Subharmonic Bifurcations Research Articles

Many routes to turbulent convection

Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R, and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 104 to 105. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R. Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained.

An example of bifurcation to homoclinic orbits

Consider the equation x ̈ − x + x 2 = −λ 1x + λ 2ƒ(t) where ƒ(t + 1) = ƒ(t) and λ = (λ 1, λ 2) is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations.

Scaling Behavior of Chaotic Flows

It is shown that in the turbulent regime of systems with period-doubling subharmonic bifurcations, the maximum Lyapunov characteristic exponent behaves like $\overline{\ensuremath{\lambda}}={\overline{\ensuremath{\lambda}}}_{0}{(\mathcal{r}\ensuremath{-}{\mathcal{r}}_{c})}^{t}$, with $t$ a universal exponent which is calculated to be $t=0.4498069\dots{}$. This result is in agreement with the available data on $\overline{\ensuremath{\lambda}}$ for a number of dynamical systems.

Fluctuations and the onset of chaos

We consider the role of fluctuations on the onset and characteristics of chaotic behavior associated with period doubling subharmonic bifurcations. By studying the problem of forced dissipative motion of an anharmonic oscillator we show that the effect of noise is to produce a bifurcation gap in the set of available states. We discuss the possible experimental observation of this gap in many systems which display turbulent behavior.

The onset spectrum of turbulence

A general theory for the fluctuation spectrum of the onset of turbulence is developed, applying to systems that approach turbulence through a cascade of subharmonic bifurcations. Applied to Rayleigh-Bénard flow, we find excellent quantitative agreement with the recent experimental data of Libchaber and Maurer.

Successive Subharmonic Bifurcations and Chaos in a Nonlinear Mathieu Equation

Successive Subharmonic Bifurcations and Chaos in a Nonlinear Mathieu Equation

A Computer Investigation of a Subharmonic Bifurcation Point in the Duffing Equation

A Computer Investigation of a Subharmonic Bifurcation Point in the Duffing Equation