International Journal of Bifurcation and Chaos

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

A variety of heterogeneous potentials are capable of localizing linear noninteracting waves. In this work, we review different examples of heterogeneous localizing potentials which were realized in experiments. We then discuss the impact of nonlinearity induced by wave interactions, in particular, its destructive effect on the localizing properties of the heterogeneous potentials.

When nonlinearity is added to an infinite system with purely discrete linear spectrum, Anderson modes become coupled with one another by terms of higher order than linear, allowing energy exchange between them. It is generally believed, on the basis of numerical simulations in such systems, that any initial wave-packet with finite energy spreads down chaotically to zero amplitude with second moment diverging as a power law of time, slower than standard diffusion (subdiffusion). We present results which suggest that the interpretation of spreading cannot be described as initially believed and that new questions arise and still remain opened. We show that an initially localized wave-packet with finite norm may generate two kinds of trajectories both obtained with nonvanishing probability.

The first kind consists of KAM trajectories which are recurrent and do not spread. Empirical investigations suggest that KAM theory may still hold in infinite systems under two conditions: (1) the linearized spectrum is purely discrete, (2) the considered solutions are square summable and not too large in amplitude. We check numerically that in appropriate regions of the parameter space, indeed many initial conditions can be found with finite probability that generate (nonspreading) infinite dimension tori (almost periodic solutions) in a fat Cantor set in (projected) phase space.

The second kind consists of trajectories which look initially chaotic and often spread over long times. We first rigorously prove that initial chaos does not necessarily imply complete spreading e.g. for large norm initial wave-packet. Otherwise, in some modified models, no spreading at all is proven to be possible, despite the presence of initial chaos in contradiction with early beliefs. The nature of the limit state is still unknown.

However, we attempt to present empirical arguments suggesting that if a trajectory starts chaotically spreading, there will necessarily exist (generally large) critical spreading distances that depend on the disorder realization where the trajectory will be sticking to a dense set of KAM tori. This effect should induce drastic slowing down of the spreading which could be viewed as "inverse Arnold diffusion" since the trajectory approaches KAM tori regions instead of leaving them. We suggest that this effect should self-organize the chaotic behavior and that at long time, the wave-packet might not be spread down to zero, but could have a limit profile with marginal chaos (with singular continuous spectrum), despite a long spatial tail. Further analytical and numerical investigations are required.

Power transmission in one-dimensional nonlinear magnetic metamaterials driven at one end is investigated numerically and analytically in a wide frequency range. The nonlinear magnetic metamaterials are composed of varactor-loaded split-ring resonators which are coupled magnetically through their mutual inductances, forming thus a magnetoiductive transmission line. In the linear limit, significant power transmission along the array only appears for frequencies inside the linear magnetoinductive wave band. We present analytical, closed form solutions for the magnetoinductive waves transmitting the power in this regime, and their discrete frequency dispersion. When nonlinearity is important, more frequency bands with significant power transmission along the array may appear. In the equivalent circuit picture, the nonlinear magnetoiductive transmission line driven at one end by a relatively weak electromotive force, can be modeled by coupled resistive-inductive-capacitive (RLC) circuits with voltage-dependent capacitance. Extended numerical simulations reveal that power transmission along the array is also possible in other than the linear frequency bands, which are located close to the nonlinear resonances of a single nonlinear RLC circuit. Moreover, the effectiveness of power transmission for driving frequencies in the nonlinear bands is comparable to that in the linear band. Power transmission in the nonlinear bands occurs through the linear modes of the system, and it is closely related to the instability of a mode that is localized at the driven site.

In this work, we revisit the question of stability of multibreather configurations, i.e. discrete breathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. We present two methods that yield quantitative predictions about the Floquet multipliers of the linear stability analysis around such exponentially localized in space, time-periodic orbits, based on the Aubry band method and the MacKay effective Hamiltonian method, and prove that by making the suitable assumptions about the form of the bands in the Aubry band theory, their conclusions are equivalent. Subsequently, we showcase the usefulness of the methods through a series of case examples including one-dimensional multi-breathers, and two-dimensional vortex breathers in the case of a lattice of linearly coupled oscillators with the Morse potential and in that of the discrete ϕ⁴ model.

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem, asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These families of periodic orbits continue existing and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario first described by Bozis and Hadjidemetriou.

