Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

Symmetry breaking Paradigm In Typical Laminar-Turbulence Transition System

A stationary cylindrical vessel containing a rotating plate near the bottle surface is partially filled with liquid. With the bottom rotating, the shape of the liquid surface would become polygon-like. This polygon vortex phenomenon is a ideal system to demonstrate the Laminar-Turbulent transition process. Within the framework of equilibrium statistical mechanics, a profound comparison with Landau's phase transition theory was applied in symmetry breaking aspect to derive the evolution equation of this system phenomenologically. Comparison between theoretical prediction and experimental data is carried out. We concluded a considerably highly matched result, while some exceptions are viewed as the natural result that the experiment break through the up-limit of using equilibrium mechanics as a effective theory, namely breaking through the Arnold Tongue. Some extremely complex Non-equilibrium approaches was desired to solve this problem thoroughly in the future. So our method could be viewed as a linear approximation of this theoretical framework.

Torus-Breakdown Near a Heteroclinic Attractor: A Case Study

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of [Formula: see text]-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to [Formula: see text] of a polynomial vector field in [Formula: see text]. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

Creation of discontinuities in circle maps

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Oscillation or not – why we can and need to know

Neural oscillations are pivotal to brain function and cognition, but they can be difficult to identify. Researchers engage in careful experimentation to identify their presence, ruling out non-oscillatory processes that could give rise to a similar response. Recently, Doelling and Assaneo have argued against these efforts on the basis that oscillators are heterogeneous, which makes the line to non-oscillators blurred and thereby meaningless to draw. Here, we offer our opposing viewpoint, arguing that we can know whether oscillations are involved, and that we need to know. First, we can know because there are unique properties that only oscillators have, which can be reliably used to identify them – the line is not blurred. These unique properties include eigenfrequency, Arnold Tongue, convergence, and independence. Second, we need to know because there are shared properties, which all oscillators or all oscillators within a subclass have. These shared properties comprise all the information we get once we know there is an oscillator, including neurophysiological, functional, and methodological properties – the fruits of decades of research. We argue that identifying oscillators is crucial for the advancement of research fields as it constrains the possible neural dynamics involved and allows us to make informed predictions on a variety of levels. While neural oscillations are the start and not the end, we have to reach that start.

Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

Modelling Oscillations in the Supply Chain: The Case of a Just-in-Sequence Supply Process from the Automotive Industry

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

Injection locking in an optomechanical coherent phonon source

Spontaneous locking of the phase of a coherent phonon source to an external reference is demonstrated in a deeply sideband-unresolved optomechanical system. The high-amplitude mechanical oscillations are driven by the anharmonic modulation of the radiation pressure force that result from an absorption-mediated free-carrier/temperature limit cycle, i.e., self-pulsing. Synchronization is observed when the pump laser driving the mechanical oscillator to a self-sustained state is modulated by a radiofrequency tone. We employ a pump-probe phonon detection scheme based on an independent optical cavity to observe only the mechanical oscillator dynamics. The lock range of the oscillation frequency, i.e., the Arnold tongue, is experimentally determined over a range of external reference strengths, evidencing the possibility to tune the oscillator frequency for a range up to 350 kHz. The stability of the coherent phonon source is evaluated via its phase noise, with a maximum achieved suppression of 44 dBc/Hz at 1 kHz offset for a 100 MHz mechanical resonator. Introducing a weak modulation in the excitation laser reveals as a further knob to trigger, control and stabilize the dynamical solutions of self-pulsing based optomechanical oscillators, thus enhancing their potential as acoustic wave sources in a single-layer silicon platform.

Period-Bubbling Transition to Chaos in Thermo-Viscoelastic Fluid Systems

We report a 6D nonlinear dynamical system for thermo-viscoelastic fluid by selecting higher modes of infinite Fourier series of flow quantities. This nonlinear system demonstrates overstable convective motion and some organized structures such as period-bubbling and Arnold tongue-like structures. Studies reveal that the stability of the conduction state does not alter for the new 6D system in comparison with the lowest order 4D system of Khayat [1995] . However, the stabilities of the convective state have some differences. The onset of unsteady convection in the 6D system is delayed for weak elasticity of the fluid. There exists a critical range of fluid elasticity where the 4D system exhibits subcritical Hopf bifurcation while the 6D system shows supercritical Hopf bifurcation, which ensures the increase of the domain of stability. In this range, catastrophic route to chaos occurs in the 4D system, whereas the 6D system exhibits intermittent onset of chaos. Comparing the two-parameter dependent dynamics for the two systems, the chaotic zones enclosed by periodic regions are suppressed in the 6D system, so the flow behaviors become more predictable. Owing to interacting thermal buoyancy and fluid elasticity, both the models exhibit period-bubbling transition to chaos, but the period-bubbling cascade in the 6D model occurs at lower Rayleigh number than the 4D model. The convergence rate of the period-bubbling process slows down compared to usual period-doubling and approaches the square root of the Feigenbaum constant asymptotically.

Quantum synchronization and correlations of two qutrits in a non-Markovian bath

We investigate quantum synchronization and correlations of two qutrits in one non-Markovian environment using the hierarchy equation method. There is no direct interaction between two qutrits and each qutrit interacts with the same non-Markovian environment. The influence of the temperature of the bath, correlation time and coupling strength between qutrits and bath on the quantum synchronization and correlations of two qutrits are studied without the Markovian, Born and rotating wave approximations. We also discuss the influence of dissipation and dephasing on the synchronization of two qutrits. In the presence of dissipation, the phase locking between two qutrits without any direct interaction can be achieved when each qutrit interacts with the common bath. Two qutrits within one common bath cannot be syncrhonized in the purely dephasing case. In addition, the Arnold tongue can be significantly broadened by decreasing the correlation time of two qutrits and bath. Markovian baths are more suitable for synchronizing qutrits than non-Markovian baths.