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.     

Visibility graph approach to the analysis of ocean tidal records

By using the recent method of the visibility graph, three time series of oceanic tide level in central Argentina were investigated. The degree distributions show a rich structure; in particular the maximum is due to the main periodic oscillations at 24 hours and 12 hours and higher harmonics. The degree distributions of the residuals (obtained removing from the original signals the cyclic components) suggest that the local effects, linked with the particular coastal conditions of the sites, are discernible for the degree k < 20, while the global effects, linked with linked with the more general and common atmospheric forcing and ocean current conditions, are visible for k > 100. Although a relationship between the spectral exponent α and the exponent of the degree distribution γ of tidal signals can be recognized, this cannot be simply stated due to the very rich and complex structure of time dynamics of tides. The present study, even if still preliminary, show the importance of the visibility graph method in investigating the complex time dynamics of observational and experimental signals.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Development of a Neuro-Fuzzy approach for treating multimodal distributions and spurious clusters

In this paper, we introduce a developed approach of neuro-fuzzy model free estimators to treat decision problems, where presence of highly multimodal distribution and spurious clusters in decision space causes problems in estimation. After critical analysis of Bart Kosko's DCL–AVQ + FAM neuro-fuzzy system, we developed our Straitjacket + F-wing system. Straitjacket neural system is an improved version of Kohonen's feature mapping according to convergence and termination conditions. F-wing fuzzy system bases on the feature map produced by Straitjacket and provides faster and more exact estimator interface than Bart Kosko's FAM, using fuzzy wing functions instead of fuzzy hyperpyramids. At the end of the paper we perform a test on artificial database comparing the efficiency of our approach with Kosko's system, discriminant analysis, hierarchic and K-mean clustering.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

Stabilization via parametric excitation of multi-dof statically unstable systems

The problem of re-stabilization via parametric excitation of statically unstable linear Hamiltonian systems is addressed. An n-degree-of-freedom dynamical system is considered, at rest in a critical equilibrium position, possessing a pair of zero-eigenvalues and n − 1 pairs of distinct purely imaginary conjugate eigenvalues. The response of the system to a small static load, making the zero eigenvalues real and opposite, simultaneous to a harmonic parametric excitation of small amplitude, is studied by the Multiple Scale perturbation method, and the stability of the equilibrium position is investigated. Several cases of resonance between the excitation frequency and the natural non-zero frequencies are studied, calling for standard and non-standard applications of the method. It is found that the parametric excitation is able to re-stabilize the equilibrium for any value of the excitation frequencies, except for frequencies close to resonant values, provided a sufficiently large excitation amplitude is enforced. Results are compared with those provided by a purely numerical approach grounded on the Floquet theory.     

Entropy estimation of the Hénon attractor

The topological entropy of the Hénon attractor is estimated using a function that describes the stable and unstable manifolds of the Hénon map. This function provides an accurate estimate of the length of curves in the attractor. The estimation method presented here can be applied to cases in which the invariant set is not hyperbolic. From the result of the length calculation, we have estimated the topological entropy h as h ～ 0.49703 for the original parameters a = 1.4 and b = 0.3 adopted by Hénon.     

The necessary and sufficient conditions for the stability of linear systems with an arbitrary delay

A system of linear differential equations with a Hurwitz matrix A and a variable delay is considered. The system is assumed to be stable if it is stable for any delay function τ(t) ≤ h. The necessary and sufficient condition for stability, expressed using the eigenvalues of the matrix A and the quantity h, is found. It is established that the function τ(t), corresponding to the critical value of h, is constant or piecewise-linear depending on to which eigenvalue of matrix A (complex or real respectively) it corresponds. In the first case, the critical values of h in systems with a variable and constant delay are identical and, in the second case, they differ very slightly.     

Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

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.     

