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A periodically forced mathematical pendulum is one of the typical
and popular nonlinear oscillators that possess complex and rich
dynamical behaviors. Although the pendulum is one of the simplest
nonlinear oscillators, yet, until now, we are still not able to
undertake a systematical study of periodic motions to chaos in such
a simplest system due to lack of suitable mathematical methods and
computational tools. To understand periodic motions and chaos in
the periodically forced pendulum, the perturbation method has been
adopted. One could use the Taylor series to expend the sinusoidal
function to the polynomial nonlinear terms, followed by traditional
perturbation methods to obtain the periodic motions of the
approximated differential system.This book discusses Hamiltonian
chaos and periodic motions to chaos in pendulums. This book first
detects and discovers chaos in resonant layers and bifurcation
trees of periodic motions to chaos in pendulum in the comprehensive
fashion, which is a base to understand the behaviors of nonlinear
dynamical systems, as a results of Hamiltonian chaos in the
resonant layers and bifurcation trees of periodic motions to chaos.
The bifurcation trees of travelable and non-travelable periodic
motions to chaos will be presented through the periodically forced
pendulum.
This book is about Lie group analysis of differential equations for
physical and engineering problems. The topics include: --
Approximate symmetry in nonlinear physical problems -- Complex
methods for Lie symmetry analysis -- Lie group classification,
Symmetry analysis, and conservation laws -- Conservative difference
schemes -- Hamiltonian structure and conservation laws of
three-dimensional linear elasticity -- Involutive systems of
partial differential equations This collection of works is written
in memory of Professor Nail H. Ibragimov (1939-2018). It could be
used as a reference book in differential equations in mathematics,
mechanical, and electrical engineering.
This is the first book focusing on bifurcation dynamics in
1-dimensional polynomial nonlinear discrete systems. It
comprehensively discusses the general mathematical conditions of
bifurcations in polynomial nonlinear discrete systems, as well as
appearing and switching bifurcations for simple and higher-order
singularity period-1 fixed-points in the 1-dimensional polynomial
discrete systems. Further, it analyzes the bifurcation trees of
period-1 to chaos generated by period-doubling, and monotonic
saddle-node bifurcations. Lastly, the book presents methods for
period-2 and period-doubling renormalization for polynomial
discrete systems, and describes the appearing mechanism and
period-doublization of period-n fixed-points on bifurcation trees
for the first time, offering readers fascinating insights into
recent research results in nonlinear discrete systems.
This book focuses on bifurcation and stability in nonlinear
discrete systems, including monotonic and oscillatory stability. It
presents the local monotonic and oscillatory stability and
bifurcation of period-1 fixed-points on a specific eigenvector
direction, and discusses the corresponding higher-order singularity
of fixed-points. Further, it explores the global analysis of
monotonic and oscillatory stability of fixed-points in
1-dimensional discrete systems through 1-dimensional polynomial
discrete systems. Based on the Yin-Yang theory of nonlinear
discrete systems, the book also addresses the dynamics of forward
and backward nonlinear discrete systems, and the existence
conditions of fixed-points in said systems. Lastly, in the context
of local analysis, it describes the normal forms of nonlinear
discrete systems and infinite-fixed-point discrete systems.
Examining nonlinear discrete systems from various perspectives, the
book helps readers gain a better understanding of the nonlinear
dynamics of such systems.
This is the first book focusing on bifurcation dynamics in
1-dimensional polynomial nonlinear discrete systems. It
comprehensively discusses the general mathematical conditions of
bifurcations in polynomial nonlinear discrete systems, as well as
appearing and switching bifurcations for simple and higher-order
singularity period-1 fixed-points in the 1-dimensional polynomial
discrete systems. Further, it analyzes the bifurcation trees of
period-1 to chaos generated by period-doubling, and monotonic
saddle-node bifurcations. Lastly, the book presents methods for
period-2 and period-doubling renormalization for polynomial
discrete systems, and describes the appearing mechanism and
period-doublization of period-n fixed-points on bifurcation trees
for the first time, offering readers fascinating insights into
recent research results in nonlinear discrete systems.
