Imprint of non-standard interactions on the CP violation measurements at long baseline experiments

By introducing an additional state feedback into classic Rikitake system, a new hyperchaotic system without equilibrium is derived. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincaré maps. Based on adaptive control and Lyapunov stability theory, we design a reduced-order projective synchronization scheme for synchronizing the hyperchaotic Rikitake system coexisting without equilibria and the original classic Rikitake system coexisting with two non-hyperbolic equilibria. Finally, numerical simulations are given to illustrate the effectiveness of the proposed synchronization scheme.     

Synchronization and parameter estimations of an uncertain Rikitake system

In this Letter we address the synchronization and parameter estimation of the uncertain Rikitake system, under the assumption the state is partially known. To this end we use the master/slave scheme in conjunction with the adaptive control technique. Our control approach consists of proposing a slave system which has to follow asymptotically the uncertain Rikitake system, refereed as the master system. The gains of the slave system are adjusted continually according to a convenient adaptation control law, until the measurable output errors converge to zero. The convergence analysis is carried out by using the Barbalat's Lemma. Under this context, uncertainty means that although the system structure is known, only a partial knowledge of the corresponding parameter values is available.     

The Rikitake Two-Disk Dynamo System and Domains with Periodic Orbits

The Rikitake two-disk dynamo system is a simplemodel to describe the earth's magnetic field. We derivethe conditions to find periodic orbits of this systemusing an ellipsoid bounding condition. We prove that the conditions cannot besatisfied.     

Adaptive synchronization of chaotic systems with less measurement and actuation

We investigate the synchronization problem between identical chaotic systems only when necessary measurement(output) and actuation(input) are needed to be implemented by the adaptive controllers. A sufficient condition is derived based on the Lyapunov stability theory and Schur complementary lemma. Moreover, the theoretic result is applied to the Rikitake system and the hyperchaotic Liu system to show its effectiveness and correctness. Numerical simulations are presented to verify the results.     

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

Based on a non‐Riemannian treatment of geometric objects, the geometric structures of fractional‐order dynamical systems are investigated. A fractional derivative describes non‐local effects across a space or a history encoded in memory features of the system. A system of fractional‐order differential equations is formulated in film space that includes fictitious forces. Film space is a geometric space whose coordinates comprise time, and the geometric quantities vary in time. Fractional‐order torsion tensors that appear are related to the dissipated energy and the energy conversions between subsystems and power of the system. The geometric treatment is then applied to damped‐harmonic and fractional oscillators and the hybrid electromechanical Rikitake system. The damped‐harmonic oscillator is characterized by two torsion tensors, whereas the fractional oscillator is characterized by one torsion tensor. Herein, the fractional order of the derivative of the metric tensor is used to characterize the damping of the fractional oscillator. The energy conversions between electromechanical subsystems in the Rikitake system are characterized by the torsion tensor. These results suggest that the non‐Riemannian geometric objects can represent the non‐local properties of fractional‐order dynamical systems.     

Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system

It is well known that equilibrium in a cosymmetric system in the general position is a member of a one-parameter family. In the present paper the Lyapunov-Schmidt method and the method of the central manifold are used to analyze bifurcations of such a family of equilibria as well as internal bifurcations: transitions of the type focus-node, node-saddle, and so on during motion along the family. A series of scenarios of branching of families of equilibria and the change in the structure of their arcs, consisting of equilibria of the same type, is described. Bifurcations of stable and unstable arcs, coalescence and decomposition of families of equilibria, bifurcation of the loss of smoothness by the family of equilibria, and branching of a small equilibrium cycle from a corner point of the family of equilibria are investigated in detail. The variability of the spectrum along a family gives rise to a variety of new phenomena that are not encountered in the classical case of an isolated equilibrium or in bifurcations of families of equilibria of a system with symmetry. They include protraction with respect to the branching parameter of the family of equilibria, Lyapunov instability of a family of equilibria with the attraction properties being preserved, and the appearance and disappearance of new stable and unstable arcs on the family of equilibria. (c) 2000 American Institute of Physics.     

快慢Lorenz-Stenflo系统分析

通过对系统的重新标度,得到了气流旋转缓慢变化时的Lorenz-Stenflo系统;基于Routh-Hurwitz准则,分析了平衡点的稳定性问题,得到了参数平面上的分岔集,这些分岔集将参数平面划分为不同的区域,在各个不同的区域对应于系统不同的解.随着参数的变化,从平衡点分岔出不同的解.此外,展示了系统的对称簇发解和对称混沌吸引子,并用快慢分析法给出了对称簇发解的产生机理. 关键词： Lorenz-Stenflo系统 快慢分析法 分岔 对称簇发     

Variability in the noise-induced modes of climate dynamics

New features of noise-induced climate variability are revealed on the basis of the three-dimensional model derived by Saltzman and Maasch. It is shown that the climate system can be highly noise excitable and it possesses the large-amplitude fluctuations even in those regions where its akin deterministic model does not contain any self-sustained oscillations. Intermittency in small- and large amplitude climate fluctuations between different basins of attraction of a limit cycle and stable equilibria substantially influencing the climate state (from warm to cold and vice versa) are found at various noise intensities. Suddenly occurring jumps between the basins of attraction of two stable equilibria corresponding to the warm and cold climate states are statistically confirmed under a certain diapason of noise intensities. The climate system undergoes transitions between its equilibria in the presence of noise in its prognostic variables. In addition, such transitions become more likely with increasing the noise intensity.     

