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.     

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.     

On the solution of the trajectory survival problem for a nonlinear dynamical system

Nonlinear control systems possessing the flatness property are encountered in many applied mathematical models. In this paper, a trajectory survival problem is considered for a specific nonlinear system that possesses the above property. A method based on the properties of the system is proposed for constructing a control that solves the trajectory survival problem when the controlled object moves to the goal set within a bounding set containing an obstacle. Results of numerical calculations of the control and the trajectory of a system with a given initial position are presented.     

Newtonian and special-relativistic predictions for the trajectory of a slow-moving dissipative dynamical system

We show that the trajectories predicted by Newtonian mechanics and special relativistic mechanics from the same parameters and initial conditions for a slow-moving dissipative dynamical system will rapidly disagree completely if the trajectories are chaotic or transiently chaotic. There is no breakdown of agreement if the trajectories are non-chaotic, in contrast to the slow breakdown of agreement between non-chaotic Newtonian and relativistic trajectories for a slow-moving non-dissipative dynamical system studied previously. We argue that, once the two trajectory predictions are completely different for a slow-moving dissipative dynamical system, special relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly study its trajectory.     

On one problem of tracking a given trajectory

A controlled system nonlinear in phase variables is considered. Under the assumption that the system is subject to an uncontrollable disturbance, an algorithm of forming the control is specified such that a given trajectory is tracked.     

The number of cells of a dynamical system

In a dynamical system with a finite number of elementary stationary points, in which just these points serve as the limiting sets of its trajectories, a component of the connection of the set of trajectory points with the common positive and common negative limiting set is called a cell. An example is constructed which shows that a dynamical system can have any finite number of cells even though the number of stationary points is fixed.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 707–714, June, 1968.     

The conservative model of a dissipative dynamical system

Let R_σ be the response operator of a dissipative dynamical system (DS) governed by the equation u_tt−σu_t−u_xx=0, x>0, where σ=σ(x)≧0. Let R_q be the response operator of a conservative DS governed by the equation u_tt−u_xx+qu=0, x>0, where q=q(x) is real. We demonstrate that for any dissipative DS there exists a unique conservative DS (the “model”) such that R_σ=R_q. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 21–35. Translated by M. I. Belishev.     

Concerning a stochastic dynamical system

We study the discrete-time dynamical system $$X_{n + 1} = 2\sigma \cos (2\pi \theta _n )g(X_n ), n \in \mathbb{Z},$$ Whereθ _n is an ergodic stationary process whose univariate distribution is uniform on the interval [0, 1], the functiong(x) is odd, bounded, increasing, and continous, and ? is the ring of integers. It is proved that under certain conditions there exists a unique stationary process that is a solution of the above equation and this process has a continous purely singular spectrum.     

The conflict triad dynamical system

A dynamical model of the natural conflict triad is investigated. The conflict interacting substances of the triad are: some biological population, a living resource, and a negative factor (e.g., infection diseases). We suppose that each substance is multi-component. The main coexistence phases for substances are established: the equilibrium point (stable state), the local cyclic orbits (attractors), the global periodic oscillating trajectories, and the evolution close to chaotic. The bifurcation points and obvious thresholds between phases are exhibited in the computer simulations.     

On a nonlinear p-adic dynamical system

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ? _p , when there exists a unique fixed point x ₀. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x ₀). We show that there exists a radius r depending on parameters of the rational function such that: when x ₀ is an attracting point then the trajectory of an inner point from the ball U _r (x ₀) goes to x ₀ and each sphere with a radius > r (with the center at x ₀) is invariant; When x ₀ is a repeller point then the trajectory of an inner point from a ball U _r (x ₀) goes forward to the sphere S _r (x ₀). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U _r (x ₀) or stays in S _r (x ₀) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U _r(x ₀) it will stay (for all the rest of time) in the sphere (outside of U _r(x ₀)) that it reached first.     

Waveform relaxation as a dynamical system

In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time map generated by the relaxation process. This is done, and -closeness of the resulting map to the time map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed.

     

On a one-dimensional version of the dynamical Marguerre-Vlasov system

A one-dimensional version of the so-called Marguerre-Vlasov system of equations describing the vibrations of shallow shells is considered. The system depends on a parameter 0 in a singular way and undergoes the effect of damping mechanisms. We show that the system converges to a nonlinear beam equation while the energy decays exponentially uniformly (on 0) as time goes to infinity.