This book focuses on bifurcation and stability in nonlinear
discrete systems, including monotonic and oscillatory stability. It
presents the local monotonic and oscillatory stability and
bifurcation of period-1 fixed-points on a specific eigenvector
direction, and discusses the corresponding higher-order singularity
of fixed-points. Further, it explores the global analysis of
monotonic and oscillatory stability of fixed-points in
1-dimensional discrete systems through 1-dimensional polynomial
discrete systems. Based on the Yin-Yang theory of nonlinear
discrete systems, the book also addresses the dynamics of forward
and backward nonlinear discrete systems, and the existence
conditions of fixed-points in said systems. Lastly, in the context
of local analysis, it describes the normal forms of nonlinear
discrete systems and infinite-fixed-point discrete systems.
Examining nonlinear discrete systems from various perspectives, the
book helps readers gain a better understanding of the nonlinear
dynamics of such systems.
This book for the first time examines periodic motions to chaos in
time-delay systems, which exist extensively in engineering. For a
long time, the stability of time-delay systems at equilibrium has
been of great interest from the Lyapunov theory-based methods,
where one cannot achieve the ideal results. Thus, time-delay
discretization in time-delay systems was used for the stability of
these systems. In this volume, Dr. Luo presents an accurate method
based on the finite Fourier series to determine periodic motions in
nonlinear time-delay systems. The stability and bifurcation of
periodic motions are determined by the time-delayed system of
coefficients in the Fourier series and the method for nonlinear
time-delay systems is equivalent to the Laplace transformation
method for linear time-delay systems.
This book examines discrete dynamical systems with memory-nonlinear
systems that exist extensively in biological organisms and
financial and economic organizations, and time-delay systems that
can be discretized into the memorized, discrete dynamical systems.
It book further discusses stability and bifurcations of time-delay
dynamical systems that can be investigated through memorized
dynamical systems as well as bifurcations of memorized nonlinear
dynamical systems, discretization methods of time-delay systems,
and periodic motions to chaos in nonlinear time-delay systems. The
book helps readers find analytical solutions of MDS, change
traditional perturbation analysis in time-delay systems, detect
motion complexity and singularity in MDS; and determine stability,
bifurcation, and chaos in any time-delay system.
The book covers nonlinear physical problems and mathematical
modeling, including molecular biology, genetics, neurosciences,
artificial intelligence with classical problems in mechanics and
astronomy and physics. The chapters present nonlinear mathematical
modeling in life science and physics through nonlinear differential
equations, nonlinear discrete equations and hybrid equations. Such
modeling can be effectively applied to the wide spectrum of
nonlinear physical problems, including the KAM
(Kolmogorov-Arnold-Moser (KAM)) theory, singular differential
equations, impulsive dichotomous linear systems, analytical
bifurcation trees of periodic motions, and almost or pseudo- almost
periodic solutions in nonlinear dynamical systems.
This book provides students and researchers with a systematic
solution for fluid-induced structural vibrations, galloping
instability and the chaos of cables. They will also gain a better
understanding of stable and unstable periodic motions and chaos in
fluid-induced structural vibrations. Further, the results presented
here will help engineers effectively design and analyze
fluid-induced vibrations.
This book examines discrete dynamical systems with memory-nonlinear
systems that exist extensively in biological organisms and
financial and economic organizations, and time-delay systems that
can be discretized into the memorized, discrete dynamical systems.
It book further discusses stability and bifurcations of time-delay
dynamical systems that can be investigated through memorized
dynamical systems as well as bifurcations of memorized nonlinear
dynamical systems, discretization methods of time-delay systems,
and periodic motions to chaos in nonlinear time-delay systems. The
book helps readers find analytical solutions of MDS, change
traditional perturbation analysis in time-delay systems, detect
motion complexity and singularity in MDS; and determine stability,
bifurcation, and chaos in any time-delay system.
This book describes system dynamics with discontinuity caused by
system interactions and presents the theory of flow singularity and
switchability at the boundary in discontinuous dynamical systems.
