Fields of investigation of L.D. Pustyl'nikov

in mathematics and in its applications

I. Qualitative theory of ordinary differential equations and dynamical systems

1) Reducibility to analytic normal form of nonautonomous transformations and differential equations in the neighbourhod of an equilibrium point ([2], [5], [53], [70])

2) The study of stability of nonautonomous conservative (and closed to them) systems in the neighbourhood of a fixed point and of an arbitrary trajectory ([3], [6], [15], [53])

3) The construction of periodic, quasiperiodic solutions, and the investigation of stability in finite and infinite-dimensional hamiltonian systems ([4], [6], [9], [15], [20], [22], [25], [28], [54], [55], [58], [65], [117])

4) Construction of strange attractors and bifurcations in infinite-dimensional systems of ordinary differential equations ([41], [102])

5) Construction of explicit and asymptotic solutions in conservative systems ([22], [40])

6) A control and an identification in systems of ordinary differential equations ([12], [149)

II. Final motions in mechanics of systems and particles

1) The study of an acceleration mechanism in models associated with Fermi-Ulam model ([1], [4], [6], [16], [27], [30], [33], [47], [53], [56], [110], [112], [116], [124])

2) Construction of oscillatory motions ([1], [4], [6], [20], [28], [34], [71], [117])

3) Final motions in n-body problem of celestial mechanics ([34], [71])

III. Application of dynamical systems to rigorous justification of laws in statistical mechanics, accelerators theory and plasma physics

1) Rigorous justification of the second law of thermodynamics in Poincarè models for nonequilibrium gas ([35], [53], [74])

2) Construction of phase transitions in Frenkel-Kontorovou model (829], [51])

3. Qualitative behaviour of trajectories of systems with $\delta$-type interaction, arising in the accelerators theory and plasma physics ([9], [15], [20], [28], [33], [117])

4) Quantum chaos ([75], [96], [109], [114])

IV. Some problems of analysis and number theory

1) Application of ergodic theory to finding strong estimates of Weyl sums and to the remainder term in the law of the distribution of the fractional parts of polynomials ([13], [39], [46], [48], [49], [50], [72], [106])

2) Probability and ergodic laws in the distribution of fractional parts of polynomials ([75], [85], [114])

3) Problems of distribution of quadratic residues and non-residues ([45], [64], [95], [104], [106])

4) New theory of generalized continued fractions (849], [50], [64], [72], [103], [104], [106], [109])

5) Classical zeta-function and Riemann hypothesis ([36], [73], [81], [84], [107], [120])

V. Probability theory ([82], [85], [101], [105])

VI. Toeplitz and Hankel matrices and their applications

1) The structure of Toeplitz and Hankel matrices ([10], [18], [31])

2) Fast computations in problems of linear algebra connected with Toeplitz and Hankel matrices ([11], 818], [31])

3) Fast prediction of random processes ([18], [31])

4) Toeplitz and Hankel integral operators ([18])

VII. Combinatorics ([94], [123])

VIII. Geometry ([111], [118], [119])

IX. Ergodic theory in partial differential equations ([83])

X. Theory of singularities ([89], [92], [98])

XI. Applications of mathematics to some fields of discrete optimization and energetics ([7], [8], [14], [17], [19], [21], [23], [24], [26], [32], [37], [38], [42], [43], [44])