Table Of ContentNonlinear Systems and Complexity
Series Editor: Albert C.J. Luo
Albert C.J. Luo
Memorized
Discrete
Systems and
Time-delay
Nonlinear Systems and Complexity
Volume 17
Series editor
Albert C.J. Luo
Southern Illinois University
Edwardsville, IL, USA
Nonlinear Systems and Complexity provides a place to systematically summarize
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niques in nonlinear systems and mathematics with engineering applications. The
Series establishes highly relevant monographs on wide ranging topics covering
fundamental advances and new applications inthe field. Topical areas include, but
are not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos
Computational methods for nonlinear systems Stability, bifurcation, chaos and
fractals in engineering Nonlinear chemical and biological phenomena Fractional
dynamics and applications Discontinuity,synchronizationand control.
More information about this series at http://www.springer.com/series/11433
Albert C.J. Luo
Memorized Discrete Systems
and Time-delay
123
Albert C.J.Luo
Department ofMechanical andIndustrial
Engineering, Schoolof Engineering
SouthernIllinois University Edwardsville
Edwardsville, IL
USA
ISSN 2195-9994 ISSN 2196-0003 (electronic)
Nonlinear Systemsand Complexity
ISBN978-3-319-42777-5 ISBN978-3-319-42778-2 (eBook)
DOI 10.1007/978-3-319-42778-2
LibraryofCongressControlNumber:2016945868
©SpringerInternationalPublishingSwitzerland2017
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Preface
Thisbookisthefirstbooktalkingaboutdiscretedynamicalsystemswithmemory.
Such memorized discrete systems extensively exist in bio-systems and financial
systems, and economical systems. In addition, time-delay systems can be dis-
cretized into the memorized discrete dynamical systems. Further, stability and
bifurcations of time-delay dynamical systems can be investigated through the
memorized dynamical systems. Currently, one has difficulty to determine periodic
motions in time-delay nonlinear systems. Thus, the discretization of time-delay
systemscanbeadopted,andthecorrespondingmemorizeddiscretesystemscanbe
used to investigate periodic motions to chaos in time-delay nonlinear systems. In
this book, systematical discussions on stability and bifurcations of memorized
nonlineardynamicalsystemsarefirstpresented,andthendiscretizationmethodsof
time-delay systems are also presented. Periodic motions to chaos in time-delay
nonlineardynamicalsystemscanbeinvestigatedthroughthememorized,nonlinear,
discrete systems.
This book consists offive chapters. The first chapter briefly discusses solutions
and stability in memorized linear discrete systems. In Chap. 2, stability and
bifurcationsoffixedpointsinmemorizednonlineardiscrete systemsarediscussed.
Discretization methods of time-delay nonlinear systems are presented in Chap. 3.
Semi-analytical methods for periodic motions in time-delay nonlinear dynamical
systemsarepresentedthroughthetheoryofmemorizeddiscretesystemsinChap.4.
In Chap. 5, bifurcation trees of periodic motions to chaos in time-delayed Duffing
oscillator are discussed through the semi-analytical method. With discrete Fourier
seriestheory,frequency–amplitudecurvescanbeobtained,whichcanbecompared
with analytical methods for periodic motions to chaos and motion complexity in
time-delay nonlinear systems.
Finally,Iwouldliketothankmystudent(SiyuanXing)forcomputingnumerical
results.Inaddition,Iwouldliketothankmyfamily’ssupportforthiswork.Ihopethe
materialspresentedhereinwouldlastlongforscienceandengineering.
Edwardsville, IL, USA Albert C.J. Luo
vii
Contents
1 Memorized Linear Discrete Systems . .... .... .... .... ..... .... 1
1.1 Basic Iterative Solutions.... .... .... .... .... .... ..... .... 1
1.2 Iterative Solutions with Distinct Eigenvalues .... .... ..... .... 4
1.3 Iterative Solutions with Multiplicity... .... .... .... ..... .... 12
1.4 Stability and Boundary..... .... .... .... .... .... ..... .... 27
1.5 1-D Memorized Linear Discrete Systems... .... .... ..... .... 46
1.5.1 1-D Linear Discrete Systems Without Memory. ..... .... 46
1.5.2 1-D Linear Discrete System with Finite-Step Memory .... 47
References.. .... .... .... ..... .... .... .... .... .... ..... .... 50
2 Memorized Nonlinear Discrete Systems... .... .... .... ..... .... 51
2.1 Definitions . .... .... ..... .... .... .... .... .... ..... .... 51
2.2 Fixed Points and Stability... .... .... .... .... .... ..... .... 55
2.3 Bifurcation and Stability Switching ... .... .... .... ..... .... 70
2.3.1 Stability and Switching... .... .... .... .... ..... .... 76
2.3.2 Bifurcations... ..... .... .... .... .... .... ..... .... 102
