A method to determine cutoff frequencies for linear acoustic waves propagating in nonisothermal media is introduced. The developed method is based on wave variable transformations that lead to Klein-Gordon equations, and the oscillation theorem is applied to obtain the turning point frequencies. Physical arguments are used to justify the choice of the largest turning point frequency as the cutoff frequency. The method is used to derive the cutoff frequencies in nonisothermal media modeled by exponential and power law temperature gradients, for which the cutoffs cannot be obtained based on known analytical solutions. An interesting result is that the acoustic cutoff frequencies calculated by the method are local quantities that vary in the media, and that their specific values at a given height determine the frequency that acoustic waves must have in order to be propagating at this height. To extend this physical interpretation of the acoustic cutoff frequency to nonisothermal media of arbitrary temperature gradients, a generalized version of the method applicable to these media is also presented.

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

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A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.

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