Form Symmetries and Reduction of Order in Difference Equations

322 Pages·2011·4.135 MB·English

by  Hassan Sedaghat

Mathematics
F ADVANCES IN DISCRETE MATHEMATICS AND ITS APPLICATIONS
Form Symmetries and Reduction of Order in Difference Equations o
o r
presents a new approach to the formulation and analysis of difference
r m
equations in which the underlying space is typically an algebraic group. d
e S
In some problems and applications, an additional algebraic or topological F S
r y o r m y m m e t r i e S
structure is assumed in order to define equations and obtain significant m
i
results about them. Reflecting the author’s past research experience, the n m

majority of examples involve equations in finite dimensional Euclidean d e
t r
spaces. i r a n d e d u c t i o n o F
F
i
F e
Covering difference equations on groups, the author first illustrates the e S
r

wide variety of possible formulations and interpretations of difference e a o d
equations that occur in concrete contexts. He then proposes a systematic n n r d e r i n i F F e r e n c e
c d
method of decomposition for recursive difference equations that uses a e
r
semiconjugate relation between maps. The final chapter goes beyond e
e e
semiconjugacy by extending the fundamental ideas based on form q d q u a t i o n S
u
symmetries to nonrecursive difference equations. u
a
c
t
t
Features i
o i
o
• Uses form symmetries to define semiconjugate factorizations of n
n
S
recursive higher-order difference equations and general nonrecursive
o
higher-order difference equations F

