這是一本關(guān)于單擺共振層及其周期運(yùn)動(dòng)到混沌的著作。周期強(qiáng)迫振動(dòng)擺是一個(gè)典型且常見的最簡(jiǎn)單的非線性振子,具有復(fù)雜和豐富的非線性動(dòng)力學(xué)行為。雖然此類周期強(qiáng)迫振動(dòng)擺是一個(gè)最簡(jiǎn)單的非線性動(dòng)力系統(tǒng),但要找到它的周期運(yùn)動(dòng)到混沌非常困難。并且這一周期強(qiáng)迫振動(dòng)擺固有的復(fù)雜動(dòng)力學(xué)行為遠(yuǎn)遠(yuǎn)超出了我們基于傳統(tǒng)線性動(dòng)力系統(tǒng)的想象。到目前為止,人們?nèi)匀徊恢来祟愔芷趶?qiáng)迫振動(dòng)鐘擺的復(fù)雜運(yùn)動(dòng)及其物理學(xué)本質(zhì)和數(shù)學(xué)理論,本書中所展示的結(jié)果將為探索周期強(qiáng)迫振動(dòng)擺的復(fù)雜非線性動(dòng)力學(xué)行為帶來一些新穎觀點(diǎn)。
Preface
1 Resonance and Hamiltonian Chaos
1.1 Stochastic layers
1.1.1 Definitions
1.1.2 Approximate criteria
1.2 Resonant separatrix layers
1.2.1 Layer dynamics
1.2.2 Approximate criteria
References
2 Hamiltonian Chaos in Pendulum
2.1 Resonance conditions
2.1.1 Conservative system
2.1.2 Resonance and energy increments
2.2 Stochastic layers
2.3 Resonant layers
2.3.1 Librational resonant layers
2.3.2 Rotational resonant layers
2.4 Numerical simulations
References
3 Parametric Chaos in Pendulum
3.1 Resonance and energy increment
3.1.1 Libration
3.1.2 Rotation
3.2 Parametric stochastic layers
3.2.1 Analytic predictions
3.2.2 Numerical predictions
3.2.3 Illustrations
3.2.4 Numerical simulations
3.3 Parametric resonant layers
3.3.1 Approximate predictions
3.3.2 Numerical illustrations
References
4 Nonlinear Discrete Systems
4.1 Definitions
4.2 Fixed points and stability
4.3 Stability switching theory
4.4 Bifurcation theory
References
5 Periodic Flows in Continuous Systems
5.1 Discretization-based methods
5.2 Discrete Fourier series
References
6 Periodic Motions to Chaos in Pendulum
6.1 Periodic motions in pendulum
6.1.1 Implicit discretization
6.1.2 Periodic motions
6.2 Bifurcation trees to chaos
6.2.1 Period-1 motions to chaos
6.2.2 Period-3 motions to chaos
6.2.3 Period-5 motions to chaos
6.3 Frequency-amplitude characteristics
6.3.1 Period-1 to period-4 motions
6.3.2 Period-3 to period-6 motions
6.3.3 Symmetric to asymmetric period-5 motions.
6.4 Bifurcation trees varying with excitation amplitude
6.4.1 Non-travelable period-1 motions to chaos
6.4.2 Non-travelable period-3 motions to chaos
6.4.3 Travelable period-1 motions to chaos
6.4.4 Travelable period-2 motions to chaos
6.5 Numerical simulations
6.5.1 Non-travelable periodic motions
6.5.2 Travelable periodic motions
References
Subject Index