Comments on “Dynamics of the general Lorenz family” by Y. Liu and W. Pang

A paper, “Dynamics of the general Lorenz family,” was published in the journal Nonlinear Dynamics. The authors investigate the general Lorenz family $\dot{x} = a(y-x)$ , $\dot{y} = dx + cy - xz$ , $\dot{z} = -bz + xy $ , considering that it contains four independent parameters. However, we show that, generically (for c≠0), this family is equivalent to the Lorenz system and, thus, the results they provide for the general Lorenz family are easily obtained from the corresponding results on the Lorenz system. Moreover, in the case c=0, it is sufficient to consider a two-parameter system.     

Comments on ＂Non-existence of Shilnikov chaos in continuous-time systems＂

A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.     

Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters \(b,\ c\) and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.     

Temporal and spatial elements of thermocapillary convection in floating zones

We investigated experimentally spatio-temporal convective flow phenomena in cylindrical liquid bridges [floating-(half-)zones] of liquids with different Prandtl-numbers (NaNO₃?Pr=7; C₂₄H₅₀?Pr=49; C₃₆H₇₄?Pr=65). The convective flow is driven by thermocapillary forces (TC-forces) and buoyancy forces. The zones were heated from above (ΔT, Ma>0) or from below (ΔT, Ma<0) to couple both effects in different ways. Optical evaluations (view from above and view from the front) in connection with thermocouple (tc) measurements (tc-tips distributed over one half of the free surface) made it possible to get very new ideas of spatio-temporal flow structures in the considered convective system. In this article we deal with some transitionary temporal phenomena accompanying the system’s way to chaotic behaviour. We present results supplementary to well-known transitions to chaos (i.e. quasi-periodic and period-doubled flow states) and introduce some very special events. Here all considerations are based on a primarily “temporal way of thinking”. We then try to illuminate several flow situations primarily from a more “spatial point of view”. Possible spatio-temporal convective flow structures are discussed by accompanying the system from a laminar flow state up to the onset of chaotic motion. Starting with former ideas of spatio-temporal flow situations we recognize 2D- and 3D-stationary flows, “pulsating” and “rotating” modes m=1 and 2, different spatial reasons for quasi-periodic and period-doubled temporal behaviour and different spatial mechanisms that cause spatio-temporal chaotic structures in the system considered. One should realize the ambiguity of a certain time-signal with respect to various spatial structures. Additionally we find out that a revision of the interpretation of very complicated Ma/Ma _c (A)-state maps now becomes necessary. These state maps show the present flow state (e.g. a time-dependent, quasi-periodic or chaotic one) depending on the geometrical parameter aspect ratio A (i.e. the zone length) and the TC-force (i.e. the Marangoni-number).     

The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches

More than a century ago, the Serbian engineer and astronomer Milutin Milankovitch presented a remarkable formulation for the thrust-line of arches that do not sustain tension, and by taking radial cuts and a polar coordinate system, he published for the first time the correct and complete solution for the theoretical minimum thickness, t, of a monolithic semicircular arch with radius R. This paper shows that Milankovitch’s solution, t/R = 0.1075, is not unique and that it depends on the stereotomy exercised. The adoption of vertical cuts which are associated with a cartesian coordinate system yields a neighboring thrust-line and a different, slightly higher value for the minimum thickness (t/R = 0.1095) than the value computed by Milankovitch. This result has been obtained in this paper with a geometric and a variational formulation. The Milankovitch minimum thrust-line derived with radial stereotomy and our minimum thrust-line derived with vertical stereotomy are two distinguishable, physically admissible thrust-lines which do not coincide with R. Hooke’s catenary that meets the extrados of the arch at the three extreme points. Furthermore, the paper shows that the catenary (the “hanging chain”) is not a physically admissible minimum thrust-line of the semicircular arch, although it is a neighboring line to the aforementioned physically admissible thrust-lines. The minimum thickness of a semicircular arch that is needed to accommodate the catenary curve is t/R = 0.1117—a value that is even higher than the enhanced minimum thickness t/R = 0.1095 computed in this paper after adopting a cartesian coordinate system; therefore, it works toward the safety of the arch.     

