Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Transitions to chaos in squeeze-film dampers

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.     

Crisis-limited chaotic dynamics in ecological systems

We review our recent efforts to understand why chaotic dynamics is rarely observed in natural populations. The study of two-model ecosystems considered in this paper suggests that chaos exists in narrow parameter ranges. This dynamical behaviour is caused by the crisis-induced sudden death of chaotic attractors. The computed bifurcation diagrams and basin boundary calculations reinforce our earlier conclusion [Chaos, Solitons & Fractals 8 (12) (1997) 1933; Int J Bifurc Chaos 8 (6) (1998) 1325] that the reason why chaos is rarely observed in natural populations is hidden within the mathematical structure of the ecological interactions and not with the problem associated with the data (insufficient length, precision, noise, etc.) and its analysis. We also argue that crisis-limited chaotic dynamics can be commonly found in model terrestrial ecosystems.     

Bifurcation of Periodic Orbits and Chaos in Duffing Equation

Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].     

On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3 solution and chaotic movement.     

Nonlinear dynamics of regenerative cutting processes—Comparison of two models

Understanding the nonlinear dynamics of cutting processes is essential for the improvement of machining technology. We study machine cutting processes by two different models, one has been recently introduced by Litak [Litak G. Chaotic vibrations in a regenerative cutting process. Chaos, Solitons & Fractals 2002;13:1531–5] and the other is the classic delay differential equation model. Although chaotic solutions have been found in both models, well known routes to chaos, such as period-doubling or quasi-periodic motion to chaos are not observed in either model. Careful analysis shows that the chaotic motion from the Litak’s model has sharper spectral peaks, a smaller correlation dimension and a smaller value for the largest positive Lyapunov exponent. Implications to the control of chaos in cutting processes are discussed.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C¹ and C² maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 1881]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

Bifurcation and chaos in discrete-time predator–prey system

The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8, 9, 10, 12, 18, 20, 22, 30, 39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.     

Crisis of interspike intervals in Hodgkin–Huxley model

The bifurcations of the chaotic attractor in a Hodgkin–Huxley (H–H) model under stimulation of periodic signal is presented in this work, where the frequency of signal is taken as the controlling parameter. The chaotic behavior is realized over a wide range of frequency and is visualized by using interspike intervals (ISIs). Many kinds of abrupt undergoing changes of the ISIs are observed in different frequency regions, such as boundary crisis, interior crisis and merging crisis displaying alternately along with the changes of external signal frequency. And there are logistic-like bifurcation behaviors, e.g., periodic windows and fractal structures in ISIs dynamics. The saddle-node bifurcations resulting in collapses of chaos to period-6 orbit in dynamics of ISIs are identified.     

The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to 7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to period-7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

On the Analysis of Stability,Bifurcation, Chaos and Chaos Control of Kopel Map

A dynamical system of two nonlinear difference equations introduced by Kopel is studied (Kopel, M., Simple and complex adjustment dynamics in Cournot Duopoly model, Chaos, Solitons and Fractals, 1996, 7 (12), 2031–2048.). In this study, we determine conditions for the stability of the fixed points and we study the bifurcations and chaos (Devany, R. L., Introduction to Chaotic Dynamical Systems. Addison–Wesley, New York, 1989.) by computing the maximum Laypunov exponents (Luhta, I. and Virtanen, I., Chaos, Solitons and Fractals, 1996, 7, 2083.). The OGY chaos control formula is used to control chaos in this system.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.