Bifurcations on a Five-Parameter Family of Planar Vector Field

In this paper we consider a five-parameter family of planar vector fields where μ = (μ ₁, μ ₂, μ ₃, μ ₄, μ ₅), which is a small parameter vector, and c(0) ≠ 0. The family X _μ represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X _μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, 1991). After making some rescalings, we obtain the truncated systems of X _μ . For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at most three limit cycles occur in the small neighborhood of the Hopf singularity. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

A numerical bifurcation study of friction effects in a slip-controlled torque converter clutch

Friction plays a key role in the efficiency and stability of the slip-controlled torque converter clutches. The effects of friction on the dynamics and stability of a slip-controlled torque converter clutch system using a bifurcation-analysis-based approach is presented in this paper. A three degree-of-freedom nonlinear driveline model with integral feedback action to control the clutch slip speed has been utilized for this study. The clutch interface friction is dependent on the slip speed and is a function of the static friction constant, μ ₀, the low velocity friction constant μ ₁, and the low velocity exponential rate, γ. Using one-parameter numerical continuation, local Hopf bifurcations of the subcritical type are observed as the friction parameters μ ₁ and γ were varied at low slip speeds. The continuation results are verified using simulations of the full nonlinear model. Stick-slip and undesirable oscillations of the model inertia elements are observed for certain parameter values. As the slip speed is increased, the bifurcation instability occurs at an increasingly higher value of μ ₁ signifying an improved tolerance of negative friction gradient at higher slip speeds. Smaller exponential rates γ are tolerated at higher slip speeds before the bifurcation instability occurs. For the range of parameter values considered, no bifurcations occur for a slip speeds higher than 3.4 and 4.5 rad/s with μ ₁ and γ as the continuation parameters, respectively. These values of slip speeds are much lower than the system’s first mode of torsional vibration of 16 Hz (≈100 rad/s).     

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T _in, T _on and T _out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions. The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.     

Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems

Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many systems exhibit non-smooth dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

Stability of an Initially, Stably Stratified Fluid Subjected to a Step Change in Temperature

The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature. The critical time t _c to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by , where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained as a function of τ(=αt/d ²) and ζ(=Z/), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing τ as τ_c under the frame of coordinates τ and ζ. For a given γ, Pr and τ_c, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τ_c increases. However, it is inversely proportional to τ_c ^3/2 as τ_c decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory produces the stability criteria to bound the available experimental data over the whole domain of time. Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work has been supported by both SK Chemicals Co. Ltd. and LG Chemical Ltd., Seoul under the Brain Korea 21 Project of the Ministry of Education. Communicated by H.J.S. Fernando     

Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source

 The Cattaneo hyperbolic and classical parabolic models of heat conduction in the laser irradiated materials are compared. Laser heating is modelled as an internal heat source, whose capacity is given by g(x,t)= I(t)(1−R)μexp(−μx). Analytical solution for the one-dimensional, semi-infinite body with the insulated boundary is obtained using Laplace transforms and the discussion of solutions for different time characteristics of the heat source capacity (constant, instantaneous, exponential, pulsed and periodic) is presented. Received on 18 May 1999     

The laminar hole pressure for Newtonian fluids

For flows with wall turbulence the hole pressure, P _H , was shown empirically by Franklin and Wallace (J Fluid Mech, 42, 33–48, 1970) to depend solely on R ₊, the Reynolds number constructed from the friction velocity and the hole diameter b. Here this dependence is extended to the laminar regime by numerical simulation of a Newtonian fluid flowing in a plane channel (gap H) with a deep tap hole on one wall. Calculated hole pressures are in good agreement with experimental values, and for two hole sizes are well represented by: (P _H −P _HS )/τ _w = √(k ² + c ² R ₊²)−k, where the Stokes hole pressure P _HS /τ _w = s (b/H)³, k, c, s are fitted constants, and τ _w is the wall shear stress.     

Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

A Near-Asymptotics analysis of the turbulence energy spectrum is presented that accounts for the effects of finite Reynolds number recently reported by Mydlarski and Warhaft [21]. From dimensional and physical considerations (following Kolmogorov and von Karman), proper scalings are defined for both low and high wavenumbers, but with functions describing the entire range of the spectrum. The scaling for low wavenumbers uses the kinetic energy and the integral scale, L, based on the integral of the correlation function. The fact that the two scaled profiles describe the entire spectrum for finite values of Reynolds number, but reduce to different profiles in the limit, is used to determine their functional forms in the “overlap” region that both retain in the limit. The spectra in the overlap follow a power law, E(k) =Ck _{−5/3 + μ}, where μ and C are Reynolds number dependent. In the limit of infinite Reynolds number, μ → 0 and C → constant, so the Kolmogorov/Obukhov theory is recovered in the limit. Explicit expressions for μ and the other parameters are obtained, and these are compared to the Mydlarski/Warhaft data. To get a better estimate of the exponent from the experimental data, existing models for low and high wavenumbers are modified to account for the Reynolds number dependence. They are then used to build a spectral model covering all the range of wavenumbers at every Reynolds number. Experimental data from grid-generated turbulence are examined and found to be in good agreement with the theory and the model. Finally, from the theory and data, an explicit form for the Reynolds number dependence of φ = ɛL/u ₃ is obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation

It is well known that saddle-node bifurcations can occur in the steady-state response of a forced single-degree-of-freedom (SDOF) nonlinear system in the cases of primary and superharmonic resonances. This discontinuous or catastrophic bifurcation can lead to jump and hysteresis phenomena, where at a certain interval of the control parameter, two stable attractors exist with an unstable one in between. A feedback control law is designed to control the saddle-node bifurcations taking place in the resonance response, thus removing or delaying the occurrence of jump and hysteresis phenomena. The structure of candidate feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback – linear, nonlinear, and a combination of linear and nonlinear – are adequate for the bifurcation control. Finally, numerical simulations are performed to verify the effectiveness of the proposed feedback control.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

A Collocation-Based Algorithm for Computing the Pull-In Range of a Class of Phase-Locked Loops

The pull-in range (ω_p) of a phase-locked loop (PLL) is defined as the maximum value of loop detuning ω_0s for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if ω_0s < ω_p. Simple approximation is available for computing ω_p for the high gain PLL where saddle-node bifurcation occurs at ω_0s = ω_p. Unlike the high gain case, a simple approximation for ω_p is not available for the low gain case where bifurcation from a separatrix cycle occurs at ω_0s = ω_p. The vector field model for a class of second-order PLLs is shown to have rotational properties, which imply the existence of a separatrix cycle. The external stability of this separatrix cycle is an indicator of the type of bifurcation (saddle-node or separatrix cycle) which terminates the limit cycle associated with the PLL's stable false lock state and the PLL pulls-in (i.e. achieve phase lock). A formula is given for determining the separatrix cycle's stability, which indicates that these paratrix cycle is externally stable for small values of closed loop gain. A collocation-based algorithm is presented for computing the PLL's separatrix cycle and the value of pull-in range frequency ω_0s = ω_p at which a stable separatrix cycle exists.     

Lagrangian Time-Scales in Homogeneous Non-Gaussian Turbulence

Lagrangian time-scales in homogeneous non-Gaussian turbulence were studied using a one-dimensional Lagrangian Stochastic Model. The existence of two time-scales τ _L and T _L , one typical of the inertial subrange and the other which is an integral property, is outlined. Variations of the ratio T _L /τ _L in the plane skewness-flatness (S, F) are shown and a connection with the statistical constraint F ≥S ² + 1 is evidenced. The Lagrangian autocorrelation function ρ(t) of particle velocity was computed for some values of (S, F). It is shown that for small times, say t < T _L , the influence of non-Gaussianity is negligible and ρ(t) presents the same behaviour as in the Gaussian case regardless of variations in (S, F).As the time increases, departures from Gaussianity are observed and autocorrelation turns out to be always larger than in the Gaussiancase. This is supported by some considerations in terms of information entropy, which is shown to decrease with increasing departures from Gaussianity. Spectral analysis of Lagrangian velocity shows that non-Gaussianity is relevant only to large scales of the stochastic process and that the expected inertial subrange decay ω⁻² is attained by spectra of all simulations, except for one case in which the model probability density function is bimodal, due to the vicinity to the statistical limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