Helical Structures in the Near Field of a Turbulent Pipe Jet

We perform a finely resolved Large-eddy simulation to study coherent vortical structures populating the initial (near-nozzle) zone of a pipe jet at the Reynolds number of 5300. In contrast to ‘top-hat’ jets featured by Kelvin-Helmholtz rings with the non-dimensional frequency S t≈0.3?0.6, no high-frequency dominant mode is observed in the near field of a jet issuing from a fully-developed pipe flow. Instead, in shear layers we observe a relatively wide peak in the power spectrum within the low-frequency range (S t≈0.14) corresponding to the propagating helical waves entering with the pipe flow. This is confirmed by the Fourier transform with respect to the azimuthal angle and the Proper Orthogonal Decomposition complemented with the linear stability analysis revealing that this low-frequency motion is not connected to the Kelvin-Helmholtz instability. We demonstrate that the azimuthal wavenumbers m=1?5 contain the most of the turbulent kinetic energy and that a common form of an eigenmode is a helical vortex rotating around the axis of symmetry. Small and large timescales are identified corresponding to “fast” and “slow” rotating modes. While the “fast” modes correspond to background turbulence and stochastically switch from co- to counter-rotation, the “slow” modes are due to coherent helical structures which are long-lived and have low angular velocities, in agreement with the previously described spectral peak at low S t.     

Periodic solution and chaotic behavior of a class of nonautonomic pendulum systems with large damping

In this paper the existence and uniqueness of the periodic solution is studied for a class of second order nonautonomic pendulum aystems x + ax + ψ(t)sinx=F(t) anil the parameter regions tor which the system in chaos is myestigated when ψ(t)=1-ελcosωt, F(t)=β +εμ(cosωt-ωsinωt) and the tamping coefficient a>0 is large. The result obtained generalize the corresponding conclusions of papers [1-8].     

FLOW-INDUCED VIBRATION OF A CIRCULAR CYLINDER AT LIMITING STRUCTURAL PARAMETERS

Transverse oscillation of a dynamically supported circular cylinder in a flow at Re=100 has been numerically simulated using a high-resolution viscous-vortex method, for a range of dynamical parameters. At the limiting case with zero values of mass, damping and elastic force, the cylinder oscillates sinusoidally at amplitudeA /D=0·47 and frequency fD/U_∞=0·156. For zero damping, the effects of mass and elasticity are combined into a new, “effective” dynamic parameter, which is different from the classic “reduced velocity”. Over a range of this parameter, the response exhibits oscillations at amplitudes up to 0·6 and frequencies between 0·15 and 0·2. From this response function, the classic response in terms of reduced velocity can be obtained for fixed values of the cylinder/fluid ratio m^*. It displays “lock-in” at very high values of m^*.     

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh(β)/π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.     

‘Generalized Burgers' equation’ for nonlinear viscoelastic waves

This paper deals with nonlinear longitudinal waves in a viscoelastic medium in which the viscoelastic relaxation function has the form K(t) = const. t^-v (0<v<1). This sort of slow relaxation may be more appropriate for polymers than the often used exponential relaxation. For a far field evolution of unindirectional waves, a “generalized Burgers' equation” is obtained, which is of a form with the second derivative in the usual Burgers' equation replaced by the derivative of real order 1 + v. The steady shock solution and self-similar pulse solution to this equation are discussed. In both cases numerical solutions are presented and analytic results are obtained for the asymptotic behaviors of the solutions. It is found that both shock and pulse solutions rise exponentially, but in their tails they have slow, algebraic decay.     

Chaos synchronization of two different chaotic complex Chen and Lü systems

This paper investigates the phenomenon of chaos synchronization of two different chaotic complex systems of the Chen and Lü type via the methods of active control and global synchronization. In this regard, it generalizes earlier work on the synchronization of two identical oscillators in cases where the drive and response systems are different, the parameter space is larger, and the dimensionality increases due to the complexification of the dependent variables. The idea of chaos synchronization is to use the output of the drive system to control the response system so that the output of the response system converges to the output of the drive system as time increases. Lyapunov functions are derived to prove that the differences in the dynamics of the two systems converge to zero exponentially fast, explicit expressions are given for the control functions and numerical simulations are presented to illustrate the success of our chaos synchronization techniques. We also point out that the global synchronization method is better suited for synchronizing identical chaotic oscillators, as it has serious limitations when applied to the case where the drive and response systems are different.     

