International Journal of Bifurcation and Chaos

In this paper we relate the methods of the theory of dynamical systems to problems of accelerator design and optimization. It is written in review article style and any prior knowledge of accelerator physics is not required.

We derive the Poincaré map for the equation of horizontal transverse motion of a test particle in a circular accelerator in the presence of any number of sextupolar nonlinearities. This map is studied within the framework of nonlinear dynamics and we identify several optimization strategies for the design of an accelerator magnet lattice. The dynamic aperture problem is analysed and we obtain, for a beam line with two sextupoles, the dependence of the dynamic aperture on the line parameters. The mechanisms of aperture reduction near rational values of the tune, ν₀ = p/q ≠ 1/3, are due to the creation of heteroclinic tangles and do not affect the stability of the design orbit. For a machine with an arbitrary number of localized sextupolar fields, we obtain the dependence of the nonlinear tune on the amplitude of the particles and the parameters of the accelerator. The predictions for the nonlinear tune agree with the numerical simulations within the maximum relative error of 6% at large amplitudes (~ dynamic aperture) and within the relative error range 1%–2% at intermediate amplitudes (~ half dynamic aperture).

The computation of local bifurcation diagrams near degenerate Hopf bifurcations of certain types using feedback system theory and harmonic balance techniques is presented. This approach also provides the analytical expressions for the defining and the nondegeneracy conditions in the so-called frequency domain counterpart. A classical graphical method is easily adapted to carry on the continuation of the oscillatory branches to depict the local bifurcation diagrams. Moreover, several higher-order harmonic balance approximations are implemented to compare the accuracy of the computed solutions. The results are presented using local bifurcation diagrams, phase portrait plots and period diagrams, with similar ones obtained by using AUTO.

Normal forms for eleven cases of bifurcations of codimension-3 are considered, basically, in systems with a symmetry, which can be reduced to one of the two three-dimensional systems. The first system is the well-known Lorenz model in a special notation, the second is the Shimizu-Morioka model. In contrast with two-dimensional normal forms which admit, in principle, a complete theoretical study, in three-dimensional systems such analysis is practically impossible, except for particular parameter values when a system is close to an integrable system. Therefore, the main method of the investigation is qualitatively-numerical. In that sense, a description of principal bifurcations which lead to the appearance of the Lorenz attractor is given for the models above, and the boundaries of the regions of the existence of this attractor are selected.

We pay special attention to bifurcation points corresponding to a formation of a homoclinic figure-8 of a saddle with zero saddle value and that of a homoclinic figure-8 with zero separatrix value. In L. P. Shil'nikov [1981], it was established that these points belong to the boundary of the existence of the Lorenz attractor. In the present paper, the bifurcation diagrams near such points for the symmetric case are given and a new criterion of existence of the nonorientable Lorenz is also suggested.

We consider the nonlinear equation f (x, Λ) = 0, f: X × ℝ³ → X, where X is a Banach space and f satisfies a Z₂-symmetry relation. We are interested in a singular point (x⁰, Λ⁰) called a double Takens–Bogdanov point, where f_x has a zero eigenvalue of geometric multiplicity 2 and algebraic multiplicity 3. We show that only three parameters are needed to recognise and compute this singular point when the Z₂-symmetry is present. Paths of the following three kinds of singular points bifurcating at this double Takens–Bogdanov point are studied: double s-breaking fold points (where f_x has a zero eigenvalue of algebraic and geometric multiplicity 2), Takens–Bogdanov points (a zero eigenvalue of geometric multiplicity 1 and algebraic multiplicity 2) and s-breaking G points (a simple zero eigenvalue and a pair of simple purely imaginary eigenvalues). The theory is developed in such a way as to produce nondegeneracy conditions and tests which may be utilised in numerical calculations. Numerical results are presented for the one-dimensional Brusselator model, described by a system of four reaction–diffusion equations.

