The nonstationary effects on a softening duffing oscillator

This paper presents a numerical study for the bifurcations of a softening Duffing oscillator subjected to stationary and nonstationary excitation. The nonstationary inputs used are linear functions of time. The bifurcations are the results of either a single control parameter or two control parameters that are constrained to vary in a selected direction on the plane of forcing amplitude and forcing frequency. The results indicate: 1. Delay (memory, penetration) of nonstationary bifurcations relative to stationary bifurcations may occur. 2. The nonstationary trajectories jump into the neighboring stationary trajectories with possible overshoots, while the stationary trajectories transit smoothly. 3. The nonstationary penetrations (delays) are compressed to zero with an increasing number of iterations. 4. The nonstationary responses converge through a period-doubling sequence to a nonstationary limit motion that has the characteristics of chaotic motion. The Duffing oscillator has been used as an example of the existence of broad effects of nonstationary (time dependent) and codimensional (control parameter variations in the bifurcation region) inputs which markedly modify the dynamical behavior of dynamical systems.     

Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System

The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.     

Period doubling bifurcation problems in the softening Duffing oscillator with nonstationary excitation

The Duffing oscillators are widely used to mathematically model a variety of engineering and physical systems. A computational analysis has been initiated to explore the effects of nonstationary excitations on the response of the softening Duffing oscillator in the region of the parameter space where the period doubling sequences occur. Significant differences between the stationary and nonstationary responses have been uncovered: (i) the stationary transitions from T to 2T, from 2T to 4T ... etc. branches at the stationary period doubling bifurcations are smooth, in nonstationary cases they exhibit jumps to the near stationary branches at the values of the control parameters greater than those for the stationary; this phenomenon is called penetration (delay or memory). The lengths of the penetrations is being compressed to zero with the increasing number of the iterations. (ii) The stationary and nonstationary responses eventually settle on different limit motions, the nonstationary has modulated components. (iii) The jumps appearing in the stationary bifurcation diagram at 2T from the upper to the lower branches of the (x, f) and (x, ), i.e., (displacement-forcing amplitude) and (displacement-forcing frequency), diagrams have been replaced by continuous transition in the nonstationary diagram climinating thus the discontinuity. Apart from these differences, some specific characteristic nonstationary responses have been observed not encountered in the stationary cases: (iv) the appearance of the window in the nonstationary limit bifurcation diagrams. (v) The nonstationary limit motions located on the upper (lower) branches of the (x, f) or (x, ) diagrams expanded rapidly to the lower (upper) branches. (vi) The stationary and nonstationary bifurcation diagrams are extremely sensitive to the initial conditions, manifested by the mirror reflections, called the flipflop phenomenon. (vii) The nonstationary limit motion has been characterized by a complex phase portrait, the appearance of the Cantor-like set of the limit motion bifurcation plot, and continuous spectral density. For the purpose of comparison, a stationary period doubling sequence T, 2T,..., 2 ⁿ T,... stationary limit motion, _ST which is known to be chaotic has been determined. A far reaching observation has been made in the process of this study: the determination of the nonstationary bifurcations, their branches and limit motions, has been independent of the calculations of the stationary ones, indicating, thus, the existence of an independent class of nonstationary (time varying) dynamics.     

Lyapunov and LMI analysis and feedback control of border collision bifurcations

Feedback control of piecewise smooth discrete-time systems that undergo border collision bifurcations is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The class of systems studied is piecewise smooth maps that depend on a parameter, where the system dimension n can take any value. The goal of the control effort in this work is to replace the bifurcation so that in the closed-loop system, the steady state remains locally attracting and locally unique (“nonbifurcation with persistent stability”). To achieve this, Lyapunov and linear matrix inequality (LMI) techniques are used to derive a sufficient condition for nonbifurcation with persistent stability. The derived condition is stated in terms of LMIs. This condition is then used as a basis for the design of feedback controls to eliminate border collision bifurcations in piecewise smooth maps and to produce the desirable behavior noted earlier. Numerical examples that demonstrate the effectiveness of the proposed control techniques are given.     