2.3.3 1-D Memorized Nonlinear Discrete Systems... ..... .... 110
References.. .... .... .... ..... .... .... .... .... .... ..... .... 114
3 Discretization of Time-delay Systems. .... .... .... .... ..... .... 115
3.1 Time-delay Systems.. ..... .... .... .... .... .... ..... .... 115
3.2 Basic Discretization Methods.... .... .... .... .... ..... .... 118
3.2.1 Forward Euler Method ... .... .... .... .... ..... .... 120
3.2.2 Backward Euler Method.. .... .... .... .... ..... .... 127
3.2.3 Trapezoidal Rule Discretization. .... .... .... ..... .... 133
3.2.4 Midpoint Method ... .... .... .... .... .... ..... .... 141
3.3 Explicit Runge–Kutta Methods... .... .... .... .... ..... .... 148
3.3.1 Runge–Kutta Method of Order 2 ... .... .... ..... .... 148
3.3.2 General Explicit Runge–Kutta Methods .. .... ..... .... 152
3.3.3 Runge–Kutta Method of Order 3 ... .... .... ..... .... 161
3.3.4 Runge–Kutta Method of Order 4 ... .... .... ..... .... 165
ix
x Contents
3.4 Implicit Runge–Kutta Methods... .... .... .... .... ..... .... 169
3.4.1 Polynomial Interpolation.. .... .... .... .... ..... .... 171
3.4.2 Implicit Runge–Kutta Methods. .... .... .... ..... .... 173
3.4.3 Gauss Method. ..... .... .... .... .... .... ..... .... 179
3.4.4 Radau Method. ..... .... .... .... .... .... ..... .... 183
3.4.5 Lobatto Method..... .... .... .... .... .... ..... .... 187
3.4.6 Diagonally Implicit RK Methods ... .... .... ..... .... 190
3.5 Multistep Methods ... ..... .... .... .... .... .... ..... .... 193
3.5.1 Adams–Bashforth Methods.... .... .... .... ..... .... 193
3.5.2 Adams–Moulton Methods. .... .... .... .... ..... .... 203
3.5.3 Explicit Adams Methods.. .... .... .... .... ..... .... 212
3.5.4 Implicit Adams Methods.. .... .... .... .... ..... .... 214
3.5.5 General Forms. ..... .... .... .... .... .... ..... .... 217
References.. .... .... .... ..... .... .... .... .... .... ..... .... 220
4 Periodic Flows in Time-delay Systems.... .... .... .... ..... .... 221
4.1 Interpolation Method . ..... .... .... .... .... .... ..... .... 221
4.2 Continuous Integration Method .. .... .... .... .... ..... .... 242
4.3 Discrete Fourier Series..... .... .... .... .... .... ..... .... 264
References.. .... .... .... ..... .... .... .... .... .... ..... .... 270
5 Time-Delayed Duffing Oscillator .... .... .... .... .... ..... .... 271
5.1 A Semi-analytical Method .. .... .... .... .... .... ..... .... 271
5.1.1 Discretization . ..... .... .... .... .... .... ..... .... 271
5.1.2 Period-1 Motions ... .... .... .... .... .... ..... .... 273
5.1.3 Period-m Motions... .... .... .... .... .... ..... .... 276
5.1.4 Finite Discrete Fourier Series .. .... .... .... ..... .... 278
5.2 Bifurcation Trees of Periodic Motions . .... .... .... ..... .... 280
5.3 Frequency–Amplitude Characteristics.. .... .... .... ..... .... 291
5.4 Numerical Illustrations ..... .... .... .... .... .... ..... .... 293
Reference .. .... .... .... ..... .... .... .... .... .... ..... .... 296
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 297
Chapter 1
Memorized Linear Discrete Systems
In this chapter, the basic iterative solutions for memorized linear discrete systems
will be presented. The iterative solutions of the memorized linear discrete systems
withdistinctandrepeatedeigenvalueswillbediscussed.Thestabilityofmemorized
lineardiscretesystemswillbediscussedfromtheoscillatory,monotonic,andspiral
convergence and divergence. The invariant, flip, and circular critical boundaries
of the stability on the direction of the specific eigenvector will be classified.
The 1-dimensional, memorized, linear discrete systems will be discussed.
1.1 Basic Iterative Solutions
Basic concepts of memorized discrete dynamical systems will be presented herein
before further comprehensive discussion on the corresponding iterative solutions
and stability.
Definition1.1 Consideramemorized,linear,discrete,dynamicalsystembased on
a linear map P:ðx ;x ;(cid:3)(cid:3)(cid:3);x Þ!ðx ;x ;(cid:3)(cid:3)(cid:3);x Þ with the corre-
k k(cid:2)1 k(cid:2)s kþ1 k k(cid:2)sþ1
sponding relation
x ¼ax þRs bx þc fork;s2Z and
kþ1 k j¼1 j k(cid:2)j þ
x ¼x forr ¼k;k(cid:2)1;(cid:3)(cid:3)(cid:3);kþ1(cid:2)s ð1:1Þ
r r
x ¼ðx ;x ;(cid:3)(cid:3)(cid:3);x ÞT 2Rn
k 1k 2k nk
where a and b are n(cid:4)n matrice and c is a constant vector function. If c = 0, the
memorized linear discrete system in Eq. (1.1) is homogeneous. Equation (1.1)
becomes
x ¼ax þRs bx fork;s2Z andx 2Rn
kþ1 k j¼1 j k(cid:2)j þ k ð1:2Þ
x ¼x forr ¼k;k(cid:2)1;(cid:3)(cid:3)(cid:3);kþ1(cid:2)s
r r
©SpringerInternationalPublishingSwitzerland2017 1
A.C.J.Luo,MemorizedDiscreteSystemsandTime-delay,
NonlinearSystemsandComplexity17,DOI10.1007/978-3-319-42778-2_1