Simulations of resistive magnetohydrodynamics in a rotating plasma are analyzed by calculating the correlation dimensions and the local intrinsic dimensions. A rotating plasma with a nonlinear oscillatory state is found to be associated with a low-dimensional attractor. The solutions are also checked for sensitivity to initial conditions, characterized by the sign of the largest Lyapunov exponent.

Dynamic bifurcations, including Hopf and period-doubling bifurcations, are found to occur in a power system dynamic model recently employed in voltage collapse studies. The occurrence of dynamic bifurcations is ascertained in a region of state and parameter space linked with the onset of voltage collapse. The work focuses on a power system model studied by Dobson & Chiang [1989]. The presence of the dynamic bifurcations, and the resulting implications for dynamic behavior, necessitate a re-examination of the role of saddle node bifurcations in the voltage collapse phenomenon. The bifurcation analysis is performed using the reactive power demand at a load bus as the bifurcation parameter. It is determined that the power system model under consideration exhibits two Hopf bifurcations in the vicinity of the saddle node bifurcation. Between the Hopf bifurcations, i.e., in the "Hopf window," period-doubling bifurcations are found to occur. Simulations are given to illustrate the various types of dynamic behaviors associated with voltage collapse for the model. In particular, it is seen that an oscillatory transient may play a role in the collapse.

In order to describe silicon carbide polytype growth under CVD/CVI conditions, a dynamical model based on iterations of the sawtooth map has been studied. A two-parameter bifurcation diagram has been computed for a simple logistic-like family, and its structure has been studied in more detail by means of kneading theory and study of the skeleton. All the routes to chaos are shown to derive from the application of a central theorem involved in the construction of the skeleton. A comparison, under a combinatorial point of view, between this mathematical model and the reported data on known SiC polytypes is carried out and discussed. Indeed, the model is able to generate any periodic or chaotic polytypic sequence.

Excess-matrices were defined in a previously published Part I as a special class of integer matrices with multifractal features. The present Part II considers compactifications of an excess-matrix E, i.e., matrices obtained by replacing submatrices of specific sizes in E by a single number according to some prescribed rule. It will be shown how these compactifications can be obtained by a direct mapping from E itself, through the derivation of a set of formulas for the associated mapping sequences (as defined in Part I). Then these formulas are used for characterizing the distribution of certain integer sets over these matrices. The evolution of this distribution over all scales of observation is described by two different fractal-dimension functions, which are scale-dependent extensions of classical practical fractal-dimension measures. Expressions for these functions are derived, and their properties discussed. Exploring the limits of these functions which relate them to the classical small-scale fractal dimension reveals some difficulty in the interpretation of the latter when applied to excess-matrices.

It is known that the index on the Mandelbrot set introduces a Fibonacci partition. In this paper we will give an interpretation of this property.

We present a theory of constructive Poincaré maps. The basis of our theory is the concept of irreducible nonlinear maps closely associated to concepts from Lie groups. Irreducible nonlinear maps are, heuristically, nonlinear maps which cannot be made simpler without removing the nonlinearity. A single irreducible map cannot produce chaos or any complex nonlinear effect. It can be implemented in an electronic circuit, and there are only a finite number of families of irreducible maps in any n-dimensional space. The composition of two or more irreducible maps can produce chaos and most of the maps studied today that produce chaos are compositions of two or more irreducible maps. The composition of a finite number of irreducible maps is called a completely reducible map and a map which can be approximated pointwise by completely reducible maps is called a reducible map. Poincaré maps from sinusoidally forced oscillators are the most familiar examples of reducible maps.

