Hassan Sedaghat

Dynamics of rational difference equations containing quadratic terms

Second order rational difference equations with quadratic terms in their numerators and linear terms in their denominators exhibit a rich variety of dynamic behaviors. It is demonstrated that depending on the parameters and initial values, there can be globally attracting fixed points, coexisting periodic solutions or chaotic trajectories.

Global behaviours of rational difference equations of orders two and three with quadratic terms

We determine the global behaviours of all solutions of the following rational difference equations These equations are related to each other via semiconjugate relations that also let us reduce them to first-order equations. Using this approach, we determine the forbidden sets of each equation explicitly and show that for initial values outside the forbidden sets, their solutions may converge to 0, or to a positive fixed point, or they may be periodic of period 2 or unbounded. In some cases, different types of solutions coexist depending on the initial values.

Monotone and oscillatory solutions of a rational difference equation containing quadratic terms

We show that the second order rational difference equation has several qualitatively different types of positive solutions. Depending on the non-negative parameter values A,B,C,α,β, all solutions may converge to 0, or they may all be unbounded. For some parameter values both cases can occur, or coexist depending on the initial values. We find converging solutions of both monotonic and oscillatory types, as well as periodic solutions with period two. A semiconjugate relation facilitates derivations of these results by providing a link to a rational first order equation.

A class of nonlinear second order difference equations from macroeconomics

where the constant A,, = Co + I0 + Go represents the sum of the minimum consumption, the “autonomous” investment and the fixed government spending in period n, and Y, is the output--GNP or national income-in period n. The net investment amount in the same period is given as Z, = ac( Y,I Y,-,). The constant c E (0, 1) represents Keynes ’ “marginal propensity to consume” or the MPC, while the coefficient CY > 0 is the “accelerator”. The linear model above improved the earlier Keynesian models and resulted in substantial new research. However, this model was soon found to be unsatisfactory, since CYC can exceed unity in typical economic settings (see, [2, Chap. 91). This fact results in exponentially divergent solutions for the linear equation, which of course, is not observed in reality. Certain nonlinear models were subsequently proposed to resolve this anomaly. For instance, rather than the linear Keynesian consumption C(Y) = cY + C’, , Samuelson considered a nonlinear consumption function [3]. Samuelson’s assumptions amount to a third order difference equation and an MPC which is itself a decreasing&n&on of output. Some years later, Hicks proposed a model in which consumption was linear, but investment and output were both piecewise linear [4]. Hicks’ model results in a second order difference equation with constant MPC, but the accelerator is not defined (or could mathematically be set equal to zero) for a certain range of output differences; the model also happens to be nonautonomous because of a time dependent “hard ceiling” on output (as well as the induced investment “floor”). There have been several other models such as continuous time models of Goodwin and Kaldor, or the more recent stochastic models which we have not mentioned because of the attributes italicized here; [5] contains brief discussions of some of these models and comprehensive bibliography.

Global attractivity in a quadratic-linear rational difference equation with delay

We investigate the global behaviour of non-negative solutions of the following rational difference equation with arbitrary delay and quadratic terms in its numerator: where all coefficients are non-negative and and γ>0. In this case, the origin is the only non-negative fixed point and we establish the asymptotic stability of the origin relative to an invariant set and behaviour of positive solutions outside that invariant set. We also state conditions implying that the invariant set is , i.e. the origin is a global attractor of all non-negative solutions.

The Impossibility of Unstable, Globally Attracting Fixed Points for Continuous Mappings of the Line

for every choice of xO E R (fan represents the n-th iterate of fa under function composition). Clearly, once xk 2 a for any k, then xn = 0 for all n > k. In particular, every solution of (1) converges to zero, regardless of the choice of xO. In this sense, the origin, which is the unique fixed point °f fa, is globally attracting. However, if xO + 0, then no matter how close xO is chosen to the origin, xn must first exceed a before ultimately reaching the origin. Hence, the origin is unstable (in fact, locally repelling). The preceding example shows that globally attracting fixed points that are not stable can easily occur in one-dimensional dynamical systems such as (1). Since fa is discontinuous at x = a, it is natural to ask whether a continuous example of an unstable global point attractor can be constructed in one dimension. As the title of this note suggests, this is not possible. To see why continuous maps are nice in this sense, we need a local or asymptotic stability result from [7, p. 47]. Complete definitions of all concepts and terminology used here can be found in [2] and [5].

Global Attractivity in a Rational Delay Difference Equation with Quadratic terms

For the following rational difference equation with arbitrary delay and quadratic terms: we determine sufficient conditions on the parameter values which guarantee that the unique non-negative fixed point attracts all positive solutions. When the fixed point is the origin (F = 0), we show that it attracts all non-negative solutions of the more general equation where . We also show that altering some of the above conditions on parameters causes the origin to not be globally attracting.

A note: all homogeneous second order difference equations of degree one have semiconjugate factorizations

We show that the second order difference equation on a group G has an i-semiconjugate factorization into a triangular system of first order equations if and only if each mapping is homogeneous of degree 1.

Asymptotic stability for difference equations with decreasing arguments

We consider general, higher order difference equations of type in which the function f is non-increasing in each coordinate. We obtain sufficient conditions for the asymptotic stability of a unique fixed point relative to an invariant interval. We also discuss various applications of our main results.

Bounded Oscillations in the Hicks Business Cycle Model and other Delay Equations

Sufficient conditions for the persistent (non-decaying) oscillations of the bounded trajectories of the delay difference equation are obtained. Applications include the Hicks equation for the trade cycle with arbitrary lag structure. The case m=2 of second order equations is discussed in greater detail with applications to certain rational recursive sequences and to a generic extension of the Hicks equation with a one-period lag.