A novel four-wing non-equilibrium chaotic system and its circuit implementation

In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps. There is little difference between this chaotic system without equilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincaré maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.     

Lagrangian relative equilibria for a gyrostat in the three-body problem

In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.      

Sequence of phase transformations in the Gd-Mn-O system

Heterogeneous equilibria in the Gd-Mn-O system at low oxygen pressures are studied by the static method in combination with X-ray diffraction analysis of quenched solid phases. The sequence of phase equilibria upon the dissociation of GdMn₂O₅ and GdMnO₃ is determined. A fragment of an isothermal section of the phase equilibrium diagram of Gd-Mn-O in composition-oxygen pressure coordinates at 1173 K is constructed.     

Bifurcation structures and dominant modes near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose-Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.     

Bifurcation of the branching of a cycle in n-parameter family of dynamic systems with cosymmetry

A study is reported of the bifurcation of the branching of a cycle (Poincare-Andronov-Hopf bifurcation) from a smooth one-dimensional submanifold of equilibria of a dynamical system that depends on a vector parameter and admits cosymmetry. The paper reports a topological classification of local phase portraits near a known equilibrium, when the system parameter is close to its critical value that corresponds to an oscillatory instability. New phenomena that are not observed in the classical case of an isolated equilibrium include a delay of cycle creation with respect to the system parameter, loss of stability by the family of equilibria without loss of attraction, and the possibility of unstable supercritical self-oscillations. (c) 1997 American Institute of Physics.     

Heterogeneous equilibria in the Tm-Mn-O system

Heterogeneous equilibria in the Tm-Mn-O system have been studied at low temperatures by the static method in combination with X-ray diffraction analysis of quenched solid phases. The sequence of phase equilibria upon the dissociation of TmMn₂O₅ and TmMnO₃ has been determined. A portion of the isothermal section of the Tm-Mn-O phase diagram has been constructed in composition-oxygen-pressure coordinates.     

Experimental investigation and thermodynamic calculation of the Bi-Ga-Sn phase equilibria

Binary thermodynamic data, successfully used for phase diagram calculations of binary systems Bi-Ga, Bi-Sn, and Ga-Sn, were used for prediction of phase equilibria in ternary Bi-Ga-Sn system. The thermodynamic functions, such as enthalpy of formation and activity, were calculated using the Redlich-Kister-Muggianu model and compared with experimental data reported in the literature. The liquidus surface, invariant equilibria and three vertical sections with molar ratio Ga:Sn=1, Bi:Sn=1 and Bi:Ga=1 of the Bi-Ga-Sn ternary system were calculated by the CALPHAD method. Alloys, situated along three calculated vertical sections, were investigated by Differential Scanning Calorimetry (DSC). The experimentally determined phase transition temperatures were compared with calculation results and good mutual agreement was noticed.     

Simple proof of electron beam stability in the deep-potential-wellequilibrium

It is well known that a strongly magnetized electron beam propagating in an evacuated conducting pipe can exist in either of two equilibria, one with a shallow potential well and the other with a deep potential well. A Lorentz transformation connecting the two classes of equilibria is derived. Thus, since shallow-potential-well equilibria are known to be stable, the deep-potential-well equilibria must also be. The stability of space charge waves in the two equilibria should follow the same pattern     

Tokamak equilibria with toroidal-current reversal in the plasma core consistent with experimental data

For the first time, tokamak equilibria with negative toroidal current flowing in the plasma core are computed consistently with available measurements from typical current-hole discharges. The equilibrium reconstruction, which leads to non-nested configurations where a system of axisymmetric magnetic islands unfolds, yields an overall good agreement between the computed and experimental plasma-pressure profiles, together with an excellent fit to motional-Stark-effect data. Therefore, considering the accuracy limits of present-day experimental results, care must be exercised when ruling out the existence of tokamak equilibria with central toroidal-current reversal, particularly if relying on reconstruction tools that cannot cope with non-nested configurations.     

Analysis of the T-point-Hopf bifurcation

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol-Duffing electronic oscillator.     

Mathematical Modelling of Malaria with Treatment

This paper proposes a Susceptible-Infective-Susceptible (SIS) model to study the malaria transmission with treatment by considering logistic growth of mosquito population. In this work, it is assumed that the treatment rate is proportional to the number of infectives below the capacity and is constant when the number of infectives is greater than the capacity. We find that the system exhibits backward bifurcation if the capacity is small and it gives bi-stable equilibria which makes the system more sensitive to the initial conditions. The existence and stability of the equilibria of the model are discussed in detail and numerical simulations are presented to illustrate the numerical results.