Thermal noise engines

In this paper, electrical heat engines driven by the Johnson-Nyquist noise of resistors are introduced. They utilize Coulomb’s law and the fluctuation–dissipation theorem of statistical physics. In these engines, resistors, capacitors and switches are the building elements. For best performance, a large number of parallel engines must be integrated to run in a synchronized fashion, and the size of an elementary engine must be at the 10 nm scale. At room temperature, in the most idealistic case, a two-dimensional ensemble of engines of 25 nm size integrated on a 2.5 × 2.5 cm silicon wafer with 10 °C temperature difference between the warm-source and the cold-sink would produce a power of about 0.5 W. Regular and coherent (correlated-cylinder states) versions of these engines are shown and both of them can operate in either four-stroke or two-stroke modes. In the idealistic case, all these engines have Carnot efficiency, which is the highest efficiency possible in any heat engine without violating the second law of thermodynamics.     

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.     

Second-order approximation of angle-ply composite laminated thin plate under combined excitations

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω₂ ? 2ω₁ and Ω₂ ? 2ω₂) in the presence of 1:2 internal resonance ω₂ ? 2ω₁ are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.     

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/q (q > 1). This result is more general than what they deal with the jump of deterministic magnitude.     

The lower and upper forcing geodetic numbers of block–cactus graphs

A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u–v path of G, where u, v ∈ D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this paper, we find out the lower and the upper forcing geodetic numbers of block–cactus graphs.     

Effect of hot ion temperature on obliquely propagating ion-acoustic solitons and double layers in an auroral plasma

Properties of obliquely propagating ion-acoustic solitons and double layers in a magnetized auroral plasma composed of hot adiabatic ions and two types of, cool and hot Maxwellian electrons are studied using Sagdeev pseudo-potential technique and assuming the quasi-neutrality condition. The new and surprising result which emerges from the model is that in contrast to the case of cold ions where ion-acoustic solitons and double layers are found for subsonic Mach numbers only, the hot ions case allows these nonlinear structures to exist for both subsonic and supersonic Mach number regimes. The double layers exist at lower angle of propagation as hot ion temperature is increased. The soliton electric field amplitudes are increased but their width and pulse duration are decreased with the increase in hot ion temperature. For the auroral zone parameters, the maximum electric field amplitude, width, pulse duration and speed for the solitons come out to be in the range ∼ (0.3–15) mV/m, ∼ (195–455) m, (7–20) ms and (22–26) km/s, respectively. The results seem to be in agreement with the Viking satellite observations in the auroral zone.     

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.     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6) ⩾ 35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

Dynamics of landslide model with time delay and periodic parameter perturbations

In present paper, we analyze the dynamics of a single-block model on an inclined slope with Dieterich–Ruina friction law under the variation of two new introduced parameters: time delay T_d and initial shear stress μ. It is assumed that this phenomenological model qualitatively simulates the motion along the infinite creeping slope. The introduction of time delay is proposed to mimic the memory effect of the sliding surface and it is generally considered as a function of history of sliding. On the other hand, periodic perturbation of initial shear stress emulates external triggering effect of long-distant earthquakes or some non-natural vibration source. The effects of variation of a single observed parameter, T_d or μ, as well as their co-action, are estimated for three different sliding regimes: β < 1, β = 1 and β > 1, where β stands for the ratio of long-term to short-term stress changes. The results of standard local bifurcation analysis indicate the onset of complex dynamics for very low values of time delay. On the other side, numerical approach confirms an additional complexity that was not observed by local analysis, due to the possible effect of global bifurcations. The most complex dynamics is detected for β < 1, with a complete Ruelle–Takens–Newhouse route to chaos under the variation of T_d, or the co-action of both parameters T_d and μ. These results correspond well with the previous experimental observations on clay and siltstone with low clay fraction. In the same regime, the perturbation of only a single parameter, μ, renders the oscillatory motion of the block. Within the velocity-independent regime, β = 1, the inclusion and variation of T_d generates a transition to equilibrium state, whereas the small oscillations of μ induce oscillatory motion with decreasing amplitude. The co-action of both parameters, in the same regime, causes the decrease of block’s velocity. As for β > 1, highly-frequent, limit-amplitude oscillations of initial stress give rise to oscillatory motion. Also for β > 1, in case of perturbing only the initial shear stress, with smaller amplitude, velocity of the block changes exponentially fast. If the time delay is introduced, besides the stress perturbation, within the same regime, the co-action of T_d (T_d < 0.1) and small oscillations of μ induce the onset of deterministic chaos.