Narrow Results

The bifurcation diagram of a truncation to six degrees of freedom of the equations for quasi-geostrophic, baroclinic flow is investigated. Period-doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The low dimension of the chaotic attractor suggests the possibility to reduce the model to three degrees of freedom. In a physically comprehensible limit of the parameters this reduction is done explicitly. The bifurcation diagram of the reduced model in this limit is compared to the diagram of the six degrees of freedom model and agrees well. A numerical implementation of the graph transform is used to approximate the three-dimensional invariant manifold away from the limit case. If the six-dimensional model is reduced to a linearisation of the invariant manifold about the Hadley state, the Lorenz-84 model is found. Its parameters can then be calculated from the physical parameters of the quasi-geostrophic model. Bifurcation diagrams at physical and traditional parameter values are compared and routes to chaos in the Lorenz-84 model are described.

Glass networks have been proposed as a model framework for gene regulation, chemical kinetics and neural networks. Their main distinguishing feature is that although the network variables evolve continuously in time, interactions between them depend discontinuously on their sign (i.e. above or below a threshold). While this is a simplification, it has tremendous analytic advantages if the approximation is reasonable in an application. This study explores and classifies bifurcations in Glass networks, and relates them to bifurcations of smooth systems. These bifurcations can often not be studied with traditional bifurcation theory, as the vector fields are discontinuous. However, the theory that has been developed for periodic orbits of Glass networks allows a natural classification for bifurcations of periodic orbits. Some of these are shown to correspond to smooth-system bifurcations, others are shown to fit into the framework of "C-bifurcations" or "border-collision bifurcations" and others are shown to allow truly ambiguous behavior, for which Filippov's theory for discontinuous vector fields is an appropriate tool. Routes to chaos are also explored, and it is demonstrated that period-doubling cascades do not occur. However, sudden transitions to chaos, which are common in Glass networks, can result in a limiting sense from compression to a point of a period-doubling cascade in corresponding networks with sigmoidal interactions as the sigmoid's gain is increased. Other phenomena such as intermittency and multistability are also discussed.

In this paper, a novel autonomous RC chaotic jerk circuit is introduced and the corresponding dynamics is systematically investigated. The circuit consists of opamps, resistors, capacitors and a pair of semiconductor diodes connected in anti-parallel to synthesize the nonlinear component necessary for chaotic oscillations. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a linear transformation of model MO15 previously introduced in [Sprott, 2010]. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. The bifurcation analysis indicates that chaos arises via the usual paths of period-doubling and symmetry restoring crisis. One of the key contributions of this work is the finding of a region in the parameter space in which the proposed (“elegant”) jerk circuit exhibits the unusual and striking feature of multiple attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). Laboratory experimental results are in good agreement with the theoretical predictions.

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

Two common routes to chaos, period-doubling and quasi-periodic, are theoretically investigated in semiconductor laser subject to optical injection. In particular, the sensitivity of the route to the injection of an additional optical signal is examined using bifurcation diagrams. Period-doubling route to chaos is found to be less sensitive to the perturbation of the second signal than the quasi-periodic route.