Blue Sky Catastrophe Research Articles

A bifurcation analysis of subsynchronous oscillations in power systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded turbine-generator system connected to an infinite busbar through a series capacitor-compensated transmission line. It reveals the existence of self-excited subsynchronous torsional oscillations of a 5-mass rotor, which may eventually lead to the destruction of the shaft or the loss of synchronism of the generator. Specifically, we show that, as the capacitor-compensation value increases and reaches a critical value, called supercritical Hopf bifurcation, the system around the operating point undergoes small single-period oscillations with constant amplitude. This in turn results in a small limit-cycle attractor. As the compensation level increases further, the amplitude of oscillation grows until a secondary Hopf bifurcation is reached. There, the oscillations characterize themselves by two incommensurate periods and bounded amplitudes, signifying the transformation of the limit cycle into two-period quasiperiodic motion called a two-torus attractor. When the capacitor-compensation level passes a third critical value, the amplitude of oscillations becomes unbounded following the destruction of the two-torus attractor and its basin of attraction in a so-called bluesky catastrophe. Interestingly, this scenario repeats itself in the vicinity of three supercritical Hopf bifurcations. The bifurcation analysis is validated with numerical solutions of the differential equations that govern the power system.

Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System. The system has five mechanical and two electrical modes. The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. The results show that adding damper windings may induce subsynchronous resonance.

THE EVOLUTION OF THE STABLE AND UNSTABLE MANIFOLD OF AN EQUILIBRIUM POINT

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter,. The eigenvalues of the linearized system are complex for 0. Thus for 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation. One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for <0 and the whole structure of these manifolds shrinks to the equilibrium as ! 0 and disappears. These considerations have interesting implications about Str ¨ omgren's conjecture in celestial mechanics and the blue sky catastrophe of Devaney.

Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies

The behavior of a driven double-well Duffing--van der Pol oscillator for a specific parametric choice $(|\ensuremath{\alpha}|=\ensuremath{\beta})$ is studied. The existence of different attractors in the system parameters $f$-$\ensuremath{\omega}$ domain is examined and a detailed account of various steady states for fixed damping is presented. The transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period-doubling regions), boundary crisis, three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during the transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instabilities allows one to predict Neimark instability in the $f$-$\ensuremath{\omega}$ plane and eventually results in the approximate predictive criteria for the chaotic region.

Simple bifurcations leading to hyperbolic attractors

We prove the existence of a new stability boundary of periodic orbits in a high-dimensional case, thereby resolving the problem on a “blue sky catastrophe” in a general one-parameter family. We additionally establish the existence of a codimension-one boundary which separates the Morse-Smale systems from systems with hyperbolic Smale-Williams-like attractors. The route across this boundary is accomplished by the disappearance of a saddle-node periodic orbit. We also study the principal bifurcations of a torus breakdown which lead to Anosov attractors and to multidimensional solenoids.

Optimal escape from potential wells-patterns of regular and chaotic bifurcation

Abstract The patterns of bifurcation governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier are studied by numerical simulation. Both the generic asymmetric single-well cubic potential and the symmetric twin-well potential Duffing oscillator are surveyed by varying three parameters: forcing frequency, forcing amplitude, and damping coefficient. The close relationship between optimal escape and nonlinear resonance within the well is confirmed over a wide range of damping. Subtle but significant differences are observed at higher damping ratios. The possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is cmppletely suppressed above a critical level of the damping ratio (about 0.12 for the asymmetric single-well oscillator). Coincidentally, at almost the same level of damping, the optimal escape condition becomes distinct from the apex in the (ω, F ) plane of the bistable regime; this corresponds to the appearance of chaotic attractors which subsume both resonant and non-resonant motions within one well. At higher damping levels, further changes occur involving conversions from chaotic-saddle to regular-saddle bifurcations. These changes in optimal escape phenomena correspond to codimension three bifurcations at exceptional points in the space of three parameters. These bifurcations are described in terms of homoclinic and heteroclinic structures of invariant manifolds, and changes in accessible boundary orbits. The same sequence of codimension three bifurcations is observed in both the twin-well Duffing oscillator and the asymmetric single-well escape equation. Within the codimension three bifurcation patterns governing escape, one particular codimension two global bifurcation involves a chaotic attractor explosion, or interior crisis, compounded with a blue sky catastrophe or boundary crisis of the exploded attractor. This codimension two bifurcation has structure containing a form of predictive power: knowledge of attractor bifurcations in part of the codimension two pattern permits inference of the attractor and basin bifurcations in the remainder. This predictive power is applicable beyond the context of escape from potential wells. Quantitative correlation of bifurcation patterns between the two equations according to simple scaling laws is tested. The unstable periodic orbits which figure most prominently in the major attractor-basin bifurcations are of periods one and three. Their linking is conveniently interpreted by a three-layer spiral horseshoe structure for the folding action in phase space within a well. The structure of this 3-shoe implies a partial ordering among order three subharmonic saddle-node bifurcations. This helps explain the sequence of codimension three bifurcations near optimal escape. Some bifurcational precedence relations are known to follow from the linking of periodic orbits in a braid on a 3-shoe. Additional bifurcational precedence relations follow from a quantitative property of generic potential wells: the dynamic hilltop saddle has a very large expanding multiplier over one cycle of forcing near fundamental resonance. This quantitative property explains the close coincidence of codimension three bifurcations near the suppression of indeterminate outcomes. An experimentalist's approach to identifying the three-layer template structure from time series data is discussed, including a consistency check involving Poincare indices. The bifurcation patterns emerging at higher damling values create favorable conditions for realizing experimental strategies to recognize optimal escape and locate it in parameter space. Strategies based solely on observations of quasi-steady behavior while remaining always within one well are discussed.

