ResearchPad - equations-of-motion https://www.researchpad.co Default RSS Feed en-us © 2020 Newgen KnowledgeWorks <![CDATA[Conservation laws by virtue of scale symmetries in neural systems]]> https://www.researchpad.co/article/elastic_article_14657 Considerations of the way in which a dynamical system changes under transformation of scale offer insight into its operational principles. Scale freeness is a paradigm that has been observed in a variety of physical and biological phenomena and describes a situation in which appropriately scaling the space and time coordinates of any evolution of the system yields another possible evolution. In the brain, scale freeness has drawn considerable attention, as it has been associated with optimal information transmission capabilities. Scale symmetry describes a special case of scale freeness, in which a system is perfectly unchanged under transformation of scale. Noether’s theorem tells us that in a system that possesses such a symmetry, an associated conservation law must also exist. Here we show that scale symmetry can be identified, and the related conserved quantities measured, in both simulations and real-world data. We achieve this by deriving a generalised equation of motion that leaves the action invariant under spatiotemporal scale transformations and using a modified version of Noether’s theorem to write the associated family of conservation laws. Our contribution allows for the first such statistical characterisation of the quantity that is conserved purely by virtue of scale symmetry.

]]>
<![CDATA[Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks]]> https://www.researchpad.co/article/5989da22ab0ee8fa60b7f6dd

In oscillating mechanical systems, nonlinearity is responsible for the departure from proportionality between the forces that sustain their motion and the resulting vibration amplitude. Such effect may have both beneficial and harmful effects in a broad class of technological applications, ranging from microelectromechanical devices to edifice structures. The dependence of the oscillation frequency on the amplitude, in particular, jeopardizes the use of nonlinear oscillators in the design of time-keeping electronic components. Nonlinearity, however, can itself counteract this adverse response by triggering a resonant interaction between different oscillation modes, which transfers the excess of energy in the main oscillation to higher harmonics, and thus stabilizes its frequency. In this paper, we examine a model for internal resonance in a vibrating elastic beam clamped at its two ends. In this case, nonlinearity occurs in the form of a restoring force proportional to the cube of the oscillation amplitude, which induces resonance between modes whose frequencies are in a ratio close to 1:3. The model is based on a representation of the resonant modes as two Duffing oscillators, coupled through cubic interactions. Our focus is put on illustrating the diversity of behavior that internal resonance brings about in the dynamical response of the system, depending on the detailed form of the coupling forces. The mathematical treatment of the model is developed at several approximation levels. A qualitative comparison of our results with previous experiments and numerical calculations on elastic beams is outlined.

]]>
<![CDATA[Higher Dimensional Gaussian-Type Solitons of Nonlinear Schrödinger Equation with Cubic and Power-Law Nonlinearities in PT-Symmetric Potentials]]> https://www.researchpad.co/article/5989d9f7ab0ee8fa60b708a4

Two families of Gaussian-type soliton solutions of the (n+1)-dimensional Schrödinger equation with cubic and power-law nonlinearities in -symmetric potentials are analytically derived. As an example, we discuss some dynamical behaviors of two dimensional soliton solutions. Their phase switches, powers and transverse power-flow densities are discussed. Results imply that the powers flow and exchange from the gain toward the loss regions in the cell. Moreover, the linear stability analysis and the direct numerical simulation are carried out, which indicates that spatial Gaussian-type soliton solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the defocusing cubic and focusing power-law nonlinear medium, while they are always unstable for all parameters in other media.

]]>