Fibonacci-type polynomial as a trajectory of a discrete dynamical system

Families of polynomials which obey the Fibonacci recursion relation can be generated by repeated iterations of a 2×2 matrix,Q ₂, acting on an initial value matrix,R ₂. One matrix fixes the recursion relation, while the other one distinguishes between the different polynomial families. Each family of polynomials can be considered as a single trajectory of a discrete dynamical system whose dynamics are determined byQ ₂. The starting point for each trajectory is fixed byR ₂(x). The forms of these matrices are studied, and some consequences for the properties of the corresponding polynomials are obtained. The main results generalize to the so-calledr-Bonacci polynomials.     

Investigation of a discrete dynamical system in a neighborhood of the invariant torus

We study the behavior of a discrete dynamical system in a neighborhood of the invariant torus for the case where the trajectories may have arbitrary structure on the torus and establish conditions under which the system can be reduced to the canonical form in the indicated neighborhood.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1676–1685, December, 1995.     

Modeling of a discrete dynamic system using a modified Petri net

Together with the proposed modification connected with the introduction of a time-dependent net, we define a criterion for optimality of a sequence of initiations of transitions, and study a method of obtaining such a sequence. We given the construction of a model of a discrete dynamic system having finite parameters. to describe it we introduce a modification of the Petri net using a determination of the state of the positions, time delay of the transitions, and variation of the conditions and rules for initiating transitions. Translated fromDinamicheskie Sistemy, No. 13, 1994. pp. 39–50.     

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.     

Lagrangian stability of a nonlinear quasi-periodic system

In this paper, we prove the Lagrangian stability of the quasi-periodic system d²x/dt²+G_x(x,t)=0, where G is quasi-periodic in both x and t, respectively.     

Choosing trajectory sections with highest information content for a dynamic system

We solve the problem of choosing the minimum number of measurement points t₁,...,t_n in an interval [a, b] which are required for constructing polynomial P_n(t) with minimum deviation from a given trajectory f (t). We also consider the problem of optimal sensor placement. The solutions of these problems for the two- and three-dimensional cases are optimal in the sense of uniform approximation of functions and have the advantages of the least squares method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 90–98, 1987.     

A new forecasting method of discrete dynamic system

Usually, a linear differential equation is used to represent continuous dynamic systems, but a linear difference equation is used to represent discrete dynamic systems. AGO is one of the most important characteristics of grey theory, and its main purpose is to reduce the random of data. A linear differential equation, instead of a linear difference equation, is used to replace the grey differential equation to analyze discrete systems in this paper. The k-order derivatives of 1-AGO data are calculated after cubic spline interpolation of them, and the model parameters are estimated by means of the deterministic convergence scheme. ARIMA models are used to analyze the leading indicator in advance, and Fourier series with suitably chosen values of parameters is used for fitting the leading indicator. The model presented in this paper is called Grey Dynamic Model GDM(1,1,1).     

Regularity of a dynamic neighborhood of a regular language

Birational maps give the main research tool for the theory of Fano varieties, as we know from the fundamental works of V.A. Iskovskikh and his school. Nowadays one can exploit them in the new approach of D. Auroux to Mirror Symmetry of Fano varieties, which is based on a certain generalization of the notion of special Lagrangian fibration suitable for Fano varieties. We present a very simple example of how a special Lagrangian fibration can be transferred by a birational map. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 209–211.     

Study on the dynamical behaviors of a two-dimensional discrete system

The generalized Ushiki map is investigated. It is theoretically proven that there are transcritical and flip bifurcations and there exists a chaotic phenomenon in the sense of Marotto. And numerical simulations not only show the consistency with the theoretical analysis but also exhibit the complex dynamical behaviors including the period-6, 5, 3 windows, intermittency mechanics, attractor merging crisis and boundary crisis. The computation of Lyapunov exponents conforms the dynamical behaviors.     

Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system

A numerical method is proposed for constructing an external polyhedral estimate for the trajectory tube of a nonlinear dynamic system described by a differential inclusion. The method is based on the approximation of cross sections of the trajectory tube (reachable sets) for an auxiliary system described by the convex hull of the graph of the differential inclusion. It produces polyhedral estimates suitable for the direct study of tubes via computer visualization and for the solution of more general problems.

     

A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

In this paper, a quasi-periodically forced nonlinear beam equation \({u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}\) with hinged boundary conditions is considered, where μ > 0, \({\varepsilon}\) is a small positive parameter, \({\phi}\) is a real analytic quasi-periodic function in t with a frequency vector ω = (ω ₁,ω ₂ . . . , ω _m ), and the nonlinearity h is a real analytic odd function of the form \({h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} \neq0, \bar{r} \in {\mathbb {N}}.}\) The above equation admits a quasi-periodic solution.     

Sensitivity analysis in discrete dynamic programming

The problem of characterizing the minimum perturbations to parameters in future stages of a discrete dynamic program necessary to change the optimal first policy is considered. Lower bounds on these perturbations are derived and used to establish ranges for the reward functions over which the optimal first policy is robust. A numerical example is presented to illustrate factors affecting the tightness of these bounds.     

Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schrödinger operator with quasi-periodic potential v_n=λv(x+nω). We assume that the frequency ω satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove—in the perturbative regime—that for large disorder λ and for most frequencies ω the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.     

Existence of a complex closed trajectory in a three-dimensional dynamical system

The DN-tracking method is used to prove the existence of a closed trajectory in a quadratic system of ordinary differential equations in three dimensions.     

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.     

Autoresonance in a dynamic system

Results on the asymptotic analysis of a model of autoresonance using a natural small parameter, the amplitude of forcing oscillations, are presented. The given forcing perturbation is rapidly oscillating oscillations with small amplitude and slow varying frequency. The goal of the paper is to reveal conditions under which the trajectory of the system goes away from the initial equilibrium point by a distance of order 1 under the influence of such a perturbation.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.     

Solvable classes of discrete dynamic programming

As finite state models to represent a discrete optimization problem given in the form of an r-ddp (recursive discrete decision process), three subclasses of r-msdp (recursive monotone sequential decision process) are introduced in this paper. They all have a feature that the functional equations of dynamic programming hold and there exists an algorithm (in the sense of the theory of computation) to obtain the set of optimal policies. (In this sense, we may call them solvable classes of discrete dynamic programming.) Besides the algorithms for obtaining optimal policies, two types of representations are extensively studied for each class of r-msdp's. Other related decision problems are also discussed. It turns out that some of them are solvable while the rest of them are unsolvable.     

Efficient discrete implementations for a dynamic problem of linear elasticity

For a dynamic three-dimensional linear elasticity problem in velocities-stresses, we construct efficient difference schemes on the basis of various additive decompositions of the original spatial operator. They include a difference scheme whose efficient implementation at the “predictor” stage has the property of complete conservativeness. Another class of efficient difference schemes is related to the representation of the operator as a product of triangular operators, that is, an operator analog of the LU-decomposition. The parallelism degree of these difference schemes is the same as of explicit schemes.