Stability Of Several Classes Of Higher Order Nonlinear Difference Equations And Applications

Difference equation (system) is a powerful modeling tool to describe discrete-timesystems in the real world. Difference equation models can be established to depict avariety of natural and social phenomena. For instance, difference equation has a widerange of applications in algorithm analysis, population dynamics and economics.Furthermore, many continuous mathematical models can be converted to theircorresponding discrete versions so as to benefit numerical simulations. In the pastdecade or so, higher-order, non-autonomous, max-type difference equations andsystems of difference equations have received considerable attention.This thesis focuses on the study of global stability and global attractivity of someinteresting difference equations (systems). Besides, by difference equation modeling, anetwork worm propagation model containing user awareness is established. The maincontributions are listed below.①Some rational difference equations with real exponents are studied. First, theasymptotic formulas for the solutions of a higher-order Stevi equation and its Max-type counterpart are found under proper conditions. Second, the global stability of thepositive equilibrium of a symmetric rational difference equation is shown, provided thatthe absolute values of all relevant exponents are bounded by unity.②The global attractivity in two difference equations is examined. First, with theaid of two alternating sequences, the global attractivity of the unique positiveequilibrium for a non-autonomous difference equation is shown under proper conditions.Second, the global attractivity of the equilibrium of a max-type extension of the abovedifference equation is proved under proper conditions.③The global attractivity in a max-type non-autonomous difference equation withparameters is addressed. First, when only one parameter is allowed to vary, somesufficient conditions for the global attractivity of its positive solution are presented.Second, if multiple parameters are allowed to vary simultaneously, some othersufficient conditions for the global attractivity of its positive solution are given.④Dynamics of two difference systems is addressed. First, the global attractivityof the positive solution of a two-dimensional difference system is shown under properconditions. Second, by introducing a part-metric on matrices, the global stability of theequilibrium of a higher order cyclic difference system is proved. ⑤By difference equation modeling, a network worm propagation model with userawareness is proposed. The dynamics of this model is analyzed by use of the stabilitytheory concerning difference equations. First, the basic reproduction numberdetermining the behavior of worm propagation on the network is calculated. Then, theasymptotic stability of the worm-free equilibrium is proved if the threshold is belowunity. Finally, the asymptotic stability of the worm equilibrium is shown provided thethreshold exceeds unity.