Flexible Accelerator Economic Systems as Coupled Oscillators

In this paper, some complex nonlinear behaviors in a four-dimensional multiscroll autonomous memristor based chaotic system are investigated. This system is derived from the three-dimensional autonomous charge-controlled Muthuswamy–Chua simplest chaotic circuit. The system can generate four different coexisting attractors for a fixed set of parameters and different initial conditions. This phenomenon is relatively rare given that we have four different attractors namely: an equilibrium point, a stable limit cycle, a 16-peak limit cycle and a strange attractor that coexist in the system within a wide range of parameters. The nonlinear phenomenon of transient chaos is studied and revealed numerically in Matlab and Pspice environments. The complex transient dynamics of this memristive system under different initial states shows that the transient time depends strongly on the initial conditions. Moreover, this model displays spiking and bursting oscillations. The bursting behavior is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the fold-Hopf type. These complex dynamical behaviors of this system are investigated by means of numerical simulations and via Pspice circuit simulations. The use of bifurcation diagrams, Lyapunov exponents diagrams, power spectrums, phase portraits, time series, isospike diagram, basin of attraction, clearly shows these complex phenomena.

We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle‐node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

We present a model study of two mass-coupled reactors containing the Belousov-Zhabotinsky reaction under chaotic conditions. The critical coupling strength is estimated for symmetry breaking when two identical low flow rate chaotic modes are coupled. Our results confirm that the critical coupling strength is directly proportional to the maximum Lyapunov exponent of the uncoupled system. The constant of proportionality is found to be somewhat larger than the theoretical value. Direct integration reveals a rich structure of dynamical behavior when the coupling strength and the flow rate in one cell are varied. Our simulations reveal domains of oscillator death, in which a stable steady state coexists with limit cycle oscillations. We introduce a simple model for predicting the dynamics of the coupled system from the dynamical behavior of the uncoupled subsystems and the dependence on the coupling strength of the Hopf bifurcation of the coupled system. The model gives good estimates of the dynamical behavior of two coupled oscillators with a small difference in one parameter at intermediate and high coupling strengths. \textcopyright{} 1996 The American Physical Society.

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters.

In this work the performance of ensembles generated by commonly used methods in a nonlinear system with multiple attractors is examined. The model used here is a spectral truncation of a barotropic quasigeostrophic channel model. The system studied here has 44 state variables, great enough to exhibit the problems associated with high state dimension, but small enough so that experiments with very large ensembles are practical, and relevant probability density functions (PDFs) can be evaluated explicitly. The attracting sets include two stable limit cycles. To begin, the basins of attraction of two known stable limit cycles are characterized. Large ensembles are then used to calculate the evolution of initially Gaussian PDFs with a range of initial covariances. If the initial covariances are small, the PDF remains essentially unimodal, and the probability that a point drawn from the initial PDF lies in a different basin of attraction from the mean of that PDF is small. If the initial covariances are so large that there is significant probability that a given point in the initial ensemble does not lie in the same basin of attraction as the mean, the initial Gaussian PDF will evolve into a bimodal PDF. In this case, graphical representation of the PDF appears to split into two distinct regions of relatively high probability. The ability of smaller ensembles drawn from spaces spanned by singular vectors and by bred vectors to capture this splitting behavior is then investigated, with the objective here being to see how well they capture multimodality in a highly nonlinear system. The performance of similarly small random ensembles drawn without dynamical constraints is also evaluated. In this application, small ensembles chosen from subspaces of singular vectors performed well, their weakest performance being for an ensemble with relatively large initial variance for which the Gaussian character of the initial PDF remained intact. This was the best case for the bred vectors because of their tendency to align tangent to the attractor, but the bred vectors were at a disadvantage in detection of the tendency of an initially Gaussian PDF to evolve into a bimodal one, as were the unconstrained ensembles.

We argue that the universality and statistical nature of the deep-ocean internal gravity-wave spectrum results from a strange attractor in the driven, dissipative internal-wave field. To explore this we construct a model which injects energy into the oceanic surface at a constant rate. A two-dimensional version of the model is explored analytically and numerically. For the numerical work we restrict our considerations to a few of the longest-wavelength modes. This few-mode system exhibits bifurcation into limit cycles, period doubling of the limit cycles, and chaotic, non-periodic behaviour associated with a strange attractor. In an appendix we present some discussion of the three-dimensional version of the model.