Based on such a theory, the authors address dynamics and motion
mechanism of engineering discontinuous systems due to interaction.
Stability and bifurcations of fixed points in nonlinear discrete
dynamical systems are presented, and mapping dynamics are developed
for analytical predictions of periodic motions in engineering
discontinuous dynamical systems. Ultimately, the book provides an
alternative way to discuss the periodic and chaotic behaviors in
discontinuous dynamical systems.
This book for the first time examines periodic motions to chaos in
time-delay systems, which exist extensively in engineering. For a
long time, the stability of time-delay systems at equilibrium has
been of great interest from the Lyapunov theory-based methods,
where one cannot achieve the ideal results. Thus, time-delay
discretization in time-delay systems was used for the stability of
these systems. In this volume, Dr. Luo presents an accurate method
based on the finite Fourier series to determine periodic motions in
nonlinear time-delay systems. The stability and bifurcation of
periodic motions are determined by the time-delayed system of
coefficients in the Fourier series and the method for nonlinear
time-delay systems is equivalent to the Laplace transformation
method for linear time-delay systems.
This important collection presents recent advances in nonlinear
dynamics including analytical solutions, chaos in Hamiltonian
systems, time-delay, uncertainty, and bio-network dynamics.
Nonlinear Dynamics and Complexity equips readers to appreciate this
increasingly main-stream approach to understanding complex
phenomena in nonlinear systems as they are examined in a broad
array of disciplines. The book facilitates a better understanding
of the mechanisms and phenomena in nonlinear dynamics and develops
the corresponding mathematical theory to apply nonlinear design to
practical engineering.
The book covers nonlinear physical problems and mathematical
modeling, including molecular biology, genetics, neurosciences,
artificial intelligence with classical problems in mechanics and
astronomy and physics. The chapters present nonlinear mathematical
modeling in life science and physics through nonlinear differential
equations, nonlinear discrete equations and hybrid equations. Such
modeling can be effectively applied to the wide spectrum of
nonlinear physical problems, including the KAM
(Kolmogorov-Arnold-Moser (KAM)) theory, singular differential
equations, impulsive dichotomous linear systems, analytical
bifurcation trees of periodic motions, and almost or pseudo- almost
periodic solutions in nonlinear dynamical systems.
This book describes system dynamics with discontinuity caused by
system interactions and presents the theory of flow singularity and
switchability at the boundary in discontinuous dynamical systems.
Based on such a theory, the authors address dynamics and motion
mechanism of engineering discontinuous systems due to interaction.
Stability and bifurcations of fixed points in nonlinear discrete
dynamical systems are presented, and mapping dynamics are developed
for analytical predictions of periodic motions in engineering
discontinuous dynamical systems. Ultimately, the book provides an
alternative way to discuss the periodic and chaotic behaviors in
discontinuous dynamical systems.
Nonlinear Systems and Methods For Mechanical, Electrical and
Biosystems presents topics observed at the 3rd Conference on
Nonlinear Science and Complexity(NSC), focusing on energy transfer
and synchronization in hybrid nonlinear systems. The studies focus
on fundamental theories and principles,analytical and symbolic
approaches, computational techniques in nonlinear physical science
and mathematics. Broken into three parts, the text covers:
Parametrical excited pendulum, nonlinear dynamics in hybrid
systems, dynamical system synchronization and (N+1) body dynamics
as well as new views different from the existing results in
nonlinear dynamics, mathematical methods for dynamical systems
including conservation laws, dynamical symmetry in nonlinear
differential equations and invex energies and nonlinear phenomena
in physical problems such as solutions, complex flows, chemical
kinetics, Toda lattices and parallel manipulator. This book is
useful to scholars, researchers and advanced technical members of
industrial laboratory facilities developing new tools and products.
