Local and Global Stability of Certain Mixed Monotone Fractional Second Order Difference Equation with Quadratic Terms

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

Estimation of Velocity of a Frictionless Motion of a Truck on an Infinitely Long Straight Rail

This paper is concerned with estimation of velocity of a frictionless motion of a truck on an infinitely long straight rail. For simplicity assume that the Truck is controlled only by the throttle producing an accelerative force per unit mass. A discrete dynamic model of first order difference equation is to describe the system. Kalman filtering technique is applied to the discrete dynamic model to estimate the velocity of the Truck at any particular time. A computer programme is developed to simulate the system

Qualitative Behavior of a Nonlinear Generalized Recursive Sequence with Delay

Difference equations are of growing importance in engineering in view of their applications in discrete time-systems used in association with microprocessors. We will check out the global stability and boundedness for a nonlinear generalized high-order difference equation with delay.

Revisiting Samuelson’s models, linear and nonlinear, stability conditions and oscillating dynamics

In this work, we reconsider the dynamics of a few versions of the classical Samuelson’s multiplier–accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations converging to the equilibrium for a wide range of values of the parameters, and give its analytic solution for all the possible cases. A delayed version proposed in the recent literature, represented by a linear third-order difference equation, is also considered. We show that also this model is able to produce converging oscillations, and give a complete analysis of the stability region of the equilibrium. A new simple nonlinear model is proposed, showing that it keeps oscillatory behavior, although coupled with other dynamics related to global effects. Our analysis confirms that the seminal work of Samuelson and simple modifications of it, may give powerful tools in the study of the business cycles.

ESTIMATION OF INTERIOR TEMPERATURE OF AN ELECTRIC OVEN

This paper is aimed at estimating interior temperature of an electric oven with respect to the jacket temperature. A discrete dynamic model of first order difference equation is described for the system. Kalman filtering technique is applied to the discrete dynamic model for estimation of the interior temperature. A computer program is written to simulate the system. It was observed that the estimates of the interior temperatures are directly proportional to estimates of the Jacket temperatures with proportionality constant of 0.0009. With this method it is therefore possible to obtain the interior temperature of the electric oven at any given time.

Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

In this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.