We study the diffusion of chaotic orbits in an N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions outside corotation support the spiral structure of the galaxy due to the phenomenon of stickiness close and along the unstable asymptotic manifolds of the unstable periodic orbits. These orbits are diffused outwards after about 13 rotations of the bar. During this time, the spiral structure is clearly visible and then it fades out gradually. The diffusion time for the majority of the chaotic orbits with initial conditions inside corotation is much longer than the age of the Universe. These orbits support mainly the outer parts of the bar. However, a part of the chaotic orbits inside corotation are diffused outwards fast and support the spiral structure.

This paper is devoted to the stability study of three Trojan planets around a central star of 1 M_Sun in a distance of 1 AU. In an earlier study, the stability of this coorbital configuration was computed if the mutual angular separation is 47.4° [Salo & Yoder, 1988]. We investigated by using numerical integrations the largeness of the stable region around these stable equilibrium points, depending on the mass parameter and the distances of the three planets. It turned out that even initial deviations in semimajor axis in the order of 5% can lead to very stable orbits. Also differences in the initial angular distance from the equilibrium points up to 70° do not mean that the orbits will be unstable. But the extensions in semimajor axis and angles depend sensitively on the masses involved, which were given the same values for our investigations in a first step. Test computations for different masses of the three planets involved show that they may be quite large without destroying the stability.

We consider the three-dimensional motion of a massless particle in a regular polygon formation of N primary bodies, one of which is located at the system's center of mass. Assuming that the central primary is a radiation source, we apply the simplified theory suggested by Radzievskii, in order to study the effect of radiation pressure in the three-dimensional dynamics of the system. We particularly study the evolution of the zero-velocity surfaces for various values of the radiation coefficient b₀ and investigate also the cases with b₀ > 1 (that is, radiation surpasses gravity) since for these cases, significant changes in the dynamics occur. We then locate numerically the onset of three-dimensional periodic motion from planar periodic motion by calculating the orbits' vertical critical stability. Many families of three-dimensional periodic motions are presented and the regions of the three-dimensional space where these motions take place, are determined. We subsequently investigate how the bifurcations from planar to three-dimensional periodic orbits are affected by the increase of the primary's radiation coefficient and how the overall dynamics of the system is affected by the value of the primaries' number N.

We study the orbits in a Manko–Novikov type metric (MN) which is a perturbed Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are found in configuration space and on a surface of section for various values of the energy E and the z-component of the angular momentum L_z. For relatively large L_z there are two permissible regions of nonplunging motion bounded by two closed curves of zero velocity (CZV), while in the Kerr metric there is only one closed CZV of nonplunging motion. The inner permissible region of the MN metric contains mainly chaotic orbits, but it contains also a large island of stability. When L_z decreases, the two permissible regions join and chaos increases. Below a certain value of L_z, most orbits escape inwards and plunge through the horizon. On the other hand, as the energy E decreases (for fixed L_z) the outer permissible region shrinks and disappears. In the inner permissible region, chaos increases and for sufficiently small E most orbits are plunging. We find the positions of the main periodic orbits as functions of L_z and E, and their bifurcations. Around the main periodic orbit of the outer region, there are islands of stability that do not appear in the Kerr metric (integrable case). In a realistic binary system, because of the gravitational radiation, the energy E and the angular momentum L_z of an inspiraling compact object decrease and therefore the orbit of the object is nongeodesic. In fact, in an extreme mass ratio inspiraling (EMRI) system the energy E and the angular momentum L_z decrease adiabatically and therefore the motion of the inspiraling object is characterized by the fundamental frequencies which are drifting slowly in time. In the Kerr metric, the ratio of the fundamental frequencies changes strictly monotonically in time. However, in the MN metric when an orbit is trapped inside an island the ratio of the fundamental frequencies remains constant for some time. Hence, if such a phenomenon is observed this will indicate that the system is nonintegrable and therefore the central object is not a Kerr black hole.

The process of relaxation of a system of particles interacting with long-range forces is relevant to many areas of physics. For obvious reasons, in Stellar Dynamics much attention has been paid to the case of r^-2 force law. However, recently the interest in alternative gravities has emerged, and significant differences with respect to Newtonian gravity have been found in relaxation phenomena. Here we begin to explore this matter further, by using a numerical model of spherical shells interacting with an r^-α force law obeying the superposition principle. We find that the virialization and phase-mixing times depend on the exponent α, with small values of α corresponding to longer relaxation times, similarly to what happens when comparing for N-body simulations in classical gravity and in Modified Newtonian Dynamics.

