Nonlinear Sciences

In this paper the chaos control in the discrete logistic map of fractional order is obtained with an impulsive control algorithm. The underlying discrete initial value problem of fractional order is considered in terms of Caputo delta fractional difference. Every Δ steps, the state variable is instantly modified with the same impulse value, chosen from a bifurcation diagram versus impulse. It is shown that the solution of the impulsive control is bounded. The numerical results are verified via time series, histograms, and the 0-1 test. Several examples are considered

Euler's three-body problem is the problem of solving for the motion of a particle moving in a Newtonian potential generated by two point sources fixed in space. This system is integrable in the Liouville sense. We consider the Euler problem with the inverse-square potential, which can be seen as a natural generalization of the three-body problem to higher-dimensional Newtonian theory. We identify a family of stable stationary orbits in the generalized Euler problem. These orbits guarantee the existence of stable bound orbits. Applying the Poincaré map method to these orbits, we show that stable bound chaotic orbits appear. As a result, we conclude that the generalized Euler problem is nonintegrable.

The time needed to exchange information in the physical world induces a delay term when the respective system is modeled by differential equations. Time delays are hence ubiquitous, being furthermore likely to induce instabilities and with it various kinds of chaotic phases. Which are then the possible types of time delays, induced chaotic states, and methods suitable to characterize the resulting dynamics? This review presents an overview of the field that includes an in-depth discussion of the most important results, of the standard numerical approaches and of several novel tests for identifying chaos. Special emphasis is placed on a structured representation that is straightforward to follow. Several educational examples are included in addition as entry points to the rapidly developing field of time delay systems.

Transition from quasiperiodicity with many frequencies (i.e., a high-dimensional torus) to chaos is studied by using N -dimensional globally coupled circle maps. First, the existence of N -dimensional tori with N≥2 is confirmed while they become exponentially rare with N . Besides, chaos exists even when the map is invertible, and such chaos has more null Lyapunov exponents as N increases. This unusual form of "chaos on a torus," termed toric chaos, exhibits delocalization and slow dynamics of the first Lyapunov vector. Fractalization of tori at the transition to chaos is also suggested. The relevance of toric chaos to neural dynamics and turbulence is discussed in relation to chaotic itinerancy.

Noise play a creative role in the evolution of periodic and complex systems which are essential for continuous performance of the system. The interaction of noise generated within one component of a chaotic system with other component in a linear or nonlinear interaction is crucial for system performance and stability. These types of noise are inherent, natural and insidious. This study investigates the effect of state-dependent noise on the bifurcation of two chaotic systems. Circuit realization of the systems were implemented. Numerical simulations were carried out to investigate the influence of state dependent noise on the bifurcation structure of the Chen and Arneodo-Coullet fractional order chaotic systems. Results obtained showed that state dependent noise inhibit the period doubling cascade bifurcation structure of the two systems. These results poses serious challenges to system reliability of chaotic systems in control design, secure communication and power systems.

Recently, there has been provided two chaotic models based on the twist-deformation of classical Henon-Heiles system. First of them has been constructed on the well-known, canonical space-time noncommutativity, while the second one on the Lie-algebraically type of quantum space, with two spatial directions commuting to classical time. In this article, we find the direct link between mentioned above systems, by synchronization both of them in the framework of active control method. Particularly, we derive at the canonical phase-space level the corresponding active controllers as well as we perform (as an example) the numerical synchronization of analyzed models.

We investigate the chaotic behaviour of multiparticle systems, in particular DNA and graphene models, by applying methods of nonlinear dynamics. Using symplectic integration techniques, we present an extensive analysis of chaos in the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity is quantified by the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair (BP) disorder on the dynamics is studied. In addition to heterogeneity due to the ratio of adenine-thymine (AT) and guanine-cytosine (GC) BPs, the distribution of BPs in the sequence is analysed by introducing the alternation index α . An exact probability distribution for BP arrangements and α is derived using Pólya counting. The value of the mLE depends on the composition and arrangement of BPs in the strand, with a dependence on temperature. We probe regions of strong chaoticity using the deviation vector distribution, studying links between strongly nonlinear behaviour and the formation of bubbles. Randomly generated sequences and biological promoters are both studied. Further, properties of bubbles are analysed through molecular dynamics simulations. The distributions of bubble lifetimes and lengths are obtained, fitted with analytical expressions, and a physically justified threshold for considering a BP to be open is successfully implemented. In addition to DNA, we present analysis of the dynamical stability of a planar model of graphene, studying the mLE in bulk graphene as well as in graphene nanoribbons (GNRs). The stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both armchair and zigzag edge GNRs, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.

The Chaplygin sleigh is a classic example of a nonholonomically constrained mechanical system. The sleigh's motion always converges to a straight line whose slope is entirely determined by the initial configuration and velocity of the sleigh. We consider the motion of a modified Chaplygin sleigh that contains a passive internal rotor. We show that the presence of even a rotor with small inertia modifies motion of the sleigh dramatically. A generic trajectory of the sleigh in a reduced velocity space exhibits two distinct transient phases before converging to a chaotic attractor. We demonstrate this through numerics. In recent work the dynamics of the Chaplygin sleigh have also been shown to be similar to that of a fish like body in an inviscid fluid. The influence of a passive degree of freedom on the motion of the Chaplygin sleigh points to several possible applications in controlling the motion of the nonholonomically constrained terrestrial and aquatic robots.

It is well known that periodic forcing of a nonlinear system, even of a two-dimensional autonomous system, can produce chaotic responses with sensitive dependence on initial conditions if the forcing induces sufficient stretching and folding of the phase space. Quasiperiodic forcing can similarly produce chaotic responses, where the transition to chaos on changing a parameter can bring the system into regions of strange non-chaotic behaviour. Although it is generally acknowledged that the timings of Pleistocene ice ages are at least partly due to Milankovitch forcing (which may be approximated as quasiperiodic, with energy concentrated near a small number of frequencies), the precise details of what can be inferred about the timings of glaciations and deglaciations from the forcing is still unclear. In this paper, we perform a quantitative comparison of the response of several low-order nonlinear conceptual models for these ice ages to various types of quasiperiodic forcing. By computing largest Lyapunov exponents and mean periods, we demonstrate that many models can have a chaotic response to quasiperiodic forcing for a range of forcing amplitudes, even though some of the simplest conceptual models do not. These results suggest that pacing of ice ages to forcing may have only limited determinism.

We demonstrate that the synergistic effect of a gauge field, Rashba spin-orbit coupling (SOC), and Zeeman splitting can generate chaotic cyclotron and Hall trajectories of particles. The physical origin of the chaotic behavior is that the SOC produces a spin-dependent (so-called anomalous) contribution to the particle velocity and the presence of Zeeman field reduces the number of integrals of motion. By using analytical and numerical arguments, we study the conditions of chaos emergence and report the dynamics both in the regular and chaotic regimes. {We observe the critical dependence of the dynamic patterns (such as the chaotic regime onset) on small variations in the initial conditions and problem parameters, that is the SOC and/or Zeeman constants. The transition to chaotic regime is further verified by the analysis of phase portraits as well as Lyapunov exponents spectrum.} The considered chaotic behavior can occur in solid state systems, weakly-relativistic plasmas, and cold atomic gases with synthetic gauge fields and spin-related couplings.