Oscillatory regulatory networks have been discovered in many regulatory pathways. Due to their enormous complexity, it is necessary to study their dynamics by means of highly simplified models. These models have received particular value because artificial regulatory networks can be engineered experimentally. In this paper, we study dynamical properties of an artificial regulatory oscillator called repressilator. We have shown that oscillations arise from the existence of an absorbing toruslike region in the phase space of the model. This geometric structure requires monotonic repression at all promoters and the absence of any regulatory connections apart from a cyclic repression loop. We show that oscillations collapse as only weak extra connections are introduced if there is imbalance between the attended concentrations and those sufficient for saturation of the promoters. We found that a pair of diffusively coupled repressilators displays synchronization properties similar to those of relaxation oscillators if the regulatory connections in the cyclic repression loop are strong. Thus, the role of strengthening these connections can be viewed as introducing time scale separation among variables. This may explain controversial synchronization properties reported for repressilators in earlier studies.

The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.

Objective: The present study aimed at observing the influence of tube phonation into water on objective voice characteristics in elderly subjects. Methods: Thirty elderly subjects with presbyphonia were randomly assigned to one of two voice exercise groups: (1) voice exercises with water resistance therapy at 4 and 8 cm of water depth (experimental group), and (2) voice exercises with vowel [a:]. Aerodynamic, electroglottographic, and acoustic voice assessments were conducted before and after exercises. Results: The experimental group showed a significant increase in contact quotient, subglottic pressure (P_sub), glottal resistance, and sound pressure level (SPL) when comparing the pre-post 8 cm and the post 4 cm-post 8 cm conditions. No significant differences were found for the control group. Moreover, significant differences for all variables (except for glottal airflow) when comparing condition post (for the control group) and condition post 8 cm (for the experimental group) were found. Conclusion: Tube phonation into water might improve vocal function in the geriatric population. This semioccluded vocal tract exercise seems to promote an increase in P_sub and vocal fold adduction immediately after exercise. This in turn might cause an increase in SPL. The effect should become manifest when a deep level of submersion (e.g., 8 cm water) is used. Shallower submersion produced negligible or no effects.

Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of z 2 + i. This is the first non-trivial example supporting the conjecture.

Abstract. We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

We give a recursive formula to count maximal small copies of the Mandelbrot set and its higher degree analogues. This formula is used to compute the asymptotic growth of the number of maximal small copies of period n.

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We present a new identity involving compositions (i.e., ordered partitions of natural numbers). The formula has its origin in complex dynamical systems and appears when counting, in the polynomial family { f c : z → z d + c}, periodic critical orbits with equivalent itineraries. We give two different proofs of the identity; one following the original approach in dynamics and another with purely combinatorial methods.