• Presents many of the main concepts and results using simple
algebraic structures, such as groups and fields
• Applies the methods to large classes of difference equations,
including nonautonomous and nonhomogeneous linear equations,
quadratic difference equations, and separable difference equations
• Includes many new results in print for the first time
With numerous examples and exercises, this book is an ideal S
E
introduction to an exciting new domain in the area of difference D
A
equations. It takes a fresh and all-inclusive look at difference equations G
H
and develops a systematic procedure for examining how these equations A
T
are constructed and solved.
HASSAN SEDAGHAT
K10389
K10389_Cover.indd 1 4/15/11 9:02 AM
F S
orm ymmetrieS
r
and eduction oF
o d
rder in iFFerence
e
quationS
K10389_FM.indd 1 4/13/11 10:27 AM
K10389_FM.indd 2 4/13/11 10:28 AM
ADVANCES IN DISCRETE MATHEMATICS AND ITS APPLICATIONS
F S
orm ymmetrieS
r
and eduction oF
o d
rder in iFFerence
e
quationS
HASSAN SEDAGHAT
VIRGINIA COMMONWEALTH UNIVERSITY
RICHMOND, USA
K10389_FM.indd 3 4/13/11 10:28 AM
CRC Press
Taylor & Francis Group
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© 2011 by Taylor & Francis Group, LLC
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Version Date: 20111012
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Contents
Preface ix
1 Introduction 1
2 Difference Equations on Groups 5
2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 One equation, many interpretations . . . . . . . . . . . . . . . 10
2.3 Examples of difference equations on groups . . . . . . . . . . 19
2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Semiconjugate Factorization and Reduction of Order 35
3.1 Semiconjugacy and ordering of maps . . . . . . . . . . . . . . 35
3.1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Coordinate transformations vs semiconjugacy . . . . . 36
3.1.3 Semiconjugacy as a map-ordering relation . . . . . . . 38
3.1.4 The semiconjugacy problem . . . . . . . . . . . . . . . 39
3.1.5 Semiconjugacy and difference equations . . . . . . . . 41
3.1.6 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Form symmetries and SC factorizations . . . . . . . . . . . . 44
3.2.1 Order-reducing form symmetries . . . . . . . . . . . . 44
3.2.2 The semiconjugate factorization theorem . . . . . . . . 46
3.3 Order-reduction types . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 The basic concept . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Nonuniqueness of factorizations and reduction types . 51
3.3.3 Reduction types (k,1) and (1,k) . . . . . . . . . . . . 53
3.3.4 Factor and cofactor chains . . . . . . . . . . . . . . . . 54
3.4 SC factorizations as triangular systems . . . . . . . . . . . . . 55
3.5 Order-preserving form symmetries . . . . . . . . . . . . . . . 57
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Homogeneous Equations of Degree One 65
4.1 Homogeneous equations on groups . . . . . . . . . . . . . . . 65
4.2 Characteristic form symmetry of HD1 equations . . . . . . . . 68
4.3 Reductions of order in HD1 equations . . . . . . . . . . . . . 70
4.4 *Absolute value equation . . . . . . . . . . . . . . . . . . . . 79
v
vi
4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Type-(k,1) Reductions 107
5.1 Invertible-map criterion . . . . . . . . . . . . . . . . . . . . . 107
5.2 Identity form symmetry . . . . . . . . . . . . . . . . . . . . . 109
5.3 Inversion form symmetry . . . . . . . . . . . . . . . . . . . . . 113
5.4 *Discrete Riccati equation of order two . . . . . . . . . . . . 115
5.5 Linear form symmetry . . . . . . . . . . . . . . . . . . . . . . 128
5.5.1 Determination criterion . . . . . . . . . . . . . . . . . 128
5.5.2 Periodic solutions . . . . . . . . . . . . . . . . . . . . 132
5.5.3 *Limitcycles . . . . . . . . . . . . . . . . . . . . . . . 135
5.6 Difference equations with linear arguments . . . . . . . . . . . 139
5.6.1 The reduction theorem . . . . . . . . . . . . . . . . . . 140
5.6.2 Economics: Modeling the business cycle . . . . . . . . 143
5.6.3 Repeated reductions of order . . . . . . . . . . . . . . 144
5.6.4 SC factorization of linear equations, factor chains . . . 148
5.6.5 *A rational equation with chaotic solutions . . . . . . 151
5.7 Field-inverse form symmetry . . . . . . . . . . . . . . . . . . . 161
5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 Type-(1,k) Reductions 169
6.1 Linear form symmetry revisited . . . . . . . . . . . . . . . . . 170
6.2 Separable difference equations . . . . . . . . . . . . . . . . . 170
6.2.1 Form symmetries of additive forms over fields . . . . . 171
6.2.2 SC factorization of linear equations, cofactor chains . 179
6.2.3 Formsymmetriesofmultiplicativeformsoversubgroups
of C 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 183
\{ }
6.3 Equations with exponential and power functions . . . . . . . 189
6.3.1 The reduction theorem . . . . . . . . . . . . . . . . . . 189
6.3.2 Reductions in orders of expow equations . . . . . . . . 192
6.3.3 Repeated reductions of order . . . . . . . . . . . . . . 195
6.3.4 Solutions of equations containing only power functions 197
6.3.5 Biology: Modeling populations . . . . . . . . . . . . . 203
6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7 Time-Dependent Form Symmetries 217
7.1 The semiconjugate relation and factorization . . . . . . . . . 217
7.2 Invertible-map criterion revisited . . . . . . . . . . . . . . . . 221
7.3 Time-dependent linear form symmetry . . . . . . . . . . . . . 223
7.4 SC factorization of linear equations . . . . . . . . . . . . . . 227
7.4.1 Factorization theorem for linear equations . . . . . . . 227
7.4.2 Constant coefficients without field extensions . . . . . 230
vii
7.4.3 Implications of the existence of constant solutions of
the Riccati equation . . . . . . . . . . . . . . . . . . . 235
7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8 Nonrecursive Difference Equations 239
8.1 Examples and discussion . . . . . . . . . . . . . . . . . . . . . 240
8.2 Form symmetries, factors, and cofactors . . . . . . . . . . . . 251
8.3 Semi-invertible map criterion . . . . . . . . . . . . . . . . . . 255
8.4 Quadratic difference equations . . . . . . . . . . . . . . . . . . 259
8.4.1 Existence and variety of real solutions . . . . . . . . . 261
8.4.2 Quadratic factors and cofactors . . . . . . . . . . . . . 276
8.4.3 Quadratic equations with a linear form symmetry . . . 279
8.5 An order-preserving form symmetry . . . . . . . . . . . . . . 285
8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
A Appendix: Asymptotic Stability on the Real Line 297
A.1 An inverse-map characterization . . . . . . . . . . . . . . . . 297
A.2 Equivalent necessary and sufficient conditions . . . . . . . . . 302
References 305
Index 309
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Preface
The field of difference equations is old, their study predating calculus and
hence differential equations. Since the timeof Fibonacci,ifnot earlier, recur-
sions and finite differences have appeared in a variety of contexts. Further,
finite differences and sums naturally lead to differential and integral calcu-
lus through the concept of limit. In modern times, difference equations have
a symbiotic relationship with their differential cousins: finite differences are
routinely used in obtaining numerical solutions of differential equations and
insights gained from the study of differential equations in turn inspire new
investigations in difference equations.
Increasingly in the last few decades, difference equations have also been
studiedindependently ofdifferentialones,leadingtointeresting newdiscover-
ies and methods of analysis. There are journalscurrently devoted specifically
to difference equations. These sources routinely publish articles of impres-
sive technical quality that reflect the depth and sophistication of present-day
researchers in this field.
The majority of articles and books on difference equations currently in
print concern topologicaland convergence issues (e.g.,the asymptotic behav-
iorsofsolutions)inmetricspaces,predominantlyfinite-dimensionalEuclidean
spaces due to their importance in scientific models. Although these spaces
easily accommodate point sequences (solutions of difference equations) they
are primarilyand naturally formulated for flows, i.e., solutions of differential
equations. Therearecomparativelyfewarticlesandevenfewerbooksthatare
devoted to the study ofdifference equations as objects or constructs on other
types of sets. Yet other andmore general types ofsets are needed inorder to
determine the most natural context for defining difference equations and im-
prove their applicabilityto solvingproblems in biology,economics, computer
science and other fields.
One mayargue that itis entirely natural to formulateandstudy difference
equations in discrete spaces where the topology is trivial but algebraic struc-
ture may exist; or even in finite “spaces” such as graphs or networks that
often contain a substantial connectivity or adjacency structure but which are
typically not amenable to analysis using limit-based techniques. Interesting
dynamic behaviors can and do occur in discrete sets but lack of an efficient
toolkit likecalculus makes studyingsuch dynamics rather difficult. It is nec-
essary to fullyutilizeexisting structures and methods (algebraic and discrete
concepts) inadditiontothe moreclassicalanalyticalandtopologicalmethods
where applicable in order to arrive at coherent and potentiallyuseful results.
ix