Synchronized oscillations on a Kuramoto ring and their entrainment under periodic driving

We consider a finite number of coupled oscillators on a ring as an adaptation of the Kuramoto model of populations of oscillators. The synchronized solutions are characterized by an integer m, the winding number, and a second integer l, with solutions of type (m, l = 0) being all stable. Following a number of recent works (see below) we indicate how the various solutions emerge as the coupling strength K is varied, presenting a perturbative expression for these for large K. The low K scenario is also briefly outlined, where the onset of synchronization by a tangent bifurcation is explained. The simplest situation involving three oscillators is described, where more than one tangent bifurcations are involved. Immediately before the tangent bifurcation leading to synchronization, the system exhibits the phenomenon of frequency- (or phase) splitting where more than one (usually two) phase clusters are involved. All the synchronized solutions are seen to be entrained by an external periodic driving, provided that the driving frequency is sufficiently close to the frequency of the synchronized population. A perturbative approach is outlined for the construction of the entrained solutions. Under a periodic driving with an appropriately limited detuning, there occurs entrainment of the phase-split solutions as well.     

Synchronization of two low-dimensional Kerr chains

Synchronization of two chaotic low-dimensional chains (α₁, α₂, α₃) and (A₁, A₂, A₃) consisting of Kerr oscillators is studied. The synchronization has been achieved by the parallel coupling of α₁ with A₁, α₂ with A₂ and α₃ with A₃. We want to find whether and when the pairs (α₁, A₁), (α₂, A₂) and (α₃, A₃) synchronize non-simultaneously (three-time synchronism). The problem of synchronization is also studied for a number of couplings between the chains lower than the number of oscillators in a single chain. Both the ring and linear geometry of synchronization is investigated. The presented results suggest a possibility of multi-time synchronism in two coupled high-dimensional chains. It seems very promising for design of some devices for advanced signal processing.     

Inverse problems for parabolic equations 2

Let u_t − u_xx = h(t) in 0 ⩽ x ⩽ π, t ⩾ 0. Assume that u(0, t) = v(t), u(π, t) = 0, and u(x, 0) = g(t). The problem is: what extra data determine the three unknown functions {h, v, g} uniquely? This question is answered and an analytical method for recovery of the above three functions is proposed.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

Homotopy analysis of unsteady boundary layer flow adjacent to permeable stretching surface in a porous medium

This communication deals with the unsteady boundary layer flow of a viscous fluid in porous medium started due to the impulsively stretching of the plane wall. The wall is assumed to be porous so that suction or injection is possible. Complete analytic solution which is uniformly valid for all the dimensionless times 0 ⩽ τ < 0 in the whole spatial region 0 ⩽ η < ∞ is obtained by a purely analytic technique, namely the homotopy analysis method. Results are discussed through graphs.     

Explicit series solution of travelling waves with a front of Fisher equation

In this paper, an analytic technique, namely the homotopy analysis method, is employed to solve the Fisher equation, which describes a family of travelling waves with a front. The explicit series solution for all possible wave speeds 0 < c < +∞ is given. Such kind of explicit series solution has never been reported, to the best of author’s knowledge. Our series solution indicates that the solution contains an oscillation part when 0 < c < 2. The proposed analytic approach is general, and can be applied to solve other similar nonlinear travelling wave problems.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method

Based on a new kind of analytic method, namely the Homotopy analysis method, an analytic approach to solve non-linear, chaotic system of ordinary differential equations is presented. The method is applied to Lorenz system; this system depends on the three parameters: σ, b and the so-called bifurcation parameter R are real constants. Two cases are considered. The first case is when R = 20.5 which corresponds to the transition region and the second case corresponds to R = 23.5 which corresponds to the chaotic region.The validity of the method is verified by comparing the approximation series solution with the results obtained using the standard numerical techniques such as Runge-Kutta method.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Heat transfer analysis of unsteady boundary layer flow by homotopy analysis method

This paper aims to present complete analytic solution to the unsteady heat transfer flow of an incompressible viscous fluid over a permeable plane wall. The flow is started due to an impulsively stretching porous plate. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The solution is uniformly valid for all time τ ∈ [0, ∞) throughout the spatial domain η ∈ [0, ∞). The accuracy of the present results is shown by giving a comparison between the present results and the results already present in the literature. This comparison proves the validity and accuracy of our present results. Finally, the effects of different parameters on temperature distribution are discussed through graphs.     

Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (u_t + 6uu_x + u_xxx)_x ± 3u_yy ± 3u_zz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Painlevé test. As by-products, the one-soliton and two-soliton solutions and four classes of specific three-soliton solutions are explicitly presented.     

Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.     