This theoretical framework provides an approach to the construction of "closed form" Poincaré maps having the properties of Poincaré maps of systems for which the Poincaré map cannot be obtained in closed form. In particular, we derive a three-dimensional ODE for which the Hénon map is the Poincaré map and show that there is no two-dimensional ODE which can be written down in closed form for which the Hénon map is the Poincaré map. We also show that the Chirikov (standard) map is a Poincaré map for a two-dimensional closed form ODE. As a result of our theory, these differential equations can be mapped into electronic circuits, thereby associating them with real world physical systems. In order to clarify our results with respect to the abstract mathematical concept of suspension, which says that every C¹ invertible map is a Poincaré map, we introduce the concept of a constructable Poincaré map. Not every map is a constructable Poincaré map and this is an important distinction between dynamical synthesis and abstract nonlinear dynamics.

We also show how to use any one-dimensional map to induce a two-dimensional Poincaré map which is a completely reducible map and hence for a very broad class of maps that includes the logistic map we derive closed form ODEs for which these one-dimensional maps are "embedded" in a Poincaré map. This provides an avenue for the study of one-dimensional maps, such as the logistic map, as two-dimensional Poincaré maps that arise from square-wave forced electronic circuits.

Transition waves are widespread in the biological and chemical sciences, and have often been successfully modelled using reaction–diffusion systems. I consider a particular system of three reaction–diffusion equations, and I show that transition waves can destabilise as the kinetic ordinary differential equations pass through a Hopf bifurcation, giving rise to either regular or irregular spatiotemporal oscillations behind the advancing transition wave front. In the case of regular oscillations, I show that these are periodic plane waves that are induced by the way in which the transition wave front approaches its terminal steady state. Further, I show that irregular oscillations arise when these periodic plane waves are unstable as reaction–diffusion solutions. The resulting behavior is not related to any chaos in the kinetic ordinary differential equations.

Dynamics of a chain of coupled Chua's circuits is investigated. Existence of spatial disorder is revealed. Dynamics of wave fronts in the limiting case of a continuous medium is studied.

The chaotic behavior of the solutions of Chua's circuit is studied in the neighborhood of a homoclinic orbit to an equilibrium point of the saddle-focus type and in a neighborhood of two heteroclinic orbits to saddle-focus points which form a closed contour.

Bifurcation phenomena of the Kuramoto–Sivashinsky equation have been studied numerically. The solutions considered are restricted to the invariant subspace of odd functions. One possible route to chaos via a period-doubling cascade is investigated in detail: The four-modal steady-state loses its stability through a Hopf bifurcation and a branch of periodic motions is created. After a symmetry breaking the periodic solution undergoes a period-doubling cascade which ends up in two antisymmetric chaotic attractors. A merging of these antisymmetric attractors to a symmetric one is observed. The chaotic branch depending on the bifurcation parameter is characterized by the values of the Lyapunov exponents. Periodic windows within the chaotic region are also detected. Finally, a further increase of the bifurcation parameter leads to a transition from the attractor into transient chaos.

In this paper we investigate the different behaviors of some satellites, subject to a spin-orbit interaction. In particular we compute, in the space of initial data, the basin of attraction of the 3:2 resonance for the Mercury–Sun and Moon–Earth coupling systems.

The results are obtained either by direct integration of the equations of motion or by a numerical implementation of the "straddle" technique to draw the boundary of the basin.

A new method for stability analysis during continuation is proposed. The method uses shifted inverse iteration to compute a relevant eigenvalue, determining stability. Some additional work has to be done to verify the above solution. Numerical experiments are performed using models of the tubular reactor and the Brusselator.

The signal recovered from the first reported experimental secure communication system via chaotic synchronization contains some inevitable noise which degrades the fidelity of the original message. In this letter we show by computer experiments that this noise can be significantly reduced by cascading the output of the receiver in the original system into an identical copy of this receiver.

A systematic approach is developed for synthesizing N-dimensional dissipative chaotic systems which possess the self-synchronization property. The ability to synthesize new chaotic systems further enhances the usefulness of synchronized chaotic systems for communications.

An extended formulation of sensitivity to initial conditions applicable to (small) finite errors and finite times is developed. It is shown that the first stages of error growth are neither exponential nor driven by the Lyapunov exponent.