It is two decades since Mitchell Feigenbaum's landmark papers on period doubling and the modern beginnings of what is now called "Chaos Theory". From the very beginning, mechanics has been a central focus for modem nonlinear dynamical systems. Fluid, structural, machine and rigid body dynamics has been a fertile field for nonlinear phenomena and chaos in particular. Early experimental evidence for chaotic phenomena in mechanics gave the new chaos theory a mark of credibility, importance, and relevance that its earlier sister field, catastrophe theory, did not achieve. The fact that mechanics…mehr
It is two decades since Mitchell Feigenbaum's landmark papers on period doubling and the modern beginnings of what is now called "Chaos Theory". From the very beginning, mechanics has been a central focus for modem nonlinear dynamical systems. Fluid, structural, machine and rigid body dynamics has been a fertile field for nonlinear phenomena and chaos in particular. Early experimental evidence for chaotic phenomena in mechanics gave the new chaos theory a mark of credibility, importance, and relevance that its earlier sister field, catastrophe theory, did not achieve. The fact that mechanics straddles both physics and engineering also meant that mechanics became a pathway for direct application of chaos theory to applied problems. So what is new in nonlinear dynamics and mechanics today? First the scope of applications in solid mechanics has broadened to cover material processing, inelasticity and fracture mechanics. In rigid body dynamics, more complex systems such as vehicles, robotics and controlled machines have come into the purview of nonlinear dynamicists. On the mathematics side of nonlinear dynamics, it is now recognized that spatio-temporal problems, hysteretic and time delay problems are the new frontiers in this field. Also the term "complexity" has been added to the lexicon of chaos theory to describe the dynamics of many interacting sub systems which can exhibit self organization and evolution. Complexity analysis has gained a foothold in biological and some social sciences as well in fluid and chemical physics.
I. Analysis Tools, Bifurcation Theory.- Bifurcations in stochastically perturbed dynamical systems.- Nonstationary responses in two degree-of-freedom nonlinear systems with 1-to-2 internal resonance.- Sinusoidally varying period-doubling bifurcations.- Wavelet approach to mechanical problems. Symplectic group, symplectic topology and symplectic scales.- On the generalized cell mapping.- Bursts.- Dynamics of a quasiperiodically-forced Mathieu oscillator.- Improved Galerkin method in the dimension reduction of nonlinear dynamical systems.- Sequences of global bifurcations and multiple chaotic transients in a mechanical driven oscillator.- Bifurcation of homoclinic orbits in autonomous systems and in chaotic blue sky catastrophe.- Wavelet approach to polynomial mechanical problems.- Jumps to resonance phenomena in nonlinear mechanics: fractal basins, chaotic transients and unpredictability.- Global stable oscillations near unstable equilibrium positions: the hilltop effect.- II. Elastica, Cables and Shell Dynamics.- Homoclinic orbits, spatial chaos and localized buckling.- Nonregular regimes of monodimensional mechanical systems with initial curvature: experiments and time series analysis.- Constrained Euler buckling: line contact solutions.- The role of homoclinic bifurcations in understanding the buckling of long thin cylindrical shells.- Dynamics of a buckled drillstring rotating in a curved oil wellbore.- The complex non-linear dynamics of imperfection sensitive shells.- III. Inelastic Materials, Material Processing.- The dynamics of chip formation in machining.- Chatter identification and control for a boring process.- Coupling and resonance phenomena in dynamic systems with hysteresis.- Maps, traps, and equilibria for a fully dissipative elastoplastic oscillator.- IV. Machine System Dynamics.- Nonlinear dynamics and the design of tuned pendulum vibration absorbers.- Dynamical behavior of chains.- Chaos in offset hydrodynamic rotor bearings.- V. Impact and Nonsmooth Systems.- Dynamics of elastic structures subjected to impact excitations.- A method for finding all possible periodic orbits in piecewise continuous mechanical systems of arbitrary dimension.- Dynamics of the impact oscillator.- Numerical and experimental investigation of nonsmooth mechanical systems.- Steady-state behaviour of a solar array system with elastic stops.- Nonlinear dynamics of mechanical systems with discontinuities.- Displacement potentials in non-smooth dynamics.- VI. Circuits, Control, Cardiac Modelling.- Driven nonlinear oscillators for modeling cardiac phenomena.- Chaos, synchronization and bifurcations in a driven R-L-diode circuit.- Transient global behavior in nonlinear experimental oscillators.- Control of chaos for pendulum systems.- VII. Vehicle and Ship Dynamics.- Delay, nonlinear oscillations and shimmying wheels.- Symmetry, generic bifurcations, and mode interaction in nonlinear railway dynamics.- A traveler's woes: some perspectives from dynamical systems.- Stability and chaos in passive-dynamic locomotion.- On the multi-degree of freedom, nonlinear dynamics of ship motions with application to the broaching problem.- VIII. Fluid-Elastic Problems.- Nonlinear dynamics of aeroelastic systems.- Stabilization of nonlinear hydroelastic structures via random parametric excitation.- Aircraft stability and control: Bifurcation analysis in the design process?.- Global dynamics in gyroscopic and aeroelastic systems.- Chaotic oscillations of a loosely supported tube in a heat-exchanger array in cross-flow.- Stablization of periodic flat-lag dynamics inrotorcraft.- A mathematical model of pendellufl flow in a single airway bifurcation.- The periodic to chaotic water wheel.- Helical waves and nonlinear dynamics of fluid/structure interactions in a cylinder row.- IX. Spatio-Temporal Chaos.- Spatio-temporal dynamics of coupled magneto-elastic system.- Spatially localized and chaotic motions of a discretely supported elastic continuum.- Optimal control of spatiotemporal chaos in coupled map lattices.- Appendix I: Addresses of Authors.- Author Index.