Analytic integrability of the Bianchi class A cosmological models with 0 ⩽ k < 1

Many works study the integrability of the Bianchi class A cosmologies with k = 1, where k is the ratio between the pressure and the energy density of the matter. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0 ⩽ k < 1. We conclude that Bianchi types VI₀, VII₀, VIII and IX can exhibit chaos whereas Bianchi type I is not chaotic and Bianchi type II is at most partially chaotic.     

Behavior of different ansätze in the generalized projection operator method

Using the recently reported generalized projection operator method for the nonlinear Schrödinger equation, we derive the generalized pulse parameters equations for ansätze like hyperbolic secant and raised cosine functions. In general, each choice of the phase factor θ in the projection operator gives a different set of ordinary differential equations. For θ = 0 or θ = π/2, the corresponding projection operator scheme is equivalent to the Lagrangian variation method or the bare approximation of the collective variable theory. We prove that because of the inherent symmetric property between the pulse parameters of a Gaussian ansätz results the same set of pulse parameters equations for any value of the generalized projection operator parameter θ. Finally we prove that after the substitution of the ansätze function, the Lagrange function simplifies to the same functional form irrespective of the ansätze used because of a special property shared by all the anätze chosen in this work.     

Delayed transitions in non-linear replicator networks: About ghosts and hypercycles

In this paper we analyze delayed transition phenomena associated to extinction thresholds in a mean field model for hypercycles composed of three and four units, respectively. Hence, we extend a previous analysis carried out with the two-membered hypercycle [see Sardanyés J, Solé RV. Ghosts in the origins of life? Int J Bifurcation Chaos 2006;16(9), in press]. The models we analyze show that, after the tangent bifurcation, these hypercycles also leave a ghost in phase space. These ghosts, which actually conserve the dynamical properties of the coalesced coexistence fixed point, delay the flows before hypercycle extinction. In contrast with the two-component hypercycle, both ghosts show a plateau in the delay as ϕ → 0, thus displacing the power-law dependence to higher values of ϕ, in which the scaling law is now given by τ ∼ ϕ^β, with β = −1/3 (where τ is the delay and ϕ = ϵ − ϵ_c, the parametric distance above the extinction bifurcation point). These results suggest that the presence of the ghost is a general property of hypercycles. Such ghosts actually cause a memory effect which might increase hypercycle survival chances in fluctuating environments.     

Lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs

The minimum semidefinite rank (msr) of a graph is the minimum rank among positive semidefinite matrices with the given graph. The OS-number is a useful lower bound for msr, which arises by considering ordered vertex sets with some connectivity properties. In this paper, we develop two new interpretations of the OS-number. We first show that OS-number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions. Our second interpretation of the OS-number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation-preserving. These interpretations not only give insight into the OS-number, but also allow us to prove some new results. For example we show that msr(G) = |G| ? 2 if and only if OS(G) = |Gzsfnc ? 2.     

Statistical properties of nucleotides in human chromosomes 21 and 22

In this paper the statistical properties of nucleotides in human chromosomes 21 and 22 are investigated. The n-tuple Zipf analysis with n = 3, 4, 5, 6, and 7 is used in our investigation. It is found that the most common n-tuples are those which consist only of adenine (A) and thymine (T), and the rarest n-tuples are those in which GC or CG pattern appears twice. With the n-tuples become more and more frequent, the double GC or CG pattern becomes a single GC or CG pattern. The percentage of four nucleotides in the rarest ten and the most common ten n-tuples are also considered in human chromosomes 21 and 22, and different behaviors are found in the percentage of four nucleotides. Frequency of appearance of n-tuple f(r) as a function of rank r is also examined. We find the n-tuple Zipf plot shows a power-law behavior for r < 4ⁿ⁻¹ and a rapid decrease for r > 4ⁿ⁻¹. In order to explore the interior statistical properties of human chromosomes 21 and 22 in detail, we divide the chromosome sequence into some moving windows and we discuss the percentage of ξη (ξ, η = A, C, G, T) pair in those moving windows. In some particular regions, there are some obvious changes in the percentage of ξη pair, and there maybe exist functional differences. The normalized number of repeats N₀(l) can be described by a power law: N₀(l) ∼ l^−μ. The distance distributions P₀(S) between two nucleotides in human chromosomes 21 and 22 are also discussed. A two-order polynomial fit exists in those distance distributions: log P₀(S) = a + bS + cS², and it is quite different from the random sequence.     

Commutativity and self-duality: Two tales of one equation

The mathematical expressions for the commutativity or self-duality of an increasing [0, 1]² → [0, 1] function F involve the transposition of its arguments. We unite both properties in a single functional equation. The solutions of this functional equation are discussed. Special attention goes to the geometrical construction of these solutions and their characterization in terms of contour lines. Furthermore, it is shown how ‘rotating’ the arguments of F allows to convert the results into properties for [0, 1]² → [0, 1] functions having monotone partial functions.     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.