Qingjie Cao

Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics.

In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load–deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

Analysis method of chaos and sub-harmonic resonance of nonlinear system without small parameters ∗

The Melnikov method is important for detecting the presence of transverse homoclinic orbits and the occurrence of homoclinic bifurcations. Unfortunately, the traditional Melnikov methods strongly depend on small parameters, which do not exist in most practical systems. Those methods are limited in dealing with the systems with strong nonlinearities. This paper presents a procedure to study the chaos and sub-harmonic resonance of strongly nonlinear practical systems by employing a homotopy method that is used to extend the Melnikov functions to the strongly nonlinear systems. Applied to a given example, the procedure shows the effectiveness via the comparison of the theoretical results and the numerical simulation.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

Threshold of Multiple Stick-Slip Chaos for an Archetypal Self-Excited SD Oscillator Driven by Moving Belt Friction

In this paper, we investigate the multiple stick-slip chaotic motion of an archetypal self-excited smooth and discontinuous (SD) oscillator driven by moving belt friction, which is constructed with the SD oscillator and the classical moving belt. The friction force between the mass and the belt is modeled as a Coulomb friction for this system. The energy introduction or dissipation during the slip and stick modes in the system is analyzed. The analytical expressions of homoclinic orbits of the unperturbed SD oscillator are derived by using a special coordinate transformation without any pronominal truncation to retain the natural characteristics, which allows us to utilize the Melnikov’s method to obtain the chaotic thresholds of the self-excited SD oscillator in the presence of the damping and external excitation. Numerical simulations are carried out to demonstrate the multiple stick-slip dynamics of the system, which show the efficiency of the prediction for stick-slip chaos of the perturbed self-excited system. The results presented herein this paper demonstrate the complicated dynamics of stick-slip periodic solutions, multiple stick-slip chaotic solutions and also coexistence of multiple solutions for the perturbed self-excited SD oscillator.

The SD oscillator and its attractors

We propose a new archetypal oscillator for smooth and discontinuous systems (SD oscillator). This oscillator behaves both smooth and discontinuous system depending on the value of the smoothness parameter. New dynamic behaviour is presented for the transitions from the smooth to discontinuous regime.

Co-Dimension Two Bifurcation

This chapter investigates the fundamental properties of the behaviour of the SD oscillator of the perturbed degenerate case. As we have already seen, there is a degenerate singularity due to the change of hyperbolicity when parameter \(\alpha \) varies crossing \(\alpha =1\), where the system is transformed from a single well dynamics to that of a double-well , in other words from a single stability to a bistability . This singularity can also be treated as the transition from non snap through buckling (\(\alpha >1\)) to a snap through buckling (\(\alpha <1\)) under a static load. We turn our attention to the interesting perturbed behaviour of the complex codimension two bifurcations of the oscillator at the degenerate equilibrium point (0, 0) near \(\alpha =1\). It is found that the universal unfolding with two parameters is the perturbed SD oscillator with a nonlinear visco-damping. This universal unfolding reveals the complicated codimension two bifurcation phenomena in the physical parameter space with homoclinic bifurcation, closed orbit bifurcation, Hopf bifurcations and also the pitchfork bifurcations demonstrated at the same time when the geometrical parameter varies.

Wada Basin Dynamics

This chapter investigates a specific point of the very intricate asymptotic behaviour of the SD oscillator , which is known as the Wada basin dynamics. The oscillator is subjected to a linear viscous damping and to a sinusoidal forcing. As described and already observed through direct numerical integration, this system may possess more than twenty coexisted low-period periodic attractors for a given set of parameters. The large number of stable orbits yields a complex structure of closely interwoven basins of attraction. We obtain the so-called Wada basins of which the boundaries are rigorously described.

The SD Oscillator

This chapter introduces briefly the smooth and discontinuous (SD) oscillator which is a simple mechanical model or a geometrical oscillator with both smooth and discontinuous dynamics depending on the value of a geometrical parameter . Like the traditional harmonic oscillator , this model is also a simple mass-spring system comprising a lumped mass linked by a pair of linear springs pinged to its rigid supports vibrating along the perpendicular bisector of the supports. The overview of the system with the definitions and the fundamental properties without detailed explanation, which will be expanded different topics discussed in the following chapters from Chaps. 3 to 6.

SD Oscillator with Friction and Impacts

This chapter extends the analysis of the SD oscillator to situations involving coupling with other nonlinearities or higher dimensional motions. The SD oscillator is a strictly one-dimensional dynamical system moving along a given line, now friction will be added as a constraint for the sliding along this line. If this constraint is removed, the system becomes two dimensional, which in turn implies that another constraint must be added to separate permanent sliding motions from jumping effects.