Monotone flows with dense periodic orbits

The main result is Theorem 1: A flow on a connected open set X ⊂ Rd is globally periodic provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32], and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally related periods (Theorem 2)

Frequency theorem for the regulator problem with unbounded cost functional and its applications to nonlinear delay equations

We study the quadratic regulator problem with unbounded cost functional in a Hilbert space. A motivation comes from delay equations, which has the feedback part with discrete delays (or, in other words, delta-like measurements, which are unbounded in $L_{2}$). We treat the problem in an abstract context of a Hilbert space, which is rigged by a Banach space. We obtain a version of the Frequency Theorem, which guarantees the existence of a unique optimal process (starting in the Banach space) and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that these functionals can be used to construct inertial manifolds for delay equations and allow to treat and extend many papers in the field of applied dynamics (especially, developments of convergence theorems and the Poincar\'{e}-Bendixson theory done by R. A. Smith, construction of inertial manifolds for delay equations with small delays done by C. Chicone, Yu. A. Ryabov and R. D. Driver) in a unified manner. We also present more concrete applications concerned with frequency-domain stability criteria, which in particular cases coincide with the well-known Circle Criterion

Geometric theory of inertial manifolds for compact cocycles in Banach spaces

We present a geometric theory of inertial manifolds for compact cocycles (non-autonomous dynamical systems), which satisfy a certain squeezing property with respect to a family of quadratic Lyapunov functionals in a Banach space. Under general assumptions we show that these manifolds posses classical properties such as exponential tracking, differentiability and normal hyperbolicity. Our theory includes and largely extends classical studies for semilinear parabolic equations by C. Foias, G. R. Sell and R. Temam (based on the Spectral Gap Condition) and by G. R. Sell and J. Mallet-Paret (based on the Spatial Averaging) and their further developments. Besides semilinear parabolic equations our theory can be applied also to ODEs, ODEs with delay (extending the inertial manifold theories of Yu. A. Ryabov, R. D. Driver and C. Chicone for delay equations with small delays), parabolic equations with delay and parabolic equations with boundary controls (nonlinear boundary conditions). In applications, the squeezing property can be verified with the aid of various versions of the Frequency Theorem, which provides optimal (in some sense) and flexible conditions. This flexibility gives a possibility in applications to obtain conditions for low-dimensional dynamics, including, in particular, developments of the Poincar\'{e}-Bendixson theory and convergence theorems from a series of papers by R. A. Smith