On bifurcations and local stability in 1-D nonlinear discrete dynamical systems
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On bifurcations and local stability in 1-D nonlinear discrete dynamical systems Albert C. J. Luo1 Received: 1 March 2020 / Revised: 4 April 2020 / Accepted: 7 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, a theory of bifurcations and local stability of fixed-points (or period-1 solutions) in one-dimensional nonlinear discrete dynamical systems is presented. The linearized discrete dynamical systems are discussed first, and the higher-order singularity and monotonic and oscillatory stability of fixed-points for one-dimensional nonlinear discrete dynamical systems are presented. The monotonic and oscillatory bifurcations of fixed-points (period-1 solutions) are presented. A few special examples in 1-dimensional maps are presented for a better understanding of the general theory for the stability and bifurcation of nonlinear discrete dynamical systems. Global analysis of period-2 motions for the sampled nonlinear discrete dynamical systems are carried out, and global illustrations of period-1 to period-2 solutions in the sampled nonlinear discrete dynamical systems are given. Keywords Bifurcation · Stability · Fixed points · Saddle-node bifurcations · Sink bifurcations · Source bifurcations
1 Introduction The bifurcation and stability in nonlinear discrete dynamical systems is an important topic in nonlinear science and engineering. With computer help, the bifurcation scenarios in nonlinear discrete dynamical systems were presented through numerical simulations. In fact, the existing lectures lack the theoretic studies of the mechanism and dynamics of bifurcation scenario. The discrete dynamical system started the differential equation discretization to get difference equations. One used such difference equations to get numerical results. The simplest difference equation is the Euler discrete map, which is obtained from discretized differential equations. For such difference equation, the numerical solutions can be obtained. To determine the converge and divergence of the difference equation, error analysis for each iteration step should be completed through the discrete systems of errors. Such convergence and divergence of the difference equations (or systems) are latter called stability in discrete dynamical systems. The Euler method is the one-step discretization method.
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Albert C. J. Luo [email protected] Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA
To improve computation accuracy and convergence, in 1883, Bashforth and Adams [1] developed the multi-step difference equations of differential equations. In 1926, Moulton [2] extended such multi-step discrete method. Thus, such a method is called the Adams–Moulton method. In 1895 Runge [3] proposed the modern one-step methods for higherorder accuracy difference equations of differential equations, and, in 1900, Heun [4] developed the fourth order Runge’s method. In 1901, Kutta [5] gave the formulation of the method with the order co
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