Irrational Frequency Research Articles

On the excess loss in FeSiAl soft magnetic composites

Aiming at the issue of inconsistent theoretical expression of excess loss in soft magnetic composites, the irrational linear frequency dependence of excess loss W ex is analyzed. By comparing and analyzing the experimental data of magnetic loss of FeSiAl core under different transverse magnetic fields with the theoretical results, it is determined that the excess loss of FeSiAl composite is proportional to the 1/2th power of the frequency. Meanwhile, it is experimentally discovered that there is a positive correlation between the hysteresis loss and the excess loss of FeSiAl magnetic core, which might provide an effective approach to reduce the excess loss of FeSiAl magnetic core.

The asymptotic solutions for the motion of a charged symmetric gyrostat in the irrational frequency case

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange’s scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body’s movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler–Poisson are solved through the utilization of Poincaré’s small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body’s position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems.

Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map

We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text].

Lissajous curves as aerial search patterns

Manned and unmanned systems are prevalent in a wide range of aerial searching applications. For aircraft whose trajectory is not or cannot be planned on-the-fly, optimal deterministic search pattern generation is a critical area of research. Lissajous curves have recently caught attention as excellent candidates for all kinds of aerial search applications, but little fundamental research has been done to understand how best to design Lissajous pattern (LP)s for this use. This paper examines the optimization of these search patterns from analytical, numerical, and data-driven perspectives to establish the state of the field in Lissajous curves for aerial search. From an analytical perspective, it was found that the average expected distance between a Lissajous searcher and a random target on a unit square approaches 0.586 as search time increases. Furthermore, an analytical approximation for the average searcher speed was found to guarantee error of no more than 22.1%. Important outcomes from the numerical optimization of Lissajous search patterns include the development of an intuitive evaluation criterion and the conclusion that irrational frequency ratios near 0.8 typically yield highest performance. Finally, while a robust predictive model for fast pattern optimization is yet out of reach, initial results indicate that such an approach shows promise.

Thermo-magnetic convection and convective magneto-gravitational parametric resonance in a horizontal ferrofluid layer under time quasi-periodic vibrations: A balance harmonic analysis

Convective instability in a horizontal ferromagnetic liquid layer with rigid boundaries is investigated in the presence of both an uniform magnetic field and a vertical quasi-periodic forcing having two incommensurate frequencies ω1 and ω2. A linearized convective instability analysis predicting the onset of convection is performed and by means of a Galerkin projection truncated to the first order, the linear dynamic of the system is reduced to a damped quasi-periodic Mathieu oscillator. Using a balance harmonic analysis, it turns out that at sufficiently low vibration frequencies, we recover the critical instability parameters related to the unmodulated case where the convection is mainly due to the magnetic mechanism. This feature means that the buoyancy mechanism does not contribute into the convective cells formation and consequently heating from below or from above does not alter the onset of convection. However, at sufficiently high vibration frequencies, both the magnetic and buoyancy mechanisms are operative and here we are dealing with a magneto-gravitational parametric resonance convection. The transition between these convective modes is occurred via bicritical states giving rise to cusp points in the magnetic Rayleigh number besides discontinuities in its corresponding critical wave number. In addition, it is shown that a proper tuning of the irrational frequency ratio ω=ω2/ω1 provides a control of the convection onset as well as the nature of its mechanism. Moreover, special attention is devoted to highlight the effect of the Prandtl number on the dynamic of the system at different values of the frequency ratio ω.

Periodic solutions with irrational frequency for a class of semilinear wave equations with variable coefficients

This paper is devoted to the study of the existence of periodic solutions for a class of semilinear wave equations with variable coefficients. The forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media is governed by this mathematical model. When the frequency is a sufficiently large irrational number with bounded partial quotients, the existence of weak solutions is established. Then, under some suitable conditions, we improve the regularity of weak solutions. Our results can also be applied to the corresponding constant coefficients wave equation.

Dynamics of a novel chaotic map

In this study, another one-dimensional sinusoidal chaotic map around the bisector is constructed, which consists of two sine functions with irrational frequency ratios but comparable amplitude and phase. This design is motivated by the process equation. Investigating the unusual dynamics of this map and contrasting them with the process equation are the goals of this work. When two sine functions are included in a map, their interaction results in complex behaviors that have not been seen for the process equation. Like the process equation, this newly created map has an endless number of fixed points, but unlike the process equation, their stability cannot be fully defined. Furthermore, the suggested chaotic map can display a biotic-like time series when escaping regions are produced; however, unlike the process equation, these biotic-like time series only consist of transient components. In other words, the introduced map converges to a fixed point or periodic solution and never becomes unbounded because of the unlimited number of stable and unstable fixed points. The parameters of the map determine how long these biotic-like transient sections last. Additionally, coexisting attractors and multi-stability are seen in this map, much like in the process equation. The significant dynamics of this map are examined with time series, cobweb, bifurcation, and Lyapunov exponent diagrams. Tests are conducted before and after the creation of the escaping regions. Additionally, two-dimensional bifurcation diagrams are used to study the simultaneous impact of many pairs of factors on the map’s dynamics.

Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Padé and log-Padé approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the Padé approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus.We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative (more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The conformally symplectic maps depend on a dissipative parameter and a drift term, which is needed to get the existence of invariant attractors. The 4D maps are obtained coupling (i) two symplectic standard maps, or (ii) two conformally symplectic standard maps, or (iii) a symplectic and a conformally symplectic standard map.Concerning the results, Padé and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.

Effect of Horizontal Quasi-Periodic Oscillation on the Interfacial Instability of Two Superimposed Viscous Fluid Layers in a Vertical Hele-Shaw Cell

We investigate the effect of horizontal quasi-periodic oscillation on the stability of two superimposed immiscible fluid layers confined in a horizontal Hele-Shaw cell. To approximate real oscillations, a quasi-periodic oscillation with two incommensurate frequencies is considered. Thus, the linear stability analysis leads to a quasi-periodic oscillator, with damping, which describes the evolution of the amplitude of the interface. Two types of quasi-periodic instabilities occur: the low-wavenumber Kelvin-Helmholtz instability and the large-wavenumber resonances. We mainly show that, for equal amplitudes of the superimposed accelerations, and for a low irrational frequency ratio, there is competition between several resonance modes allowing a very large selection of the wavenumber from lower to higher values. This is a way to control the sizes of the waves. Furthermore, increasing the frequency ratio has a stabilizing effect for both types of instability whose thresholds are found to correspond to quasi-periodic solutions using the frequency spectrum. For a ratio of the two superimposed displacement amplitudes equal to unity and less than unity, the number of resonances and competition between their modes also become significant for the intermediate values of the ratio of frequencies. The effects of other physical and geometrical parameters, such as the damping coefficient, density ratio, and heights of the two fluid layers, are also examined.

Arithmetic phase transitions for mosaic Maryland model

We give a precise description of spectral types of the mosaic Maryland model with any irrational frequency, which provides a quasi-periodic unbounded model with non-monotone potential having arithmetic phase transition.

A novel chaotic map with a shifting parameter and stair-like bifurcation diagram: dynamical analysis and multistability

In this paper, the behavior of a 1D chaotic map is proposed which includes two sine terms and shows unique dynamics. By varying the bifurcation parameter, the map has a shift, and the system’s dynamics are generated around the cross points of the map and the identity line. The irrational frequency of the sine term makes the system have stable fixed points in some parameter intervals by increasing the bifurcation parameter. So, the bifurcation diagram of the system shows that the trend of the system’s dynamics changes in a stair shape with slope one by changing the bifurcation parameter. Due to the achieving multiple steady states in some intervals of the parameter, the proposed system is known as multistable. The multistability dynamics of the map are investigated with the help of cobweb diagrams which reveal an interesting asymmetry in repeating parts of the bifurcation diagram.

The periodic solutions of a symmetric charged gyrostat for a slightly relocated center of mass

This paper aims to investigate the rotatory dynamical motion of a charged symmetric gyrostat when its center of mass is slightly displaced from the axis of dynamic symmetry as a novel case. A uniform electromagnetic field, due to the presence of point charge located on the axis of symmetry, and a gyrostatic moment are acted on the gyrostatic motion. Equations of motion are derived and solved analytically using the small parameter method of Poincaré for irrational frequencies’ case of the examined motion of the gyrostat. The approximate formulas of the rotating Euler’s angles of the gyrostat are obtained. The time histories of the achieved results and the Euler’s ones are graphed to display the excellent influence of the chosen parameters of the gyrostat on the motion. Moreover, the phase plane diagrams of these results are graphed also to reveal the stability of the gyrostatic motion at any time. The attained results can be used to improve the dynamical behavior of several engineering applications, particularly those that employ the gyroscopic theory such as satellites, submarines, compasses, and automatic pilots on aircraft and ships.

Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

<p style='text-indent:20px;'>In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in <inline-formula><tex-math id="M1">\begin{document}$ E $\end{document}</tex-math></inline-formula> on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.</p>

Analytical Solution and Quasi-Periodic Behavior of a Charged Dilaton Black Hole

With the vast breakthrough brought by the Event Horizon Telescope, the theoretical analysis of various black holes has become more critical than ever. In this paper, the second-order asymptotic analytical solution of the charged dilaton black hole flow in the spinodal region is constructed from the perspective of dynamics by using the two-timing scale method. Through a numerical comparison with the original charged dilaton black hole system, it is found that the constructed analytical solution is highly consistent with the numerical solution. In addition, several quasi-periodic motions of the charged dilaton black hole flow are numerically obtained under different groups of irrational frequency ratios, and the phase portraits of the black hole flow with sufficiently small thermal parameter perturbation display good stability. Finally, the final evolution state of black hole flow over time is studied according to the obtained analytical solution. The results show that the smaller the integral constant of the system, the greater the periodicity of the black hole flow.

The Motion of a Rigid Body with Irrational Natural Frequency

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

Low Code Coding Approach for ofdm Systems Coded with the Progressive Transmission

Orthogonal Frequency Division Multiplexing (OFDM) offers several benefits such as flexible wrapping, mail sign interference, reduced complexity and more. The only way to match parallel channels is to have genetic diversity in multipath channels. Various coded of OFDM systems reported using some form of channel coding or pre-coding to improve system performance. The execution of irrational orthogonal frequency division multiplexing (OFDM) on blurred channels is usually better than introducing some coding. Various coding schemes for off-OFDM have been reported in the literature.

Stability of the 3-torus solution in a ring of coupled Duffing oscillators

The dynamics of the ring of unidirectionally coupled single-well Duffing oscillators is analyzed in numerical simulation for identical nodal oscillators. The research is concentrated on the existence of the stable 3D torus attractor in this system. It is shown that 3-frequency quasi-periodicity can be robustly stable in wide range of parameters of the system under consideration. As an explanation of this stability, the conjecture on the coexistence and superposition of two independent effects characterized with irrational frequencies, i.e., the classical Newhouse, Ruelle and Takens scenario and rotating wave flow, is formulated.

Response of linear parametric amplifiers with arbitrary direct and parametric excitations

The effect of parametric amplification, where parametric excitation is added to direct excitation, on the performance of a dynamical system is investigated. To model the system, the linear forced Mathieu equation is considered. In order to obtain the response of the system without restrictions such as requiring the system parameters to be small or the parametric excitation frequency to be exactly twice the direct excitation frequency, the method of varying amplitudes (MVA) is used. Employing the MVA, the response of the system and its maximum value for arbitrary values of the system parameters including the excitation frequencies are derived. It is shown that when the ratio between the excitation frequencies is rational, the maximum response of the system is highly dependent on the phase angle between the excitations. However, when the frequency ratio is irrational, the maximum response of the system is not dependent on the phase angle. Consequently, large values of response can be attained by using an irrational frequency ratio regardless of the phase angle. MVA results for peak responses show good agreement with numerical results obtained by direct integration of the equation of motion.

The dynamical motion of a gyrostat for the irrational frequency case

This work outlines on the three dimensional motion of a rigid body about a fixed point according to Lagrange’s case under the action of a gyrostatic moment and a Newtonian force field. It is considered that the center of mass of the body is shifted slightly with respect to the principal axis of dynamic symmetry. Equations of motion are derived using the principal equation of the angular momentum and are solved using the Poincaré method of small parameter to achieve the asymptotic solutions for the case of irrational frequencies. Euler’s angles characterizing the position of the body at any instant are obtained. The diagrammatic representations of the obtained solutions and Euler’s angles are represents through some plots which reflect the good effect of the applied moments on the motion and its impact on the stability of the body. The numerical solutions are obtained using Runge-Kutta algorithms from fourth order. The comparison between the asymptotic solutions and the numerical ones reveal high consistency between them which reveal the good accuracy of the used perturbation method.

Chaos and ergodicity in an entangled two-qubit Bohmian system

We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the ‘nodal point-X-point complex’ mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is manifested after a short time. We then study the mixing properties of ensembles of Bohmian trajectories with initial conditions satisfying Born’s rule. The trajectory points are initially distributed in two sets S1 and S2 with disjoint supports but they exhibit, over the course of time, abrupt mixing whenever they encounter the nodal points of the wavefunction. Then a substantial fraction of trajectory points is exchanged between S1 and S2, without violating Born’s rule. Finally, we provide strong numerical indications that, in this system, the main effect of the entanglement is the establishment of ergodicity in the individual Bohmian trajectories as : different initial conditions result to the same limiting distribution of trajectory points.