Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

The multi-directional reach test in children with Down syndrome

Objective: This study investigated the limits of stability (LOS) and the movement patterns during reaching by applying the Multi-Directional Reach Test (MDRT) in children with Down syndrome (DS) aged 7–12 years old. Methods: Thirty children with DS and 30 age and gender typical development (TD) matched children, aged 7–12 years old were recruited. Each child was asked to reach as far as possible during standing in four directions using a self-selected movement pattern. The movement patterns were classified by two experienced pediatric physical therapists. Results: The reach distance in children with DS aged 7–9 years old was significantly shorter than TD children aged 7–9 years old for the forward and backward directions. Also, the reach distance in DS children aged 7–9 years old was significantly smaller than that of TD children aged 10–12 years old for all directions. For children with DS aged 10–12 years old, the reach distance was significantly less than that of TD children only in the backward direction. All children with DS in this study adopt a hip and mixed strategy during forward and backward reaching. In contrast, TD children adopt an adult-like movement pattern. Conclusion: The boundary of stability in an anteroposterior (AP) direction of children with DS aged 7–12 years old was lesser than the matched TD children, especially for the backward direction. These findings may assist therapists in detecting postural control and balance problems in children with DS.

Numerical Study of Motion and Stability of Falling Columnar Crystals

Understanding of the flow field and falling patterns of ice crystals is fundamental to cloud physics and radiative transfer, and yet the complex shape hampers a comprehensive understanding. In order to create better understanding of falling patterns of columnar crystals, this study utilizes a computational fluid dynamics package and explicitly simulates the motion as well as the flow fields. Three modes of patterns (i.e., strong damping, fluttering, and unstable modes) were identified in the space of inverse aspect ratio (q) and Reynolds number (Re). The boundary of stability depicts the “L” shape as found in a previous experimental study. This study newly found that the range of Re for stable motion increases with a decrease in q. Decomposition of hydrodynamic torques indicates that, for stable mode, the pressure and viscous torques acting on the lower prism faces counteract the rotation when the inclination angle becomes 0°. The unstable motion was attributed to the pressure torque acting on the upper prism faces, which is associated with eddies that lag behind the oscillating boundary. Observed Re–q relationships of columns suggest that the strong damping mode is most likely to occur in the atmosphere, but the fluttering mode is also possible. Furthermore, the time scales of oscillation and damping were parameterized as a function of q and Re. The impact of the fluttering on the riming process is limited at the beginning, which supports the current formulation in numerical weather and climate models.

Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control

This paper presents an investigation on the behaviour of con- ventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC mo- tor, mounted on the pendulum’s arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynam- ical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf’s theory, we find a new analytical formula for the first Lyapunov’s value at the boundary of stability. It enables one to study in detail the bifurcation behaviour of the above dynamic system. We check the validity of our analytical results on the first Lyapunov’s value by numerical simulations. Hence, we find some new results.

Simply-connected vortex-patch shallow-water quasi-equilibria

We examine the form, properties, stability and evolution of simply-connected vortex-patch relative quasi-equilibria in the single-layer $f$-plane shallow-water model of geophysical fluid dynamics. We examine the effects of the size, shape and strength of vortices in this system, represented by three distinct parameters completely describing the families of the quasi-equilibria. Namely, these are the ratio $\gamma = L/L_D$ between the horizontal size of the vortices and the Rossby deformation length; the aspect ratio $\lambda $ between the minor to major axes of the vortex; and a potential vorticity (PV)-based Rossby number $\mathit{Ro}= q^{\prime }/f$, the ratio of the PV anomaly $q^{\prime }$ within the vortex to the Coriolis frequency $f$. By defining an appropriate steadiness parameter, we find that the quasi-equilibria remain steady for long times, enabling us to determine the boundary of stability $\lambda _c=\lambda _c(\gamma ,\mathit{Ro})$, for $0.25 \leq \gamma \leq 6$ and $\delimiter "026A30C \mathit{Ro}\delimiter "026A30C \leq 1$. By calling two states which share $\gamma ,\delimiter "026A30C \mathit{Ro}\delimiter "026A30C $ and $\lambda $ ‘equivalent’, we find a clear asymmetry in the stability of cyclonic ($\mathit{Ro}> 0$) and anticyclonic ($\mathit{Ro}<0$) equilibria, with cyclones being able to sustain greater deformations than anticyclones before experiencing an instability. We find that ageostrophic motions stabilise cyclones and destabilise anticyclones. Both types of vortices undergo the same main types of unstable evolution, albeit in different ranges of the parameter space, (a) vacillations for large-$\gamma $, large-$\mathit{Ro}$ states, (b) filamentation for small-$\gamma $ states and (c) vortex splitting, asymmetric for intermediate-$\gamma $ and symmetric for large-$\gamma $ states.