Research on the price game and the application of delayed decision in oligopoly insurance market

In the oligopoly insurance market, we assumed that some oligarchs make two-period delay decisions in bounded rationality and expectation, respectively, and others make decisions with bounded rationality without the condition of delay. There also exist two cases in which only one oligarch makes a delayed decision and two oligarchs make delayed decisions at the same time. Based on the analysis of these situations, we established the corresponding dynamic price game models. We then performed a numerical simulation to the complexity state of the system with different conditions such as stability, bifurcation, and chaos, and analyzed the profits of different oligarchs when the system is in different states. The results showed that when only one oligarch makes a delayed decision, with the decrease in the price weight of period t and increase in that of periods t?1 and t?2, the system??s stable region in the direction of the price adjustment of the oligarch with a delayed decision gets smaller. However, when there are two oligarchs with a delayed decision in the system, in the case where the delay parameters of oligarch 1 remain unchanged and the price parameters of different periods of oligarch 2 change, the system??s stable region in the direction of the price adjustment of oligarch 1 does not have the obvious change as that when only one oligarch makes a delayed decision. This showed that the sensibility of one oligarch in the direction of its own price adjustment is lower than other oligarchs. In addition, in the same system with delay and when the system is in chaos, the total profit of the oligarchs is obviously less than that when the system is in a stable state. However, the use of a delayed decision may not enhance the oligarch??s competitive advantages. Finally, the variable feedback control method is used to effectively control the chaos in the system.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

Dynamical behaviors of the periodic parameter-switching system

In this paper, a periodic parameter-switching system about Lorenz oscillators is established. To investigate the bifurcation behavior of this system, Poincaré mapping of the whole system is defined by suitable local sections and local mappings. The location of the fixed point and the parameter values of local bifurcations are calculated by the shooting method and Runge–Kutta method. Then based on the Floquent theory, we conclude that the period-doubling and saddle-node bifurcations play an important role in the generation of various periodic solutions and chaos. Meanwhile, upon the analysis of the equilibrium points of the subsystems, we explore the mechanisms of different periodic switching oscillations.     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

Shil’nikov chaos in the 4D Lorenz–Stenflo system modeling the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere

The Lorenz–Stenflo system serves as a model of the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere. In the present paper, we study the Shil’nikov chaos which arises in the 4D Lorenz–Stenflo system. The analytical and numerical results constitute an application of the Shil’nikov theorems to a 4D system (whereas most results present in the literature deal with applying the Shil’nikov theorems to 3D systems), which allows for the study of chaos along homoclinic and heteroclinic orbits arising as solutions to the Lorenz–Stenflo system. We verify the observed chaos via competitive modes analysis—a diagnostic for chaotic systems. We give an analytical test, completely in terms of the model parameters, for the Smale horseshoe chaos near homoclinic orbits of the origin, as well as for the case of specific heteroclinic orbits. Numerical results are shown for other cases in which the general analytical method becomes too complicated to apply. These results can be extended to more complicated higher-dimensional systems governing plasmas, and, in particular, may be used to shed light on period-doubling and Smale horseshoe chaos that arises in such models.     

Life Expectancy Calculations of Transient Chaotic Behaviour Using Approximate 1D Maps

In engineering practice, chaotic oscillations are often observed which disappear suddenly. This phenomenon is often referred to as transient chaos. The duration of these oscillations varies stochastically. In this work two methods are presented for the simple estimation of the expected length of the chaotic behaviour. As an example, the Lorenz system is considered at some specific parameter values.     

On the boundedness of solutions to the Lorenz-like family of chaotic systems

This paper deals with a class of three-dimensional autonomous nonlinear systems which have potential applications in secure communications, and investigates the localization problem of compact invariant sets of a class of Lorenz-like chaotic systems which contain T system with the help of iterative theorem and Lyapunov function theorem. Since the Lorenz-like chaotic system does not have y in the second equation, the approach used to the Lorenz system cannot be applied to the Lorenz-like chaotic system. We overcome this difficulty by introducing a cross term and get an interesting result, which includes the most interesting case of the chaotic attractor of the Lorenz-like systems. Furthermore, the results obtained in this paper are applied to study complete chaos synchronization. Finally, numerical simulations show the effectiveness of the proposed scheme.