Vortex ring generation due to the coalescence of a water drop at a free surface

 Results are presented of an experimental investigation of vortex ring formation by a fluid drop contacting a free surface with negligible velocity. The pool fluid is mixed with fluorescein dye, and a laser sheet is used to illuminate a plane of the flow. A series of representative images is recorded by a CCD camera and speculation is made regarding specific sources of vorticity flux through the free surface. Two scaling analyses previously presented by other investigators are demonstrated to be equivalent under the assumptions of this experiment, and they provide the motivation for a series of test runs in which the duration of the coalescence process, τ_*, is related to variations in drop diameter L and fluid surface tension σ. Experimental results are in agreement with the analyses, showing τ_*∼σ^-1/2 and τ_*∼L ^3/2. Received: 22 December 1995 / Accepted: 15 October 1996     

Coherent structures in countercurrent axisymmetric shear flows

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied.The forward velocity U1 and the velocity ratio R=(U1-U2)/(U1+U2),where U2 denotes the suction velocity,are consldered as the control parameters.Two kinds of vortex structures,i.e.,axisymmetric and helical structures,were discovered with respect to different reginmes in the R versus U1 diagram .In the case of U1 rangjing from 3 to 20m/s and R from 1 to 3,the axisymmetric structures plan an important role.Based on the dynamical behaviors of axisymmetric structures,a critical forward velocity U1cr=6.8m/s was defined,subsequently,the subcritical velocity regime:U1&gt;U1cr and the supercritical velocity regime:U1&lt;U1er,In the subcritical velocity regine,the flow system contains shear layer self-excited oscillations in a certain range of the velocity ratio with respect to any forward velocity.In the supercritical velocity regime,the effect of the velocity ratio could be explained by the relative movement and the spatial evolution of the axisymmetric structure undergoes the following stages:(1) Kelvin-Helmholtz instability leading to vortex rolling up,(2) first time vortex agglomeration.(3) jet colunn self-excited oscillation,(4) shear layer self-excited oscillation,(5)“ordered tearing“,(6) turbulence in the case of U1&lt;4m/s (the “ordered tearing“ does not exist when U1&gt;4m/s),correspondingly,the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows:(1) Hopt bifurcation,(5) chaos(“weak turbulence“)in the case of U1&lt;4m/s(superharmonic bifurcation does not exist when U1&gt;4m/s).The proposed new terms,superharmonic and reversed superbarmonic bifurcations,are characterized of the frequency doubling rather than the period doubling.A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3,There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension.The scenario of the spatial evolution of helical structures could be described as follows:the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form.Correspondingly,the dynamical system evolves directly into 2-tiorus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor.In the case of U1=2m/s,the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows:(1)2-torus(Hopf bifurcation),(2) limit cycle(reversed Hopf bifurcation),(3) strange attractor (subbarmonic bifurcation).     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Bifurcation analysis of predator-prey systems with constant rate harvesting using non-standard discretization

We formulate and apply non-standard discretization methods that enable us to study the saddle, elliptic and parabolic cases of the predator-prey system with constant rate harvesting as difference dynamical systems. Our models have the same qualitative features as their corresponding continuous models. By choosing appropriate bifurcation parameters, we combine analytical and numerical investigations to produce interesting global bifurcation diagrams, including saddle-node, Hopf and Bogdanov-Takens bifurcations.