Bifurcations phenomena in a nonlinear oscillator: Approximate analytical studies versus computer simulation results

The behavior of a driven single-well oscillator with quadratic nonlinearity is studied and attention is focused on exploring a relationship between classic concepts of the approximate theory of nonlinear oscillations and the complex bifurcations and escape phenomena obtained by computer simulations. An analytical analysis of the Hill's type variational equation leads to approximate estimates for the blue-sky catastrophe and period doubling instability and eventually results in the approximate predictive criteria for the main escape region. The theoretical study and computer simulations are extended to the 1 2 subharmonic resonance and the analysis brings a new insight into the nonlinear phenomena, which occur in the higher frequency region.

The bifurcation of the ?blue sky catastrophe? on two-dimensional manifolds

The bifurcation of the ?blue sky catastrophe? on two-dimensional manifolds

Capsizing of a Ship in Quartering Seas

In the first paper, the authors pointed out that a new mode of capsizing, which accompanied the period doubling bifurcation phenomenon, had been observed and that about 25 % of the total capsizing had been classified into such a new capsizing mode. Since the period doubling bifurcation phenomenon was regarded as a precursor of the chaos in the nonlinear dynamical systems, and the chaos was regarded as a precursor of the unconditional capsizing, which was equivalent to the blue sky catastrophe or the boundary crisis in the softening spring system, the chaos and fractal in the symmetrical capsize equation was examined in the authors' second paper. Although the mysterious nonlinear phenomena appeared in the symmetrical capsize equation were examined numerically, a cascade of the period bifurcations was not observed in the numerical study.In the present paper, the authors have examined the asymmetrical capsize equation as well as the symmetrical one. As a result, the cascade of the Feigenbaum bifurcation, which begins from the period doubling bifurcation and ends at the unconditional capsizing through the chaos, has been observed for both asymmetrical and symmetrical capsize equations. Further it has been clarified that in the symmetrical equation the cascade of the period bifurcations begins after the break of the symmetry of solution, and that the period bifurcation with the distinct difference of the adjoining amplitude, such as observed in the model tests, appears only in the asymmetrical equation. The differences of the fractal capsize boundaries in the initial value plane and the control space between the symmetrical and asymmetrical equations have been also examined, and the danger of the biased ship is emphasized by the numerical results. In the examination of the safe basin in the initial value plane, a cell-to-cell mapping method developed by Hsu has been atempted to confirm its effectiveness and accuracy. A numerical analysis of the fold bifurcation which approximates the capsizing boundary in the control space, and also a numerical Melnikov analysis which approximates the beginning of the fractal metamorphosis in the safe basin in the initial value plane have been carried out. An approximate analytical expression for the flip bifurcation is obtained.The idea that the appearance of the period bifurcation in the roll motion of a ship implies the imminent danger of the capsizing has proved to be correct.

A chaotic blue sky catastrophe in forced relaxation oscillations

The chaotic attractor of a periodically forced Van der Pol oscillator (Shaw variant) is observed in digital simulation, and is made to vanish in a blue sky catastrophe by increasing a constant (bias) term in the force. The detailed bifurcation diagram, based on extensive simulations, reveals the involvement of the homoclinic outset of a nearby limit cycle of saddle type.