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## Resonance and marginal instability of switching systems

We analyze the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Chitour, Mason and Sigalotti (2012) stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. The concept of resonance originated with the same authors is modified. A characterization of marginal instability under some mild assumptions on the system is provided. These assumptions can be verified algorithmically and are believed to be generic. Finally, we analyze possible types of fastest asymptotic growth of trajectories. An example of a marginally unstable pair of matrices with non-polynomial growth is given.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

Nonlinear differential dynamic model of the relation between the branches of production was proposed. Mathematically, this model is expressed as a system of first-order ODE. Dynamic variables of the model – the value of the output of each branch of production. Each differential equation of the system includes independent growth and diminution of finished goods; growth and decline of production related to the production of allied industries. Two models were proposed: a model with Malthusian products growth (model with no restrictions on the amount of product), the model with the Verhulst limiting of the growth of output. The equilibrium points of dynamical systems, system stability were determined as well as the qualitative analysis of dynamic systems was made.

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. Stephen Smale has done pioneering work in differential topology, global analysis, dynamical systems, nonlinear functional analysis, numerical analysis, theory of computation and machine learning as well as applications in the physical and biological sciences and economics. In sum, Stephen Smale has manifestly broken the barriers among the different fields of mathematics and dispelled some remaining prejudices. He is indeed a universal mathematician. Smale has been honored with several prizes and honorary degrees including, among others, the Fields Medal(1966), The Veblen Prize (1966), the National Medal of Science (1996) and theWolf Prize (2006/2007).

We define, calculate and analyze irregularity indices λISSN of daily series of the International Sunspot Number ISSN as a function of increasing smoothing from *N* = 162 to 648 days. The irregularity indices λ are computed within 4-year sliding windows, with embedding dimensions *m* = 1 and 2. λISSN displays Schwabe cycles with ~5.5-year variations ("half Schwabe variations" HSV). The mean of λISSN undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio *R* of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and, importantly, smoothing parameter *N*. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of *R* as a function of *N*: Q1, with increasing HSV behavior and *R* value as *N* is increased, occurs before 1915–1930; and Q2, with decreasing HSV behavior and *R* value as *N* is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods *T*1 and *T*2 (*T*1 = 11 years, and intermediate *T*2 is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When *m* = 1, HSV appears in the absence of *T*2 variations. When *m* = 2, Q1 occurs when *T*2 variations are present, whereas Q2 occurs when *T*2 variations are suppressed. We propose that the HSV behavior of the irregularity index of ISSN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.

Generalized error-locating codes are discussed. An algorithm for calculation of the upper bound of the probability of erroneous decoding for known code parameters and the input error probability is given. Based on this algorithm, an algorithm for selection of the code parameters for a specified design and input and output error probabilities is constructed. The lower bound of the probability of erroneous decoding is given. Examples of the dependence of the probability of erroneous decoding on the input error probability are given and the behavior of the obtained curves is explained.