Bispectral and trispectral characterization of transition to chaos in the Duffing oscillator

Higher-order spectral (bispectral and trispectral) analyses of numerical solutions of the Duffing equation with a cubic stiffness are used to isolate the coupling between the triads and quartets, respectively, of nonlinearly interacting Fourier components of the system. The Duffing oscillator follows a period-doubling intermittency catastrophic route to chaos. For period-doubled limit cycles, higher-order spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. However, when the Duffing oscillator becomes chaotic, global behavior of the cubic nonlinearity becomes dominant and quadratic nonlinear interactions are weak, while cubic interactions remain strong. As the nonlinearity of the system is increased, the number of excited Fourier components increases, eventually leading to broad-band power spectra for chaos. The corresponding higher-order spectra indicate that although some individual nonlinear interactions weaken as nonlinearity increases, the number of nonlinearly interacting Fourier modes increases. Trispectra indicate that the cubic interactions gradually evolve from encompassing a few quartets of Fourier components for period-1 motion to encompassing many quartets for chaos. For chaos, all the components within the energetic part of the power spectrum are cubically (but not quadratically) coupled to each other.

Crime in Rural Australia

Contemporary rural crime is more varied and sophisticated than it once was. The new forms range from agricultural crimes, such as the theft of water designated for agricultural production, to environmental crimes such as the illegal dumping of waste. They take place side by side with “traditional” rural crimes such as cattle duffing while “urban” crimes such as drug and alcohol abuse and violent assaults are also prevalent, and on the rise. Crime in Rural Australia covers them all. It brings together leading academics who examine the major dimensions of crime and justice in rural and regional Australia including: •the extent of rural crime •farm crime •violence •juvenile crime •policing •Indigenous crime and justice •crime prevention •drugs •fear of crime, and •sentencing and punishment. It includes vignettes on rural policing and the stock squad from the perspectives of the NSW police. An ideal text for rural crime and criminology courses, Crime in Rural Australia will also be of interest to criminal justice practitioners, policy-makers, and criminology scholars. Three of the editors, Dr Elaine Barclay, Dr John Scott and Associate Professor Russell Hogg, are associated with the Centre for Rural Crime at the University of New England. Professor Joseph F. Donnermeyer is the International Research Co-ordinator for the Rural Crime Centre and is a leading US scholar on rural crime at Ohio State University.