Synthesis of a control in a non-linear dynamical system based on decomposition

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.     

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.

     

One-Time Pad as a nonlinear dynamical system

The One-Time Pad (OTP) is the only known unbreakable cipher, proved mathematically by Shannon in 1949. In spite of several practical drawbacks of using the OTP, it continues to be used in quantum cryptography, DNA cryptography and even in classical cryptography when the highest form of security is desired (other popular algorithms like RSA, ECC, AES are not even proven to be computationally secure). In this work, we prove that the OTP encryption and decryption is equivalent to finding the initial condition on a pair of binary maps (Bernoulli shift). The binary map belongs to a family of 1D nonlinear chaotic and ergodic dynamical systems known as Generalized Luröth Series (GLS). Having established these interesting connections, we construct other perfect secrecy systems on the GLS that are equivalent to the One-Time Pad, generalizing for larger alphabets. We further show that OTP encryption is related to Randomized Arithmetic Coding – a scheme for joint compression and encryption.     

On representing a black box as a dynamical system

Necessary conditions are derived for a black box to be representable as a dynamical system governed by ordinary differential equations. Sufficiency of these conditions depends primarily on the set of inputs. It is shown that the conditions are indeed sufficient for the set of constant inputs and under suitable hypotheses for the set of smooth inputs. These conditions cannot be verified by a finite number of input-output observations.     

The construction of discontinuous controls for non-linear dynamical system with uncertainties

The properties of the solutions of a family of non-linear parametric problems are investigated in the neighbourhood of regular and irregular values of the parameter. Rules are proposed for constructing feedback-type controlling for non-linear dynamical systems with perturbations. An example is presented.     

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.     

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型;利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指教值说明了控制的有效性.     

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.     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

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.     

A discrete dynamical system as a model of a neural network with generalized Hebbian synapses

We study the dynamical behavior of a discrete time dynamical system which can serve as a model of a learning process. We determine fixed points of this system and basins of attraction of attracting points. This system was studied by Fernanda Botelho and James J. Jamison in [A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547] but authors used its continuous counterpart to describe basins of attraction.     

On a system of non-linear differential equations

Summary E. Kasner(1925) considered a system of non-linear differential equations in three variables. The authors of this paper have extended this system to n variables by means of a linear transformation with complex coefficients. The formulas obtained for the solutions of the differential equations give a simplification of Kasner’s solutions. Entrata in Redazione il 26 agosto 1968.     

Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective

We conjecture that the well-known oscillations (3- to 5-yr and 10-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude—a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.