This book contains selected papers of NSC08, the 2nd Conference on
Nonlinear Science and Complexity, held 28-31 July, 2008, Porto,
Portugal. It focuses on fundamental theories and principles,
analytical and symbolic approaches, computational techniques in
nonlinear physics and mathematics. Topics treated include * Chaotic
Dynamics and Transport in Classic and Quantum Systems * Complexity
and Nonlinearity in Molecular Dynamics and Nano-Science *
Complexity and Fractals in Nonlinear Biological Physics and Social
Systems * Lie Group Analysis and Applications in Nonlinear Science
* Nonlinear Hydrodynamics and Turbulence * Bifurcation and
Stability in Nonlinear Dynamic Systems * Nonlinear Oscillations and
Control with Applications * Celestial Physics and Deep Space
Exploration * Nonlinear Mechanics and Nonlinear Structural Dynamics
* Non-smooth Systems and Hybrid Systems * Fractional dynamical
systems
"Machine Tool Vibrations and Cutting Dynamics" covers the
fundamentals of cutting dynamics from the perspective of
discontinuous systems theory. It shows the reader how to use
coupling, interaction, and different cutting states to mitigate
machining instability and enable better machine tool design. Among
the topics discussed are; underlying dynamics of cutting and
interruptions in cutting motions; the operation of the machine-tool
systems over a broad range of operating conditions with minimal
vibration and the need for high precision, high yield micro- and
nano-machining.
Nonlinear Dynamics of Complex Systems describes chaos, fractal and
stochasticities within celestial mechanics, financial systems and
biochemical systems. Part I discusses methods and applications in
celestial systems and new results in such areas as low energy
impact dynamics, low-thrust planar trajectories to the moon and
earth-to-halo transfers in the sun, earth and moon. Part II
presents the dynamics of complex systems including bio-systems,
neural systems, chemical systems and hydro-dynamical systems.
Finally, Part III covers economic and financial systems including
market uncertainty, inflation, economic activity and foreign
competition and the role of nonlinear dynamics in each.
Fractional Dynamics and Control provides a comprehensive overview
of recent advances in the areas of nonlinear dynamics, vibration
and control with analytical, numerical, and experimental results.
This book provides an overview of recent discoveries in fractional
control, delves into fractional variational principles and
differential equations, and applies advanced techniques in
fractional calculus to solving complicated mathematical and
physical problems.Finally, this book also discusses the role that
fractional order modeling can play in complex systems for
engineering and science.
Discontinuity in Nonlinear Physical Systems explores recent
developments in experimental research in this broad field,
organized in four distinct sections. Part I introduces the reader
to the fractional dynamics and Lie group analysis for nonlinear
partial differential equations. Part II covers chaos and complexity
in nonlinear Hamiltonian systems, important to understand the
resonance interactions in nonlinear dynamical systems, such as
Tsunami waves and wildfire propagations; as well as Lev flights in
chaotic trajectories, dynamical system synchronization and DNA
information complexity analysis. Part III examines chaos and
periodic motions in discontinuous dynamical systems, extensively
present in a range of systems, including piecewise linear systems,
vibro-impact systems and drilling systems in engineering. And in
Part IV, engineering and financial nonlinearity are discussed. The
mechanism of shock wave with saddle-node bifurcation and rotating
disk stability will be presented, and the financial nonlinear
models will be discussed.
This important collection presents recent advances in nonlinear
dynamics including analytical solutions, chaos in Hamiltonian
systems, time-delay, uncertainty, and bio-network dynamics.
Nonlinear Dynamics and Complexity equips readers to appreciate this
increasingly main-stream approach to understanding complex
phenomena in nonlinear systems as they are examined in a broad
array of disciplines. The book facilitates a better understanding
of the mechanisms and phenomena in nonlinear dynamics and develops
the corresponding mathematical theory to apply nonlinear design to
practical engineering.
Dynamical System Synchronization (DSS) meticulously presents for
the first time the theory of dynamical systems synchronization
based on the local singularity theory of discontinuous dynamical
systems. The book details the sufficient and necessary conditions
for dynamical systems synchronizations, through extensive
mathematical expression. Techniques for engineering implementation
of DSS are clearly presented compared with the existing techniques.
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