We study numerically statistical distributions of sums of orbit coordinates, viewed as independent random variables in the spirit of the Central Limit Theorem, in weakly chaotic regimes associated with the excitation of the first (k = 1) and last (k = N) linear normal modes of the Fermi–Pasta–Ulam-α system under fixed boundary conditions. We show that at low energies (E = 0.19), when k = 1 linear mode is excited, chaotic diffusion occurs characterized by distributions that are well approximated for long times (t > 10⁹) by a q-Gaussian Quasi-Stationary State (QSS) with q ≈ 1.4. On the other hand, when k = N mode is excited at the same energy, diffusive phenomena are absent and the motion is quasi-periodic. In fact, as the energy increases to E = 0.3, the distributions in the former case pass through shorterq-Gaussian states and tend rapidly to a Gaussian (i.e. q → 1) where equipartition sets in, while in the latter we need to reach up to E = 4 to see a sudden transition to Gaussian statistics, without any passage through an intermediate QSS. This may be explained by different energy localization properties and recurrence phenomena in the two cases, supporting the view that when the energy is placed in the first mode weak chaos and "sticky" dynamics lead to a more gradual process of energy sharing, while strong chaos and equipartition appear abruptly when only the last mode is initially excited.

The space-time phases of a complex dynamic system are the probability distributions for state as a function of space and time which arise by evolving initial probability distributions from the distant past. Toom proved important results about space-time phases for a class of majority voter probabilistic cellular automata (PCA). Here, variants of the majority voter PCA are presented which are proved to exhibit a variety of types of space-time phase. These examples are expected to serve as useful steps on the way to a general theory of space-time phases.

We study a minimal model for the flow of granular material on a conveyor belt consisting of a staircase-like array of K vertically vibrated compartments. Applying a steady inflow rate Q to the top compartment, we determine the maximum value Q_cr(K) for which a continuous flow through the system is possible. Beyond Q_cr(K), which depends on the vibration strength and the dimensions of the system, a dense cluster forms in one of the first compartments and obstructs the flow. We find that the formation of this cluster is already announced belowQ_cr(K) by the appearance of an oscillatory density profile along the entire length of the conveyor belt, with a distinct two-compartment wavelength.

These model predictions concerning the breakdown of the granular flow admit an elegant explanation in terms of bifurcation theory. In particular, the subcritical oscillatory pattern is shown to be a side effect of the period doubling bifurcation by which the uniform density profile (associated with a smooth particle flow) becomes unstable. The effect turns out to be robust enough to survive the presence of a reasonable amount of noise and even certain qualitative modifications to the flux model. The density oscillations may therefore well be of practical value and provide a warning signal for imminent clustering on actual conveyor belts.

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis & Zachilas, 1994] in order to visualize them. As examples, we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition, we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

This paper considers the processes of torus formation and reconstruction through smooth and nonsmooth bifurcations in a pulse-width modulated DC/DC converter with multilevel control. When operating in a regime of high corrector gain, converters of this type can generate structures of stable tori embedded one into the other and with their basins of attraction delineated by intervening repelling tori. The paper illustrates the coexistence of three stable tori with different resonance behaviors and shows how reconstruction of these tori takes place across the borders of different dynamical regimes. The paper also demonstrates how pairs of attracting and repelling tori emerge through border-collision torus-birth and border-collision torus-fold bifurcations.

The dynamic behavior of bidirectionally coupled Duffing-type circuits is numerically and experimentally investigated. Two related phenomena are explored such as anti-phase synchronization and inverse π-lag synchronization. Anti-phase synchronization is observed for the first time in coupled Duffing-type systems, while the inverse π-lag synchronization phenomenon in mutually coupled identical oscillators is a new type of synchronization. Inverse π-lag synchronization can be characterized by the vanishing of the sum of two relevant periodic signals. The first one is the signal of the first circuit x₁ and the second is the signal of the second circuit x₂ with a time lag τ_min. This time lag is equal to T/2, where T is the period of x₁ and x₂. The measurements and numerical computations suggest that these two synchronization phenomena are correlated with the symmetry under the transformation, x → -x, y → -y and t → t + T/2, which the Duffing equation exhibits.

In this work, we introduce and investigate the modified projective lag synchronization (MPLS) of two nonidentical hyperchaotic complex nonlinear systems. The idea of an active control technique based on complex Lyapunov function with lag in time is used for an approach to investigate MPLS of hyperchaotic attractors of these systems. For illustration, this approach is applied to hyperchaotic complex Chen and Lü systems. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve MPLS.

Due to the considerable complexity involved in the numerical solution of problems of phase change materials, the development of fast parallel algorithms to solve them is very useful. This large complexity can be reduced by developing faster methods for solving, and suitably modifying programming techniques. We consider the phase field model of the system of p.d.e's