I. Analysis Tools, Bifurcation Theory.- Bifurcations in stochastically perturbed dynamical systems.- Nonstationary responses in two degree-of-freedom nonlinear systems with 1-to-2 internal resonance.- Sinusoidally varying period-doubling bifurcations.- Wavelet approach to mechanical problems. Symplectic group, symplectic topology and symplectic scales.- On the generalized cell mapping.- Bursts.- Dynamics of a quasiperiodically-forced Mathieu oscillator.- Improved Galerkin method in the dimension reduction of nonlinear dynamical systems.- Sequences of global bifurcations and multiple chaotic transients in a mechanical driven oscillator.- Bifurcation of homoclinic orbits in autonomous systems and in chaotic blue sky catastrophe.- Wavelet approach to polynomial mechanical problems.- Jumps to resonance phenomena in nonlinear mechanics: fractal basins, chaotic transients and unpredictability.- Global stable oscillations near unstable equilibrium positions: the hilltop effect.- II. Elastica, Cables and Shell Dynamics.- Homoclinic orbits, spatial chaos and localized buckling.- Nonregular regimes of monodimensional mechanical systems with initial curvature: experiments and time series analysis.- Constrained Euler buckling: line contact solutions.- The role of homoclinic bifurcations in understanding the buckling of long thin cylindrical shells.- Dynamics of a buckled drillstring rotating in a curved oil wellbore.- The complex non-linear dynamics of imperfection sensitive shells.- III. Inelastic Materials, Material Processing.- The dynamics of chip formation in machining.- Chatter identification and control for a boring process.- Coupling and resonance phenomena in dynamic systems with hysteresis.- Maps, traps, and equilibria for a fully dissipative elastoplastic oscillator.- IV. Machine System Dynamics.- Nonlinear dynamics and the design of tuned pendulum vibration absorbers.- Dynamical behavior of chains.- Chaos in offset hydrodynamic rotor bearings.- V. Impact and Nonsmooth Systems.- Dynamics of elastic structures subjected to impact excitations.- A method for finding all possible periodic orbits in piecewise continuous mechanical systems of arbitrary dimension.- Dynamics of the impact oscillator.- Numerical and experimental investigation of nonsmooth mechanical systems.- Steady-state behaviour of a solar array system with elastic stops.- Nonlinear dynamics of mechanical systems with discontinuities.- Displacement potentials in non-smooth dynamics.- VI. Circuits, Control, Cardiac Modelling.- Driven nonlinear oscillators for modeling cardiac phenomena.- Chaos, synchronization and bifurcations in a driven R-L-diode circuit.- Transient global behavior in nonlinear experimental oscillators.- Control of chaos for pendulum systems.- VII. Vehicle and Ship Dynamics.- Delay, nonlinear oscillations and shimmying wheels.- Symmetry, generic bifurcations, and mode interaction in nonlinear railway dynamics.- A traveler's woes: some perspectives from dynamical systems.- Stability and chaos in passive-dynamic locomotion.- On the multi-degree of freedom, nonlinear dynamics of ship motions with application to the broaching problem.- VIII. Fluid-Elastic Problems.- Nonlinear dynamics of aeroelastic systems.- Stabilization of nonlinear hydroelastic structures via random parametric excitation.- Aircraft stability and control: Bifurcation analysis in the design process?.- Global dynamics in gyroscopic and aeroelastic systems.- Chaotic oscillations of a loosely supported tube in a heat-exchanger array in cross-flow.- Stablization of periodic flat-lag dynamics inrotorcraft.- A mathematical model of pendellufl flow in a single airway bifurcation.- The periodic to chaotic water wheel.- Helical waves and nonlinear dynamics of fluid/structure interactions in a cylinder row.- IX. Spatio-Temporal Chaos.- Spatio-temporal dynamics of coupled magneto-elastic system.- Spatially localized and chaotic motions of a discretely supported elastic continuum.- Optimal control of spatiotemporal chaos in coupled map lattices.- Appendix I: Addresses of Authors.- Author Index.
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