Characterization of chaotic attractors at bifurcations in Murali–Lakshmanan–Chua's circuit and one-way coupled map lattice system

In the present paper we study certain characteristic features associated with bifurcations of chaos in a finite dimensional dynamical system – Murali–Lakshmanan–Chua (MLC) circuit equation and an infinite dimensional dynamical system – one-way coupled map lattice (OCML) system. We characterize chaotic attractors at various bifurcations in terms of σ_n(q) – the variance of fluctuations of coarse-grained local expansion rates of nearby orbits. For all chaotic attractors the σ_n(q) versus q plot exhibits a peak at q=q_α. Additional peaks, however, are found only just before and just after the bifurcations of chaos. We show power-law variation of maximal Lyapunov exponent near intermittency and sudden widening bifurcations. Linear variation is observed for band-merging bifurcation. We characterize weak and strong chaos using probability distribution of k-step difference of a state variable.     

Natural Volterra Runge-Kutta methods

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V ₀-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4.     

Nonlinear vibration and characteristics of flexible plate-strips with non-symmetric boundary conditions

Regular and chaotic vibrations together with bifurcations of flexible plate-strips with non-symmetric boundary conditions, are investigated through the Bubnov–Galerkin method and a finite difference method of error O(h⁴). Particular attention is paid to non-symmetric boundary conditions. Lyapunov exponents are estimated via Bennetin’s method. Some new examples of routes from regular to chaotic dynamics, and within chaotic dynamics are illustrated and discussed. The phase transitions from chaos to hyperchaos, and a novel phenomenon of a shift from hyperchaos to hyperhyper chaos is also reported.     

Dynamics of coupled non-identical systems with period-doubling cascade

We discuss the structure of bifurcation diagram in the plane of parameters controlling period-doublings for the system of coupled logistic maps. The analysis is carried out by computing the charts of dynamical regimes and charts of Lyapunov exponents giving showy and effective illustrations. The critical point of codimension two at the border of chaos is found. It is a terminal point for the Feigenbaum critical line. The bifurcation analysis in the vicinity of this point is presented.      

Chaotic vibrations of flexible infinite length cylindrical panels using the Kirchhoff–Love model

Treated as continuous deformable systems with an infinite number of degrees of freedom, flexible infinite length cylindrical panels subject to harmonic load are studied. Using the finite difference method with respect to spatial coordinates, the continuous system is reduced to lumped one governed by ordinary differential equations. These equations are transformed to a normal form and then solved numerically using the fourth order Runge–Kutta method. In order to trace and explain vibrational behaviour, dependencies w_max(q₀) and Lyapunov exponents are calculated for panels with parameter value k_x = 48. The corresponding charts of the control parameters {q₀, ω_q} are also reported. Novel scenarios yielding chaotic dynamics exhibited by cylindrical panels are illustrated and discussed.     

The Stokes phenomenon for the q-difference equation satisfied by the Ramanujan entire function

We show connection formulae between the origin and infinity for local solutions of the q-difference equation satisfied by the Ramanujan entire function. These solutions are given by the Ramanujan entire function, the q-Airy function, and the divergent basic hypergeometric series ₂ φ ₀(0,0;?;q,x). We use two different q-Borel–Laplace resummation methods to obtain our connection formulae.     

Traveling Waves and Space-Time Chaos in the Kuramoto–Sivashinsky Equation

The transition to space-time chaos in the Kuramoto–Sivashinsky equation through cascades of traveling wave bifurcations in accordance with the Feigenbaum–Sharkovskii–Magnitskii universal bifurcation scenario is analyzed analytically and numerically. It is proved that the bifurcation parameter is the traveling wave propagation velocity along the spatial axis, which does not explicitly occur in the original equation.     

Phase-locking and chaos in a silent Hodgkin–Huxley neuron exposed to sinusoidal electric field

Neuronal firing patterns are related to the information processing in neural system. This paper investigates the response characteristics of a silent Hodgkin–Huxley neuron to the stimulation of externally-applied sinusoidal electric field. The neuron exhibits both p:q phase-locked (i.e. a periodic oscillation defined as p action potentials generated by q cycle stimulations) and chaotic behaviors, depending on the values of stimulus frequencies and amplitudes. In one-parameter space, a rich bifurcation structure including period-adding without chaos and phase-locking alternated with chaos suggests frequency discrimination of the neuronal firing patterns. Furthermore, by mapping out Arnold tongues, we partition the amplitude–frequency parameter space in terms of the qualitative behaviors of the neuron. Thus the neuron’s information (firing patterns) encodes the stimulus information (amplitude and frequency), and vice versa.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

On implicit Runge-Kutta methods with a stability function having distinct real poles

Because of their potential for offering a computational speed-up when used on certain multiprocessor computers, implicit Runge-Kutta methods with a stability function having distinct poles are analyzed. These are calledmultiply implicit (MIRK) methods, and because of the so-calledorder reduction phenomenon, their poles are required to be real, i.e., only real MIRK's are considered. Specifically, it is proved that a necessary condition for aq-stage, real MIRK to beA-stable with maximal orderq+1 is thatq=1, 2, 3 or 5. Nevertheless, it is shown that for every positive integerq, there exists aq-stage, real MIRK which is stronglyA ₀-stable with orderq+1, and for every evenq, there is aq-stage, real MIRK which isI-stable with orderq. Finally, some useful examples of algebraically stable real MIRK's are given.This work was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665-5225.     

