Period-doubling Regions Research Articles

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

A Robust Molecular Network Motif for Period-Doubling Devices.

Life is sustained by a variety of cyclic processes such as cell division, muscle contraction, and neuron firing. The periodic signals powering these processes often direct a variety of other downstream systems, which operate at different time scales and must have the capacity to divide or multiply the period of the master clock. Period modulation is also an important challenge in synthetic molecular systems, where slow and fast components may have to be coordinated simultaneously by a single oscillator whose frequency is often difficult to tune. Circuits that can multiply the period of a clock signal (frequency dividers), such as binary counters and flip-flops, are commonly encountered in electronic systems, but design principles to obtain similar devices in biological systems are still unclear. We take inspiration from the architecture of electronic flip-flops, and we propose to build biomolecular period-doubling networks by combining a bistable switch with negative feedback modules that preprocess the circuit inputs. We identify a network motif and we show it can be "realized" using different biomolecular components; two of the realizations we propose rely on transcriptional gene networks and one on nucleic acid strand displacement systems. We examine the capacity of each realization to perform period-doubling by studying how bistability of the motif is affected by the presence of the input; for this purpose, we employ mathematical tools from algebraic geometry that provide us with valuable insights on the input/output behavior as a function of the realization parameters. We show that transcriptional network realizations operate correctly also in a stochastic regime when processing oscillations from the repressilator, a canonical synthetic in vivo oscillator. Finally, we compare the performance of different realizations in a range of realistic parameters via numerical sensitivity analysis of the period-doubling region, computed with respect to the input period and amplitude. Our mathematical and computational analysis suggests that the motif we propose is generally robust with respect to specific implementation details: functionally equivalent circuits can be built as long as the species-interaction topology is respected. This indicates that experimental construction of the circuit is possible with a variety of components within the rapidly expanding libraries available in synthetic biology.

Investigation of Period-Doubling Islands in Milling With Simultaneously Engaged Helical Flutes

This paper investigates the stability of a milling process with simultaneously engaged flutes using the state-space TFEA and Chebyshev collocation methods. In contrast to prior works, multiple flute engagement due to both the high depth of cut and high step-over distance are considered. A particular outcome of this study is the demonstration of a different stability behavior in comparison to prior works. To elaborate, period-doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or nonzero helix angle are simultaneously cutting. We also demonstrate stability differences that arise due to the parity in the number of flutes, especially at full radial immersion. In addition, we study other features induced by helical tools such as the waviness of the Hopf lobes, the sensitivity of the period-doubling islands to the radial immersion, as along with the orientation of the islands with respect to the Hopf lobes.

A study on the numerical convergence of the discrete logistic map

Clocking convergence is an important tool for investigating various aspects of iterative maps, especially chaotic maps. In this work, we revisit to the numerical convergence of the discrete logistic map xn+1=rxn(1-xn) gauged with a finite computational accuracy. Most of the previous studies of the discrete logistic map have been made for r∈[3,4] and r∈[-2,-1] due to the rich complexity of the map in these regions. In this work, we consider regions with simple fixed points, i.e. r=[-1,3] for which no particular geometric structures are known, as well as the period-doubling regions. We numerically investigate the speed of convergence in these regions to expose underlying complexity. The convergence speed is mapped to the phase space with different finite precisions. Patterns generated through this map are investigated over r. Numerical results show that there exists an interesting geometric pattern in r∈[-1,3] when convergence is gauged with a finite computational precision and also show that this pattern cascades into the period-doubling areas.

Nonlinear dynamics of planetary gears using analytical and finite element models

Vibration-induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional (2D) lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The 2D finite element model is developed from a unique finite element-contact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing rules, however, are not valid in the chaotic and period-doubling regions.

Effect of contrast on the frequency response of synchronous period doubling

At temporal frequencies between approximately 30 and 70 Hz, the flicker electroretinogram (ERG) of the cone system can exhibit an alternation in response amplitude from cycle to cycle that has been termed synchronous period doubling. This phenomenon has been attributed to a nonlinear feedback mechanism at an early retinal locus. The purpose of the present study was to define the effect of stimulus contrast on period doubling in order to better understand the nature of the underlying mechanism. ERGs were recorded from three visually normal subjects in response to sinusoidal flicker ranging from 20 to 100 Hz, using stimulus contrasts of 37.7, 56.5, 75.4, and 94.2%. Period doubling was quantified as: (1) the amplitude of an harmonic component of the ERG waveform that was 1.5 times the stimulus frequency, and (2) the difference between the mean trough-to-peak amplitudes on even and odd cycles of the ERG waveform. Amplitudes were converted to responsivity by dividing by stimulus contrast. By both measures, subjects showed discrete regions of period doubling that were displaced to lower temporal frequencies as stimulus contrast was increased. The temporal frequency shift of period doubling with altered stimulus contrast can be accounted for quantitatively by postulating a neural threshold for the nonlinear feedback signal that is presumed to generate synchronous period doubling.