It is known that the unidirectional coupling between a double-scroll system (drive) in the oscillation regime with another system (response) in the steady-state, gives rise to more complex dynamics such as multiscroll attractors [Dana et al., 2008]. This paper shows the way to move from n-scroll to 2n-scroll in a bidirectional coupling scheme between two jerk oscillators. The coupling considered can be interpreted as a perturbation to each oscillator’s amplitude through a signal proportional to the amplitude of the other. Indeed, starting from a situation of two oscillators which can generate double-scroll dynamics individually, we perform a bidirectional coupling of both oscillators which results in more complex dynamics such as multistability and 4-scroll chaotic attractor in the coupled system (not observed in an isolated oscillator). The underlined coupling scheme provides an efficient method for generating higher-order multiscroll attractors. The route to the 4-scroll chaotic attractor is explained with the help of bifurcation diagrams, plots of several phase portraits, and basins of attractions. Indirect validation of the theoretical analysis is carried out by using the PSpice simulation software.

Extensive numerical simulations of the rf-biased Josephson junction are presented. It is shown that, as the amplitude is increased for fixed damping and frequency, an apparently endless sequence of bifurcating-chaotic trees separated by periodic solutions exists. The state diagram characterized in the amplitude-damping plane at fixed frequency (..cap omega.. = 0.65) shows a complex set of solutions. A detailed study of the transition in the high-damping limit is presented, indicating the bounds for asymmetric, bifurcating, and chaotic solutions for low ..beta../sub c/. It is shown that strange attractors are common to trapped and free-running chaotic solutions for nearby amplitudes due to intermittency among the various basins of attraction. Return maps at high amplitudes are found to be essentially one dimensional. Experimental consequences of our simulations are presented in terms of equivalent noise temperatures at low frequencies, resulting in maximum noise temperatures of the order of 10/sup 6/--10/sup 7/ K.

Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot

A concrete example with three limit cycles in Zeeman’s class 29 for three dimensional Lotka–Volterra competitive systems

We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.

Coupling oscillations and switches in genetic networks

It is usually believed that strong diffusive coupling in one of the dynamical variables is well-suited for imposing synchronization of oscillators. But it was recently shown that weak diffusive coupling, counter-intuitively, can lead to dephasing of coupled neural oscillators. In this paper, we investigate how diffusively coupled oscillators become dephasing. For this we study a system of coupled neural oscillators on a limit cycle generated through a homoclinic bifurcation. We examine the asymptotic behavior of diffusive coupling as the control parameter approaches the critical value for which the homoclinic bifurcation occurs. In this study, we show that the gradient of phase velocity near the limit cycle is essential in generating dephasing through diffusive interaction.

The dynamics of a perturbed system consisting of a particle embedded in a strongly nonlinear potential V(ϕ; r, η) whose shape can be varied continuously as a function of r in the range -1 < r < 1, and coupled to an harmonic oscillator is analyzed. The perturbations are made of the coupling and damping forces. When they are removed, the subsystem formed by the anharmonic oscillator contains saddle points connected by homoclinic loops. Thus, the whole unperturbed system has saddle-centre points whose stable and unstable manifolds coincide in three-dimensional manifold. We concentrate our analysis of the perturbed system in the regions near these saddle-centre points. First, the Melnikov theory is used to investigate the presence of horseshoes chaos in the dynamics. Due to some discrepancies of the Melnikov theory, we next rederive the boundary between the regions of regular and irregular motions in the parameters space by using regular perturbation expansion. It is found here that for small values of the natural pulsation ω of the linear oscillator, complicated behavior intensifies in the system as the absolute value of the shape parameter r approaches zero. On the contrary, chaotic motion intensifies according as r decreases from values close to 1 to values close to -1, for large ω. The numerical analysis, which includes computation of maximal Lyapunov exponent, bifurcation diagrams and Poincaré sections, shows that the system exhibits transient stochastic behavior, but ultimately settles down on a simple set which is either a fixed point or a limit cycle. The dependence of this transient behavior on the system parameters agrees qualitatively well with the analytical predictions.

We consider chains of identical diffusive weakly coupled oscillation systems with different conditions on coupling on a chain bound. We suppose that every interacting oscillator undergoes Andronov–Hopf bifurcation, and the coefficient of coupling is proportional to the supercriticality value. For this case on a stable integral manifold of the system a normal form is constructed for which, in case of three oscillators interact, we can study the simplest stationary states and their phase transformations. As the coupling parameter changes, for the uniform stationary state corresponding to the uniform cycle of the problem there can be two cases, in the former case it loses stability with the emergence of two stable nonuniform states, and in the latter one it merges with two unstable nonuniform states and transfers to them its stability. For the stationary state corresponding to oscillations in antiphase two cases can be also distinguished. In the first one, this stationary state becomes stable due to the contraction of a stable limit cycle of the system into it (Andronov–Hopf bifurcation), and in the second case it becomes stable after branching from it an unstable limit cycle. If there are four oscillators in the chain, the system of phase difference for a small coupling coefficient was studied.