Three-dimensional Oscillator Research Articles

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

Approximate analytical and numerical solutions of the three-dimensional Logunov–Tavkhelidze equation are found for the spherically symmetric case. Solutions are obtained in momentum and relativistic configuration representations. The wave functions in the relativistic configuration representation have additional zeroes compared to the wave functions of the nonrelativistic harmonic oscillator in the coordinate representation.

On the Stability of Control of an Inertial Pendulum-Type System

To obtain information about the inertial trihedron on board a moving object, it suffices to observe the trajectory of a three-dimensional isotropic oscillator with the center of suspension on this object. In the most general case, the trajectory of such an oscillator is an ellipse that does not have an angular velocity relative to the inertial space. In this paper, we construct an oscillator control stabilizing an elliptical trajectory with a nonzero quadrature and prove the stability of this control.

Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.

Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos

The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.

Relative Fisher information for Morse potential and isotropic quantum oscillators

The relative Fisher information associated with quantum-mechanical states for two prototypical models is presented; the radial density functions that characterize the Morse oscillator and the d-dimensional isotropic quantum oscillators are analyzed in terms of this relative information by specifying the ground state as the reference. We find that the information increases as the eigenvalue (i.e., the degree of the associated Laguerre polynomials) increases for the Morse density function, but for the isotropic quantum harmonic models, it decreases with the dimension and increases almost linearly with the increasing quantum numbers. We present also the explicit expression for the information associated with the angular density as a function of the orbital quantum number and the magnetic quantum number for the three-dimensional isotropic quantum oscillator.

Stabilization of a multi-tethered lighter-than-air rigid-body sphere undergoing vortex-induced vibrations in uniform flow

We consider the stabilization of a multi-tethered lighter-than-air sphere undergoing vortex-induced vibrations in uniform flow. Motivated by recent engineering applications, we extend a single-tether model coupled with a three-dimensional wake oscillator to incorporate a four-tether configuration with general rigid-body rotations for the light sphere. The fundamental bifurcation structure of the coupled model includes periodic self-excited limit cycles, which culminate with quasiperiodic solutions with increased reduced velocity. Stabilization of limit-cycle oscillations is achieved for optimal mass ratio and tether length parameters which are obtained by asymptotic analysis of reduced-order models. The resulting reduction in transverse and rotational amplitudes is at the expense of a decreasing range of the periodic bifurcation region for large tether inclination angles, whereas in addition to rotational amplitude reduction, regularization of nonstationary solutions occurred for a medium inclination angle.

On Discontinuous Piecewise Linear Models for Memristor Oscillators

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

Arnol’d resonance webs and Chenciner bubbles from a three-dimensional piecewise-constant hysteresis oscillator

In this study, we analyze quasiperiodic oscillations generated by a three-coupled autonomous piecewise-constant hysteresis oscillator. These oscillations generate three-dimensional tori and Arnol’d resonance webs in which two-dimensional tori-partial-entrainment regions extend in a number of directions in a manner similar to webs in three-dimensional tori-generating regions. Detailed Lyapunov analysis clearly shows that hysteresis phenomena significantly erode Chenciner bubbles. This phenomenon is significant because full-mode entrainment regions do not occur as a result of simple phase-locking of two-dimensional tori. To clarify the mechanism underlying the hysteresis phenomenon observed in electric circuits, we analyze a two-coupled delayed logistic map as one of the simplest discrete dynamics that generate Arnol’d resonance webs and confirm the generation of hysteresis around Chenciner bubbles in a manner similar to that in an autonomous oscillator. By analyzing the Chenciner bubbles produced under these discrete dynamics, we partially explain hysteresis as a result of subcritical Neimark–Sacker bifurcations.

Dynamics of dipoles and vortices in nonlinearly coupled three-dimensional field oscillators.

The dynamics of a pair of harmonic oscillators represented by three-dimensional fields coupled with a repulsive cubic nonlinearity is investigated through direct simulations of the respective field equations and with the help of the finite-mode Galerkin approximation (GA), which represents the two interacting fields by a superposition of 3+3 harmonic-oscillator p-wave eigenfunctions with orbital and magnetic quantum numbers l=1 and m=1, 0, -1. The system can be implemented in binary Bose-Einstein condensates, demonstrating the potential of the atomic condensates to emulate various complex modes predicted by classical field theories. First, the GA very accurately predicts a broadly degenerate set of the system's ground states in the p-wave manifold, in the form of complexes built of a dipole coaxial with another dipole or vortex, as well as complexes built of mutually orthogonal dipoles. Next, pairs of noncoaxial vortices and/or dipoles, including pairs of mutually perpendicular vortices, develop remarkably stable dynamical regimes, which feature periodic exchange of the angular momentum and periodic switching between dipoles and vortices. For a moderately strong nonlinearity, simulations of the coupled-field equations agree very well with results produced by the GA, demonstrating that the dynamics is accurately spanned by the set of six modes limited to l=1.

Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with Oh symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh–Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions. We show how to obtain suitable symmetry-adapted basis sets.

Co-existing hidden attractors in a radio-physical oscillator system

The term ‘hidden attractor’ relates to a stable periodic, quasiperiodic or chaotic state whose basin of attraction does not overlap with the neighborhood of an unstable equilibrium point. Considering a three-dimensional oscillator system that does not allow for the existence of an equilibrium point, this paper describes the formation of several different coexisting sets of hidden attractors, including the simultaneous presence of a pair of coinciding quasiperiodic attractors and of two mutually symmetric chaotic attractors. We follow the dynamics of the system as a function of the basic oscillator frequency, describe the bifurcations through which hidden attractors of different type arise and disappear, and illustrate the form of the basins of attraction.

Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect

In this paper, the dynamical behavior of a three-dimensional (3D) autonomous oscillator (Pehlivan and Uyarogˇlu, 2012) is further investigated in some detail (including local stability and bifurcation structures analysis) in order to reveal bursting dynamics. This system has the unusual feature of involving three equilibria asymmetrically distributed. We find that for specific parameters, the system exhibits periodic and chaotic bursting oscillations. By rescaling its dynamical variables, we show that this system evolves on two time scales, corresponding respectively to fast and slow dynamics that lie at the basis of bursting dynamics. The mechanism underlying this phenomenon is drawn via the bifurcation analysis of the fast subsystem with respect to the slow subsystem variable. This reveals that the bursting oscillations found there result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. This class of bursting has been referred to as “source/sink bursting of node/focus type”. In addition, an analog circuit is designed and implemented to realize experimentally the periodic and chaotic bursting oscillations. Good agreement is shown between the numerical and the experimental results. Further, the commensurate fractional-order version of the chaotic slow–fast system is studied using stability theorem of fractional-order systems and numerical simulations. By tuning the fractional-order, the slow–fast system displays a wide variety of dynamical behaviors ranging from chaotic bursting to fixed point dynamics via continuous chaotic spiking. Because of the asymmetric distribution of the system equilibrium points, this transition includes peculiar phenomena such as transient chaos and the coexistence of point and chaotic attractors.

Singular center as a nongravitational black hole

We consider the interpretation of a singular center in quantum mechanics as a black hole in application to the exactly solvable problem of the three-dimensional oscillator with a complex-valued angular momentum, also known as the generalized Calogero problem.

The multivariate Hahn polynomials and the singular oscillator

Karlin and McGregorʼs d-variable Hahn polynomials are shown to arise in the -dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d = 2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operators, orthogonality relation, generating function, recurrence relations, bispectrality (difference equations) and explicit expression in terms of the univariate Hahn polynomials. The extension of these results to an arbitrary number of variables is presented at the end of the paper.

Design of time-delayed connection parameters for inducing amplitude death in high-dimensional oscillator networks

The present paper studies time-delayed-connection induced amplitude death in high-dimensional oscillator networks. We provide two procedures for design of a coupling strength and a transmission delay: these procedures do not depend on the topology of oscillator networks (i.e., network structure and number of oscillators). A graphical procedure based on the Nyquist criterion is proposed and then is numerically confirmed for the case of five-dimensional oscillators, called generalized Rössler oscillators, which have two pairs of complex conjugate unstable roots. In addition, for the case of high-dimensional oscillators having two unstable roots, the procedure can be systematically carried out using only a simple algebraic calculation. This systematic procedure is numerically confirmed for the case of three-dimensional oscillators, called Moore-Spiegel oscillators, which have two positive real unstable roots.

Three-dimensional Dirac oscillator in a thermal bath

The thermal properties of the three-dimensional Dirac oscillator are considered. The canonical partition function is determined, and the high-temperature limit is assessed. The degeneracy of energy levels and their physical implications on the main thermodynamic functions are analyzed, revealing that these functions assume values greater than the one-dimensional case. So that at high temperatures, the limit value of the specific heat is three times bigger.

An Ultra Compact Millimeter-Wave VCO in 3-D IC Technology

In this letter, an ultra compact three-dimensional (3-D) voltage-controlled oscillator (VCO) is presented. By utilizing the through silicon via (TSV) as inductors and variable bridge topology in the LC tank, the core area of VCO is reduced significantly with an improved tuning range. Implemented in a standard 90 nm CMOS process, the VCO consists of the top and bottom layers connected by TSVs using an in-house developed 3-D IC technology. The VCO is capable of operating at 43.3-43.9 GHz with a footprint of 0.0022 mm3, and a core area of only 0.007 mm2 which is about one to two orders of magnitude smaller than that in previous works. The measured phase noise is -90.8 dBc/Hz (1 MHz offset) at 43.3 GHz with an output power of -12.6 dBm. Under a supply voltage of 1.2 V, the VCO consumes a dc current of 15.4 mA.

An exactly solvable three-dimensional nonlinear quantum oscillator

Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the s-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.

Generators of quasiperiodic oscillations with three-dimensional phase space

Considering a family of three-dimensional oscillators originating in the field of radio-engineering, the paper describes three different mechanisms of torus formation. Particular emphasis is paid to a process in which a saddle-node bifurcation eliminates a stable cycle and leaves the system to find a stationary state between a saddle cycle and a pair of equilibrium points of unstable focus/stable node and unstable node/stable focus type.

GeSi Raman spectra vs. local clustering/anticlustering: Percolation scheme and ab initio calculations

We formalize within the percolation scheme that operates along the linear chain approximation, i.e., at one dimension (1D), an intrinsic ability behind Raman scattering to achieve a quantitative insight into local clustering/anticlustering in an alloy, using GeSi as a case study. For doing so, we derive general expressions of the individual fractions of the six GeSi percolation-type oscillators [1×(Ge-Ge), 3×(Ge-Si), 2×(Si-Si)], which monitor directly the Raman intensities, via a relevant order parameter κ. This is introduced by adapting to the 1D oscillators of the GeSi-diamond version of the 1D-percolation scheme, i.e., along a fully consistent 1D treatment, the approach originally used by Verleur and Barker for the three-dimensional (3D) oscillators of their 1D-cluster scheme applying to zincblende alloys [H. W. Verleur and A. S. Barker, Phys. Rev. 149, 715 (1966)], a somehow problematic one in fact, due to its 3D–1D ambivalence. Predictive κ-dependent intensity-interplays between the Ge0.5Si0.5 Raman lines are confronted with existing experimental data and with ab initio Raman spectra obtained by using (32-atom) disordered supercells matching the required κ values, with special attention to the Ge-Si triplet and to the Si-Si doublet, respectively.