Periodic Orbits On Discrete-Time Dynamical Systems

In this paper, we study the existence of periodic solution of first order linear difference equation and nonlinear difference equations with delayed feedback and obtain a series of new results about the existence of periodic solutions, which generalize the conclusions of the corresponding difference equations with discrete variables. The paper is composed of three chapters.In the first chapter, we introduce the background of problem-researching, the recent development of the research in this field and what we have done in this paper. Furthermore, we illustrate the importance of our work in theory and practice.In the second chapter, we consider the difference equation arising as a discrete-time network of single neuron, whereβis the internal decay, g is a signal function with McCulloch-Pitts nonlinearity. We can obtain the following theorems and corollary.THEOREM2.1 Letβ∈(1,∞).Then (1) has a stationary state, Periodic orbits of period2,4. Suppose there is a positive integer k such that Then, (1) has a periodic orbit of period 2k.Theorem2.3 Letβ∈(1,∞), k∈N(1), then difference (1) has a periodic solution of period 2 k + 1 if and only ifCorollary2.4 Ifβ2 k 2β2 k2 + 2β2 1≥0 holds, (1) has periodic orbits of period 2 k ,2 k 2, ,4,2 and a stationary. Ifβ2 k +1 2β2 k1 1≥0holds, (1) has periodic orbits of period2 k + 1,2 k+ 3,. We obtain better theorems than corresponding those of [3,4]. In the third chapter, we consider the discrete-time system describing the dynamic interaction of two identical neurons, whereβ∈(0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. System (2) can be regarded as the discrete analog of the following artificial neuron network of two neurons with delayed feedback: where ddxt and ddyt are replaced by the forward difference x ( n + 1) x ( n)and y ( n + 1) y ( n). By algebra theory, we get a theorem.Theorem3.1 Letβ∈(0, 12] andσ< (11 +ββ) 2( 1k + 3β22 kβ+3). Then there existsΦ0 = ( 0 ,φ0 )∈Xσ+ ,+, such that the solution with initial valueΦ0 is periodic with the minimal period4( k + 1). Moreover, there exists a positive integer m such that for anyΦ= ( ,φ)∈Xσ. and...