Dynamic behavior of two-phase systems in physical equilibrium

The steady state and dynamic behavior of two-phase systems in physical equilibrium is investigated. The autonomous and non-autonomous systems are considered. The pseudo-arclength bifurcation technique reveals steady state multiplicity patterns not previously observed, including isola and mushroom patterns. It is shown that degenerate singular points of codimension 2, which violate the non-singularity and transversality conditions of the classical Hopf theorem exist. The effect of the forcing amplitude and frequency on the behavior of the non-autonomous system is investigated at a number of chosen positions of the center of forcing. It is found that the forced system is very sensitive to the position of the center of forcing relative to Hopf bifurcation points of the unforced system. The excitation diagram shows that a period doubling region may exist at the top of a 2/1 resonance horn. It is shown that a Hopf bifurcation curve of the stroboscopic map is originated at bifurcation points having double −1 Floquet multipliers.

Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies

The behavior of a driven double-well Duffing--van der Pol oscillator for a specific parametric choice $(|\ensuremath{\alpha}|=\ensuremath{\beta})$ is studied. The existence of different attractors in the system parameters $f$-$\ensuremath{\omega}$ domain is examined and a detailed account of various steady states for fixed damping is presented. The transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period-doubling regions), boundary crisis, three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during the transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instabilities allows one to predict Neimark instability in the $f$-$\ensuremath{\omega}$ plane and eventually results in the approximate predictive criteria for the chaotic region.

Chaos Induced Transition

To study the coherent nature of chaos, two models are proposed. Model 1 is a simple nonlinear system [Formula: see text] and Model 2 is a linear harmonic oscillator [Formula: see text], which are driven by a chaotic force f(t). The chaotic force f(t) is defined by [Formula: see text] for nτ < t ≤ (n + 1)τ(n = 0, 1, 2, …), where yn+1 is a chaotic sequence of a map F(y; r) with the bifurcation parameter r: yn+1 = F(yn; r) (-0.5 ≤ yn ≤ 0.5) and ŷn = yn - < y0>. In Model 1 the relaxation process of this system and the τ- and r-dependence of the stationary distribution of x are discussed. It is shown that the small change of the bifurcation parameter r causes the drastic change of the stationary distribution. In Model 2, resonance phenomena are investigated near the period 3 window of the logistic map, in particular, in the intermittent chaos region and the period doubling region. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y; r).

Simulation studies of the dynamic behavior of semiconductor lasers with Auger recombination

We apply the phase portrait analysis to semiconductor lasers with Auger recombination and extend the analysis to high modulation frequency ωm. On the two-dimensional bifurcation diagram of modulation depth and modulation frequency, there are seven regions: digital pulsing regions, analog modulation region, period doubling regions, chaos regions, and one multiloop region. It is found that Auger recombination tends to suppress chaos for ωm<ωr, the relaxation frequency. However, for ωm>ωr, chaotic behavior becomes prominent. Furthermore, in the pulsing region, the maximum pulsation frequency is limited to a value around ωr even though ωm may be twice or three times ωr. A normalized two-dimensional bifurcation diagram defining the digital pulse region and the analog modulation region is presented for the purpose of locating the suitable region for analog and digital operation of semiconductor lasers.

THE TOPOLOGICAL ENTROPY OF ONE-DIMENSIONAL UNIMODAL MAPS

On the basis of computational calculation for the logistic map, we analyse the inverse orbital structure of one-dimensional unimodalmap and prove the different recurrence formulae which show that the total number of inverse orbit N(n) changes with inverting time n at different parameters. By this way, we analytically obtain h(f)=0 within the period-doubling region; h(f) = logamp on m = 3 + 21 period point of U -sequence RLR21, where amp is the largest real root of equation am-2am-2-1 = 0; hj(f) = (1/2)j log2 on the boundary between 2j-1 band and 2j band, from this result we find the scaling exponent of topological entropy near the accumulation point μ∞, t = 0.449806…. By means of above conclusion, we obtain the topological entropy within each "window" in chaotic region and on U-sequence RLaRb period points, we also get the relation of hR*Q(f)=(l/2)hQ(f). Because we carry out our proofs on the basis of M.S.S. rule and "*" composition law, the results in this paper are universal to all the one-dimensional unimodal maps.