Low-Dimensional Chaos in a Hydrodynamic System

Electric fields yield chaos in microflows

“…3C) shows a dramatic change including significant spreading of the orbits throughout the phase map. Spreading of attractor orbits of chaotic flow has been observed experimentally in the multiple periodic-to-chaotic regime transitions in Taylor-Couette flows (3,22). The phenomenon is also evident in classical dynamical systems attractors such as the Rössler attractor and the differential-delay equation (Mackey-Glass (4).…”

“…2A). Periodic windows sandwiched between aperiodic regimes have been observed experimentally in, for example, the Belousov-Zabotinsky reaction (21) and in moderately high Reynolds number TaylorCouette flow (3,16,22). They are also well known as in mathematical models with one-dimensional mappings such as the Rössler attractor (4).…”

Multistability, period doubling and traveling waves suppression by noise excitation in a nonlinear self-oscillatory medium with periodic boundary conditions

“…Примером автоколебательной среды может служить колебательная химическая реакция в большом объеме, различные части которого совершают колебания с различными амплитудами и фазами [1]. Периодические, квазипериодические и хаотические автоколеба-ния были обнаружены экспериментально и численно во многих распределенных системах различной природы и их математических моделях: в потоках жидкости [2,3], в плазме [4,5], в химических реакциях [1,6,7,8], в оптических системах [9,10], в биофизических объектах и живых тканях [11,12]. В силу разнообразия задач, связанных с автоколебательными сре-дами, и множества наблюдаемых явлений, несмотря на большое количество публикаций, поведение автоколебательных сред все еще не является в достаточной степени изученным.…”

Summer study program in geophysical fluid dynamics : order and disorder in turbulent shear flow

“…(~o u l l e t and I were l e d t o a s i m i l a r c o n d i t i o n on t h e c o m p l e t e l y s t a b l e bands of modes i n o r e d e r t o a r r i v e a t a f o r m l i k e ( 2 ) o r ( 5 ) .) Then C ( 2 ) i s i n d e p e n d e n t of t h e m a g n i t u d e of i t s wavevector a r g u m e n t s and is a f u n c t i o n o n l y o f t h e a n g l e s between them.…”

“…Indeed, assuming that global solutions t o the Navier-Stokes equations exist, i t is highly unlikely that for large Reynolds numbers any a t t r a c t o r w i l l have low dimension. There are theorems (Ruelle C61;Foias,Manley,Temm and Treve [7]; Hyman and Nicolaenko [5]) which place an 2pper bound on the dimension of a t t r a c t o r s for p a r t i a l d i f f e r e n t i a l equations i n f i n i t e geometries. For the Navier-Stokes equations, t h i s bound is equivalent t o the i n t u i t i v e notion of Landau who argued t h a t it would be sufficient t o resolve the flow f i e l d down t o the viscous dissipation scale for Reynolds numbers of beticem lo3~:.onir.. 10 , the range i n which tlwbulence is f i r s t seen, one would require up t o a b i l l i o n modes to resolve the system.…”