tailieunhanh - STABILITY FOR DELAYED GENERALIZED 2D DISCRETE LOGISTIC SYSTEMS CHUAN JUN TIAN AND GUANRONG CHEN

STABILITY FOR DELAYED GENERALIZED 2D DISCRETE LOGISTIC SYSTEMS CHUAN JUN TIAN AND GUANRONG CHEN Received 28 August 2003 and in revised form 19 February 2004 This paper is concerned with delayed generalized 2D discrete logistic systems of the form xm+1,n = f (m,n,xm,n ,xm,n+1 ,xm−σ,n−τ ), where σ and τ are positive integers, f : N2 × R3 → 0 R is a real function, which contains the logistic map as a special case, and m and n are nonnegative integers, where N0 = {0,1,.} and R = (−∞, ∞). Some sufficient conditions for this system to be stable and exponentially stable. | STABILITY FOR DELAYED GENERALIZED 2D DISCRETE LOGISTIC SYSTEMS CHUAN JUN TIAN AND GUANRONG CHEN Received 28 August 2003 and in revised form 19 February 2004 This paper is concerned with delayed generalized 2D discrete logistic systems of the form Xm 1 n f m n Xm n Xm n 1 Xm-ữ n-T where Ơ and T are positive integers f N0 X R3 R is a real function which contains the logistic map as a special case and m and n are nonnegative integers where No 0 1 . andR - TO to . Some sufficient conditions for this system to be stable and exponentially stable are derived. 1. Introduction In engineering applications particularly in the fields of digital filtering imaging and spatial dynamical systems 2D discrete systems have been a subject of focus for investigation see . 1 2 3 4 5 6 and the references cited therein . In this paper we consider the delayed generalized 2D discrete systems of the form xm 1 n f m n xm n Xm n 1 xm-ơ n-r where Ơ and T are positive integers m and n are nonnegative integers and f No X R3 R is a real function containing the logistic map as a special case where R -TO to R3 R X R X R No 0 1 . and N2 No X No m n I m n 0 1 . . Obviously if f m n X y z ựm nX 1 - x - am ny f m n X y z m nx 1 - z - am ny f m n X y z 1 - px2 - ay or f m n X y z bm nX - am ny - Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 4 2004 279-290 2000 Mathematics Subject Classification 39A10 URL http S1687183904308101 280 Stability for 2D logistic systems then system becomes respectively xm 1 n am nxm n 1 pm nxm n 1 xm n xm 1 n am nxm n 1 pm nxm n 1 xm ơ n-r xm 1 n axm n 1 1 xm or xm 1 n am nxm n 1 bm nxm n pm nxm ơ n r 0. Systems and are regular 2D discrete logistic systems of different forms and particularly system has been studied in the literature 2 4 5 6 . If am n 0 pm n p and n n0 is fixed then system becomes the 1D logistic system xm 1 n0 pxm n0 1 xm n0 where