Abstract
The Lorenz system is a simplified model of Rayleigh-Bénard convection, a thermally driven fluid convection between two parallel plates. Two additional physical ingredients are considered in the governing equations, namely, rotation of the model frame and the presence of a density-affecting scalar in the fluid, in order to derive a six-dimensional nonlinear ordinary differential equation system. Since the new system is an extension of the original three-dimensional Lorenz system, the behavior of the new system is compared with that of the old system. Clear shifts of notable bifurcation points in the thermal Rayleigh parameter space are seen in association with the extension of the Lorenz system, and the range of thermal Rayleigh parameters within which chaotic, periodic, and intermittent solutions appear gets elongated under a greater influence of the newly introduced parameters. When considered separately, the effects of scalar and rotation manifest differently in the numerical solutions; while an increase in the rotational parameter sharply neutralizes chaos and instability, an increase in a scalar-related parameter leads to the rise of a new type of chaotic attractor. The new six-dimensional system is found to self-synchronize, and surprisingly, the transfer of solutions to only one of the variables is needed for self-synchronization to occur.
Highlights
Seven decades after Henri Poincaré’s initial glimpse at chaos in the three-body problem in 1893,1 Edward Lorenz discovered a strange attractor in the numerical solutions to a deceptively simple set of three ordinary di erential equations (ODEs), which was initially conceived to examine the problem of weather forecasting.[2]
Since the analogous destabilization of the nontrivial xed points via a Hopf bifurcation in the original Lorenz system is closely preceded by the heteroclinic bifurcation ushering the onset of chaos,[29] the observed shifts in the neutral stability curve raise the possibility that given a larger s, a higher critical rT is needed for the onset of chaos in the new system, which is con rmed to be the case in the subsequent analyses of numerical solutions
The Lorenz systems are not precise models of real uid convection, some conceptual insights can still be gained from studying our new ODE system
Summary
Seven decades after Henri Poincaré’s initial glimpse at chaos in the three-body problem in 1893,1 Edward Lorenz discovered a strange attractor in the numerical solutions to a deceptively simple set of three ordinary di erential equations (ODEs), which was initially conceived to examine the problem of weather forecasting.[2]. The governing equations from which the Lorenz system was originally derived describe the thermally driven convection of a uid.[5] Geophysical uids in nature are, rarely without impurities such as particulate matter in the atmosphere or salt particles dissolved in seawater These particles can be incorporated into the model as a density-a ecting scalar in the governing equations, from which a ve-dimensional ODE system can be derived.[8] Like the three-dimensional Lorenz system it encompasses, the vedimensional system exhibits chaos through heteroclinic explosion.[9]. Another important factor to consider in the context of geophysical uid convection is planetary rotation. For this reason, when a new system such as our new sixdimensional ODE system emerges, it is worthwhile to clarify whether self-synchronization occurs in the new system and if so, how much information from the driver is needed
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