Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

89 Pages·2020·29.32 MB·English

by  Siyuan Xing, Albert C J Luo

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Synthesis Lectures on Series ISSN: 2573-3168 N
G
• L
Mechanical Engineering U
O
Sequential Bifurcation Trees to Chaos in
Sequential
Nonlinear Time-Delay Systems
Siyuan Xing, California Polytechnic State University S
E
Q
Albert C.J. Luo, Southern Illinois University
U
E Bifurcation Trees
N
T
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In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear A
L
dynamical systems is presented for a better understanding of global behaviors and motion transitions for BI
F
one periodic motion to another one. A 1-dimensional (1-D), time-delayed, nonlinear dynamical system UR
C
is considered as an example to show how to determine the global sequential scenarios of the bifurcation A to Chaos in Nonlinear
T
I
trees of periodic motions to chaos. All stable and unstable periodic motions on the bifurcation trees can O
N
be determined. Especially, the unstable periodic motions on the bifurcation trees cannot be achieved from T
R
E
the traditional analytical methods, and such unstable periodic motions and chaos can be obtained through E
S
a specific control strategy. T
O Time-Delay Systems
The sequential periodic motions in such a 1-D time-delayed system are achieved semi-analytically, and C
H
A
the corresponding stability and bifurcations are determined by eigenvalue analysis. Each bifurcation tree of O
S
a specific periodic motion to chaos are presented in detail. The bifurcation tree appearance and vanishing IN
N
are determined by the saddle-node bifurcation, and the cascaded period-doubled periodic solutions are O
N
determined by the period-doubling bifurcation. From finite Fourier series, harmonic amplitude and L
I
N
harmonic phases for periodic motions on the global bifurcation tree are obtained for frequency analysis. E
A
Numerical illustrations of periodic motions are given for complex periodic motions in global bifurcation R
T
I
trees. The rich dynamics of the 1-D, delayed, nonlinear dynamical system is presented. Such global M
E
sequential periodic motions to chaos exist in nonlinear dynamical systems. The frequency-amplitude -D
E Siyuan Xing
analysis can be used for re-construction of analytical expression of periodic motions, which can be used for L
A
Y
motion control in dynamical systems S
YS Albert C.J. Luo
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M
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Sequential Bifurcation Trees
to Chaos in Nonlinear
Time-Delay Systems

Synthesis Lectures on
Mechanical Engineering
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pertainingtothisdiversedisciplineofmechanicalengineering.TheseriespresentsLectures
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SequentialBifurcationTreestoChaosinNonlinearTime-DelaySystems
SiyuanXingandAlbertC.J.Luo
www.morganclaypool.com
ISBN:9781681739427 paperback
ISBN:9781681739434 ebook
ISBN:9781681739441 hardcover
DOI10.2200/S01038ED1V01Y202008MEC031
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONMECHANICALENGINEERING
Lecture#31
SeriesISSN
Print2573-3168 Electronic2573-3176
Sequential Bifurcation Trees
to Chaos in Nonlinear
Time-Delay Systems
Siyuan Xing
CaliforniaPolytechnicStateUniversity,SanLuisObispo,CA
Albert C.J. Luo
SouthernIllinoisUniversity,Edwardsville,IL
SYNTHESISLECTURESONMECHANICALENGINEERING#31
M
&C Morgan &cLaypool publishers
ABSTRACT
Inthisbook,theglobalsequentialscenarioofbifurcationtreesofperiodicmotionstochaosin
nonlineardynamicalsystemsispresentedforabetterunderstandingofglobalbehaviorsandmo-
tiontransitionsforoneperiodicmotiontoanotherone.A1-dimensional(1-D),time-delayed,
nonlinear dynamical system is considered as an example to show how to determine the global
sequentialscenariosofthebifurcationtreesofperiodicmotionstochaos.Allstableandunstable
periodic motions on the bifurcation trees can be determined. Especially, the unstable periodic
motionsonthebifurcationtreescannotbeachievedfromthetraditionalanalyticalmethods,and
suchunstableperiodicmotionsandchaoscanbeobtainedthroughaspecificcontrolstrategy.
The sequential periodic motions in such a 1-D time-delayed system are achieved semi-
analytically,andthecorrespondingstabilityandbifurcationsaredeterminedbyeigenvalueanal-
ysis. Each bifurcation tree of a specific periodic motion to chaos are presented in detail. The
bifurcation tree appearance and vanishing are determined by the saddle-node bifurcation, and
thecascadedperiod-doubledperiodicsolutionsaredeterminedbytheperiod-doublingbifurca-
tion.FromfiniteFourierseries,harmonicamplitudeandharmonicphasesforperiodicmotions
on the global bifurcation tree are obtained for frequency analysis. Numerical illustrations of
periodic motions are given for complex periodic motions in global bifurcation trees. The rich
dynamicsofthe1-D,delayed,nonlineardynamicalsystemispresented.Suchglobalsequential
periodicmotionstochaosexistinnonlineardynamicalsystems.Thefrequency-amplitudeanal-
ysis can be used for re-construction of analytical expression of periodic motions, which can be
usedformotioncontrolindynamicalsystems.
KEYWORDS
1-dimensionaltime-delayedsystem,globalsequentialscenarioofbifurcationtrees,
implicitmapping,mappingstructures,nonlinearfrequency-amplitudes