadratic dynamical system are shown in the figure. 4. APPLICATIONS. The study of limit cycles and Lyapunov quantities is of importance for the theory and for the practice. The methods of Lyapunov quantities can be used in engineering mechanics for the solution of important problem on the behavior of dynamical system with parameter close to the boundary of stability domain [2], for example, in studying the excitation of oscillations in electronic machinery [11, 12]. Quadratic dynamical systems describe different real biological systems [13, 14] such as the Lotka-Volterra predator-prey model. They also describe many chemical reactions [15, 16]. Investigation of limit cycles for the discrete dynamical

systems is important for consideration the mathematical model of the Phase-Locked Loop (PLL) (see e.g. [17-19]).

Number of publications

Number of publications

5. STATISTICS. The investigation of limit cycles has hundred years history. Thousands scientific works (books and papers), devoted to this Publications about limit cycles problem, are published every year. Here we give a and Hilbert's 16th problem statistical graph of the number of publications, devoted to Hilbert's 16th problem and limit cycles 100 for the last five years, what illustrates the popularity 80 60 of this Publications about limit cycles in 40 area of quadratic systems 20 scientif 0 50 ic `05 `06 `07 `08 `09 40 Year explora 30 20 tion. 10

Number of publications

0 The problem is still far from being resolved even for `05 `06 `07 `08 `09 the simple cases. This can be shown by statistical Year graph of Publications about limit cycles the number of publications, devoted to limit cycles in 1200 quadratic systems, for the last five years. 1000 800

As was remarked, the investigation of limit cycles is of importance not only in mathematics but in biology, 400 physics, chemistry, engineering (and so on). In the 200 statistical graph is shown the huge number of 0 publications on limit cycles (in different areas of `05 `06 `07 `08 `09 Year science) for the last five years, it is obvious that this number increases permanently. There statistical data are from the site www.sciencedirect.com. 600

6. MY WORK: THE OBTAINED RESULTS. My work is connected with investigation of limit cycles since 2008. The results obtained are the following: - Methods were developed and algorithms were created for efficient computation of Lyapunov quantities, what allowed to obtain a general form of the 4th Lyapunov quantity 1 L = (106920g g f g g +16200 f 2 f g g g + ... 4 518400 30 02 11 20 13 11 20 11 02 13

45 pages ...− 42525 f

f + 78975 f g ). 30 07 30 16

(the whole expression can be found on the site https://sites.google.com/site/okuznetsovamath/home/4lyapqu.pdf) and a general form of the 5th

Lyapunov quantity for Lienard equation; - New simple analytical conditions of existence of large limit cycles were obtained. They describe the domain, extending the well-known domain of Shi; - Complete survey, devoted to limit cycles in quadratic systems, was prepared; - Investigation of coexistence of small and large limit cycles for cubic systems was fulfilled; - New configuration of large limit cycles was found and the conditions of its existence were described analytically.

Finally, it should be remarked that I continue the work of my colleagues: N.V. Kuznetsov [20] and E.V. Kudryashova [19], who defended their PhD dissertations in the University of Jyväskylä in 2008 and 2009, respectively. 7. LITERATURE: 1. Arnold V.I. Experimental mathematics. Moscow. Faziic. 2005. 63 p. 2. N. Bautin On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. N.S. 30(72). 1952. P. 181-196 (Russian); Transl. Amer. Math. Soc. 100(1). 1954. P. 397-413. 3. Petrovskii I.G. & Landis E.M. On the number of limit cycles of the equation dy/dx=P(x;y)/Q(x;y), where P and Q are polynomials of the second degree. 1955. Mat. Sb. N.S. 37(79). P. 209-250; Amer. Math. Soc. Transl. 16(2). P. 177-221. 4. Shi S. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica A 23. 1980. P. 153-158. 5. Ilyashenko Yu. Finiteness Theorems for Limit Cycles. Translations of Mathematical Monographs 94. American Mathematical Society, Providence, RI. 1991. 6. Poincare H. Memoire sur les courbes definies par lesequations diffeentielles. J. de. Mathematiques Pures et Appliquees. № 4. Vol. 1. 1885. P. 167-244. 7. Lyapunov A.M. The General Problem of the Stability of Motion. Gostekhteorizdat, Moscow. 1950. Taylor and Francos. 1992. 8. Leonov G.A. Limit cycles of Lienard system with discontinuous coefficients. Doklady Akademii Nauk. Vol. 426(1). 2009. P. 47-50. 9. Leonov G.A. Separatrix method of large limit cycles localization. 2010 [in print] 10. Serebryakova N.N. On the Behavior of a Dynamical System with One Degree of Freedom Near the Points of the Boundary of the Domain of Stability, where the Safe Boundary Becomes Dangerous. Izv. AN SSSR, Mech. Mach. No. 2. 1959. P. 178-182. 11. Andronov A.A., Vitt E.A., Khaikin S.E. Theory of Oscillators. Pergamon Press, Oxford. 1966. 12. T. Eirola, G.A. Leonov, F.F. Rodyukov, G. Söderbacka Stability of Mathematical Models for Systems of Synchronous and Asynchronous Machines.Technische Mechanik. B. 16. H. 3. 1996. P. 237-244. 13. D.S. Jones, B.D. Sleeman Differential equations and mathematical biology. Chapman & Hall/ CRC Mathematical Biology and Medicine Series. 2003. 14. T. Eirola, A.V. Osipov, G. Söderbacka Chaotic Regimes in a Dynamical System of the Type Many Predators One Prey. Helsinki University of Technology Institute of Mathematics Research Reports. A368. 1996. 15. Aris R. On some dynamical diagrams of chemical reaction engineering. Chaos And Interdisciplinary Journal of Nonlinear Science. Vol. 9 (1). 1999. P. 3–12. 16. S. Strovatz Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Cambridge. Mass. Westview Press. 2005. 17. E.V. Kudryashova, N.V. Kuznetsov, G.A. Leonov, P. Neittaanmäki, S.M. Seledzhi Analysis and synthesis of clock generator. International conference on Physics and Control, Book of selected papers. World Scientific. 2009. 18. N.V. Kuznetsov, G.A. Leonov, P. Neittaanmäki, S.M. Seledzhi Analysis and design of computer architecture circuits with controllable delay line. International Conference on Informatics in Control, Automation and Robotics, Proceedings. Vol. 3. Signal processing, Systems Modeling and Control. 2009.

19. Kudryashova E.V. Cycles in Continuous and Discrete Dynamical Systems: Computations, Computer-assisted proofs, and Computer Experiments. University of Jyväskylä. 2009. 20. Kuznetsov N.V. Stability and Oscillations of Dynamical Systems: Theory and Applications. University of Jyväskylä. 2008.