An Impulsive Multi-delayed Feedback Control Method for Stabilizing Discrete Chaotic Systems

An impulsive multi-delayed feedback control strategy to control the period-doubling bifurcations and chaos in an n dimensional discrete system is proposed. This is an extension of the previous result in which the control method is applicable to the one-dimensional case. Then the application of the control method in a discrete prey–predator model is studied systematically, including the dynamics analysis on the prey–predator model with no control, the bifurcations analysis on the controlled model, and the bifurcations and chaos control effects illustrations. Simulations show that the period-doubling bifurcations and the resulting chaos can be delayed or eliminated completely. And the periodic orbits embedded in the chaotic attractor can be stabilized. Compared with the existed methods, a milder condition is needed for the realization of the proposed method. The condition may be considered as a generic case and we may state that almost all periodic orbits can be stabilized by the proposed method. Besides, the idea of impulsive control makes the implementation of the proposed control method easy. The impulsive interval is embodied in the analytical expression of the stability condition, hence can be chosen qualitatively according to the real needs, which is an extension of the existed related results. The introduction of multi-delay enlarges the domain of the control parameters and makes the selection of the control parameters have many choices, and hence become flexible.     

Classification and unfoldings of 1:2 resonant Hopf Bifurcation

In this paper, we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1:2 ratio. This corresponds to a Hopf-Hopf mode interaction with 1:2 resonance, as occurs in the context of dissipative mechanical systems. Using an approach based on Liapunov-Schmidt reduction and singularity theory, we give a framework in which to study these problems and their perturbations in two cases: no distinguished parameter, and one distinguished (bifurcation) parameter. We give a complete classification of the generic cases and their unfoldings. (Accepted May 26, 1995) – Communicated by M. Golubitsky     

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

Alternate pacing of border-collision period-doubling bifurcations

Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise C ¹ map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations are not well understood. This paper considers a particular class of border-collision bifurcations, i.e., border-collision period-doubling bifurcations. We apply a subharmonic perturbation to the bifurcation parameter, which is also known as alternate pacing, and we investigate the response under such pacing near the original bifurcation point. The resulting behavior is characterized quantitatively by a gain, which is the ratio of the response amplitude to the applied perturbation amplitude. The gain in a border-collision period-doubling bifurcation has a qualitatively different dependence on parameters from that of a classical period-doubling bifurcation. Perhaps surprisingly, the differences are more readily apparent if the gain is plotted versus the perturbation amplitude (with the bifurcation parameter fixed) than if plotted versus the bifurcation parameter (with the perturbation amplitude fixed). When this observation is exploited, the gain under alternate pacing provides a useful experimental tool to identify a border-collision period-doubling bifurcation.     

Development of Dominating Waves From Small Disturbances in Falling Viscous-Liquid Films

For wavy liquid films, the principle of selection of the periodic solutions realized experimentally as regular waves is justified. By means of numerical methods, the bifurcations of the families of steady periodic waves and the attractors of the corresponding nonstationary problem are systematically studied. A comparison of the bifurcations and the attractors shows that, when several periodic solutions exist for a given wave number, the solution with the maximum wave amplitude and the maximum phase velocity develops from small initial disturbances (the dominating wave regime). With wave number variation, near the bifurcation points the attractor passes discontinuously from one family to another. This passage is accompanied by the appearance of two-periodic solutions in small neighborhoods of these points. The relations between the calculated parameters of the dominating waves are in a good agreement with all the available experimental data.     

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.     