On the second weight of generalized Reed-Muller codes

Not much is known about the weight distribution of the generalized Reed-Muller code RM _q (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m – 1)(q – 1) we then find the first s + 1 – (m – 1)(q–1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.     

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods withq1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist.This research has been supported by project PB89-0351 (Dirección General de Investigación Científica y Técnica) and by project No. 20-32354.91 of Swiss National Science Foundation.     

Chaotic convergence of the decision-directed blind equalization algorithm

Classically, adaptive equalization algorithms are analyzed in terms of two possible steady state behaviors: convergence to a fixed point and divergence to infinity. This twofold scenario suits well the modus operandi of linear supervised algorithms, but can be rather restrictive when unsupervised methods are considered, as their intrinsic use of higher-order statistics gives rise to nonlinear update expressions. In this work, we show, using different analytical tools belonging to dynamic system theory, that one of the most emblematic and studied unsupervised approaches – the decision-directed algorithm – is potentially capable of presenting behaviors, like convergence to limit-cycles and chaos, that transcend the aforementioned dichotomy. These results also indicate theoretical possibilities concerning step-size selection and initialization.     

Construction of steiner quadruple systems having a prescribed number of blocks in common

Let q_υ=υ(υ–1)(υ–2)/24 and let I_υ={0, 1, 2, …, q_{υ–14}∪{qυ}–12, q_υ–8, q_υ}, for υ?8 Further, let J[υ] denote the set of all k such that there exists a pair of Steiner quadruple systems of order υ having exactly k blocks in common. We determine J[υ] for all υ=2ⁿ, n?2, with the possible exception of 7 cases for υ=16 and of 5 cases for each υ?32. In particular we show: J[υ]?I_υ for all υ≡2 or 4 (mod 6) and υ?8, J[4]={1}, J[8]=I₈={0, 2, 6, 14}, I₁₆?{103, 111, 115, 119, 121, 122, 123}?J[16], and I_υ? {q_υ–h:h=17, 18, 19, 21, 25}?J[υ] for all υ=2ⁿ, n?5.     

Poisson statistics via the Chinese Remainder Theorem

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ, where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z/q₁q₂Z that are created via the Chinese Remainder Theorem from subsets of Z/q₁Z and Z/q₂Z (for q₁,q₂ coprime), and give criteria for when the spacings modulo q₁q₂ are Poisson. Moreover, we also give some examples when the spacings modulo q₁q₂ are not Poisson, even though the spacings modulo q₁ and modulo q₂ are both Poisson.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

Heegaard–Floer homologies of (+1) surgeries on torus knots

We compute the Heegaard–Floer homology of $S^{3}_{1}(K)$ (the (+1) surgery on the torus knot T _p,q ) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsváth–Szabó d-invariant. We relate the result to known knot invariants of T _p,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (?1) surgeries on torus knots. This relation is best seen at the level of τ functions.     

Symmetries of explicit Runge-Kutta methods

A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in Khashin (Can. Appl. Math. Q. 17(1), 555–569, 2009, Numer. Algorithm, 61(2), 1–11, 2012). The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher’s algebras. Here we introduce auxiliary varieties ? _D and prove that they are projective algebraic varieties (Theorem 3.2). In some cases they are completely described (Theorem 3.5). On the set of the 2-standard matrices (Definition 4.4) (RK methods with the property b ₂ = 0) the one-dimensional symmetries are introduced. These symmetries allow to reduce consideration of the RK methods to the methods with c ₂ = 2c ₃/3, that is c ₂can be removed from the list of unknowns. We formulate a hypothesis on how this method can be generalized to the case b ₂ = b ₃ = 0 where two-dimensional symmetries appear.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators

We consider in this paper the synchronization dynamics of coupled chaotic Van der Pol–Duffing systems. We first find that with the judicious choose of the set of initial conditions, the model exhibits two strange chaotic attractors. The problem of synchronizing chaos both on the same and different chaotic orbits of two coupled Van der Pol–Duffing systems is investigated. The stability boundaries of the synchronization process between two coupled driven Van der Pol model are derived and the effects of the amplitude of the periodic perturbation of the coupling parameter on these boundaries are analyzed. The results are provided on the stability map in the (q, K) plane.