Ice Lines in Circumbinary Protoplanetary Disks

I find the location of the ice line in circumbinary disks heated by steady mass accretion and stellar irradiation, comparing the position with the minimum stable semimajor axis, interior to which planetary orbits are unstable. I show that there is a critical binary separation for which binaries with separations larger than this critical value have ice lines that lie interior to the boundary of stability. The critical separation for an equal-mass binary of 1 M⊙ stars is ≈ 1.04 AU, scaling weakly with mass accretion rate and Rosseland mean opacity of the disk. For a steady mass accretion rate of Ṁ ~ 10−8 M⊙ yr−1 and a Rosseland mean opacity of κR ~ 1 cm2 g−1, I show that ≳ 80% of all binary systems with component masses M☆ ≲ 2.0 M⊙ have ice lines interior to the boundary of stability. This suggests that rocky planets should not be common in these systems. Searching for planets around binaries with separations larger than the critical separation with Kepler or microlensing will provide a test of this prediction.

Quasi-geostrophic shallow-water doubly-connected vortex equilibria and their stability

We examine the form, properties, stability and evolution of doubly-connected (two-vortex) relative equilibria in the single-layer $f$-plane quasi-geostrophic shallow-water model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio $\gamma = L/ {L}_{D} $ between the horizontal size of the vortices and the Rossby deformation length; the area ratio $\alpha $ of the smaller to the larger vortex; and the minimum distance $\delta $ between the two vortices. We vary $0\lt \gamma \leq 10$ and $0. 1\leq \alpha \leq 1. 0$, determining the boundary of stability $\delta = {\delta }_{c} (\gamma , \alpha )$. We also examine the nonlinear development of the instabilities and the transitions to other near-equilibrium configurations. Two modes of instability occur when $\delta \lt {\delta }_{c} $: a small-$\gamma $ asymmetric (wave 3) mode, which is absent for $\alpha \gtrsim 0. 6$; and a large-$\gamma $ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near ${\delta }_{c} $ may be classified as follows: (i) vacillations about equilibrium for $\gamma \gtrsim 2. 5$; (ii) partial straining out, associated with the small-$\gamma $ mode, where either one or both of the vortices get smaller for $\gamma \lesssim 2. 5$ and $\alpha \lesssim 0. 6$; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the large-$\gamma $ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path.

Study on the Morphologies and Formational Mechanism of Poly(hydroxybutyrate-co-hydroxyvalerate) Ultrafine Fibers by Dry-Jet-Wet-Electrospinning

Dry-jet-wet-electrospinning (DJWE) was carried out to study the formational mechanism of poly(hydroxybutyrate-co-hydroxyvalerate) electrospun fibers. Morphological comparison between normal electrospinning (NE) and DJWE was investigated. The results showed that jet could solidify quickly in DJWE to avoid bead collapse or fiber coherence. Jet structures could be maintained at very low collection distance. Beanpod-like beads, which were named as primary beads, could be seen at the boundary of stability and instability section and divided into spindle-like beads with longer collection distance. Bead-free electrospun fibers from DJWE had few bonding points among each other, and fast solidification and double-diffusion led to rough and shriveled fiber surface. DJWE mats were higher hydrophobic than that from NE due to more loose structure and higher surface porosity. Higher bead ratio on the surface and rounder bead structure resulted in higher hydrophobicity.