Complete dynamical analysis of a neuron model

In-depth understanding of the generic mechanisms of transitions between distinct patterns of the activity in realistic models of individual neurons presents a fundamental challenge for the theory of applied dynamical systems. The knowledge about likely mechanisms would give valuable insights and predictions for determining basic principles of the functioning of neurons both isolated and networked. We demonstrate a computational suite of the developed tools based on the qualitative theory of differential equations that is specifically tailored for slow–fast neuron models. The toolkit includes the parameter continuation technique for localizing slow-motion manifolds in a model without need of dissection, the averaging technique for localizing periodic orbits and determining their stability and bifurcations, as well as a reduction apparatus for deriving a family of Poincaré return mappings for a voltage interval. Such return mappings allow for detailed examinations of not only stable fixed points but also unstable limit solutions of the system, including periodic, homoclinic and heteroclinic orbits. Using interval mappings we can compute various quantitative characteristics such as topological entropy and kneading invariants for examinations of global bifurcations in the neuron model.     

Experimental study of regular and chaotic transients in a non-smooth system

This paper focuses on thoroughly exploring the finite-time transient behaviors occurring in a periodically driven non-smooth dynamical system. Prior to settling down into a long-term behavior, such as a periodic forced oscillation, or a chaotic attractor, responses may exhibit a variety of transient behaviors involving regular dynamics, co-existing attractors, and super-persistent chaotic transients. A simple and fundamental impacting mechanical system is used to demonstrate generic transient behavior in an experimental setting for a single degree of freedom non-smooth mechanical oscillator. Specifically, we consider a horizontally driven rigid-arm pendulum system that impacts an inclined rigid barrier. The forcing frequency of the horizontal oscillations is used as a bifurcation parameter. An important feature of this study is the systematic generation of generic experimental initial conditions, allowing a more thorough investigation of basins of attraction when multiple attractors are present. This approach also yields a perspective on some sensitive features associated with grazing bifurcations. In particular, super-persistent chaotic transients lasting much longer than the conventional settling time (associated with linear viscous damping) are characterized and distinguished from regular dynamics for the first time in an experimental mechanical system.     

Controlling Hopf�CHopf interaction bifurcations of?a?two-degree-of-freedom self-excited system with?dry?friction

The feedback control problem of designing Hopf?CHopf interaction bifurcations into a dry friction system at a pre-specified parameter point is addressed. A new bifurcation criterion without using eigenvalues is established to preferably determine the control gains. Numerical simulation shows that the torus solution of Hopf?CHopf interaction bifurcation can be created in the friction system at a desired parameter location.     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system

This paper analyzes the hyperchaotic behaviors of the newly presented simplified Lorenz system by using a sinusoidal parameter variation and hyperchaos control of the forced system via feedback. Through dynamic simulations which include phase portraits, Lyapunov exponents, bifurcation diagrams, and Poincaré sections, we find the sinusoidal forcing not only suppresses chaotic behaviors, but also generates hyperchaos. The forced system also exhibits some typical bifurcations such as the pitchfork, period-doubling, and tangent bifurcations. Interestingly, three-attractor coexisting phenomenon happens at some specific parameter values. Furthermore, a feedback controller is designed for stabilizing the hyperchaos to periodic orbits, which is useful for engineering applications.     

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods. The project supported by the National Natural Science Foundation of China     

On the existence and bifurcation of minimal normal modes

Minimal normal modes (MNMs) are defined as non-linear normal modes which give a true minimum to Jacobi's Principle of Least Action. It is shown that for a certain class of two degree of freedom non-linear conservative systems, MNMs generically occur in pairs. The nature of both generic and non-generic bifurcations of MNMs is derived and illustrative examples are given.     

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.     

Big bang bifurcations and Allee effect in Blumberg’s dynamics

This paper concerns dynamics and bifurcations properties of a class of continuous-defined one-dimensional maps, in a three-dimensional parameter space: Blumberg’s functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon, associated with the stability of a fixed point. A central point of our investigation is the study of bifurcations structure for this class of functions. We verified that under some sufficient conditions, Blumberg’s functions have a particular bifurcations structure: the big bang bifurcations of the so-called “box-within-a-box” type, but for different kinds of boxes. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct attractors. This work contributes to clarify the big bang bifurcation analysis for continuous maps. To support our results, we present fold and flip bifurcations curves and surfaces, and numerical simulations of several bifurcation diagrams.