Nikov Type Research Articles

Existence of heteroclinic orbits of the Shil'nikov type in a 3D quadratic autonomous chaotic system

In this paper, the existence of heteroclinic orbits of Shil'nikov type in a three-dimensional quadratic autonomous system is proved. Four heteroclinic orbits and four critical points together constitute two cycles simultaneously. The dynamical behaviors of the system are also studied.

Robust heteroclinic cycles in the one-dimensional complex Ginzburg–Landau equation

Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one-dimensional complex Ginzburg–Landau equation (CGL) on the unit, spatially periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions as t → ± ∞ . A careful analysis of the connections is made using a projection onto five complex Fourier modes. It is shown first that the time-periodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincaré maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in specific parameter regions where the cycles are found to be of Shil’nikov type. This criterion is also applied to a much higher-mode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shil’nikov–Hopf or blow-out bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatio-temporal intermittency in situations modelled by the CGL are discussed.

From scale invariance to deterministic chaos in DNA sequences: towards a deterministic description of gene organization in the human genome

We use the continuous wavelet transform to perform a space-scale analysis of the AT and GC skews (strand asymmetries) in human genomic sequences, which have been shown to correlate with gene transcription. This study reveals the existence of a characteristic scale ℓ c≃25±10 kb that separates a monofractal long-range correlated noisy regime at small scales (ℓ<ℓ c ) from relaxational oscillatory behavior at large-scale (ℓ>ℓ c ). We show that these large scale nonlinear oscillations enlighten an organization of the human genome into adjacent domains ( ≈400 kb ) with preferential gene orientation. When using classical techniques from dynamical systems theory, we demonstrate that these relaxational oscillations display all the characteristic properties of the chaotic strange attractor behavior observed nearby homoclinic orbits of Shil'nikov type. We discuss the possibility that replication and gene regulation processes are governed by a low-dimensional dynamical system that displays deterministic chaos.

From scale invariance to deterministic chaos in DNA sequences: towards a deterministic description of gene organization in the human genome

We use the continuous wavelet transform to perform a space-scale analysis of the AT and GC skews (strand asymmetries) in human genomic sequences, which have been shown to correlate with gene transcription. This study reveals the existence of a characteristic scale ℓ c ≃25±10 kb that separates a monofractal long-range correlated noisy regime at small scales (ℓ<ℓ c ) from relaxational oscillatory behavior at large-scale (ℓ>ℓ c ). We show that these large scale nonlinear oscillations enlighten an organization of the human genome into adjacent domains ( ≈400 kb ) with preferential gene orientation. When using classical techniques from dynamical systems theory, we demonstrate that these relaxational oscillations display all the characteristic properties of the chaotic strange attractor behavior observed nearby homoclinic orbits of Shil'nikov type. We discuss the possibility that replication and gene regulation processes are governed by a low-dimensional dynamical system that displays deterministic chaos.

Quasi-Periodic Solutions for Two-Level Systems

We consider the Schrodinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2ɛ between the two unperturbed energy levels is small, and we study the problem of finding quasi-periodic solutions of a related generalized Riccati equation. We prove the existence of quasi-periodic solutions of the latter equation for a Cantor set ℰ of values of ɛ around the origin which is of positive Lebesgue measure: such solutions can be obtained from the formal power series by a suitable resummation procedure. The set ℰ can be characterized by requesting infinitely many Diophantine conditions of Mel’nikov type.

Bogdanov-Takens bifurcation points and Sil'nikov homoclinicity in a simple power-system model of voltage collapse

The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reactive loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov-Takens bifurcation point and hence homoclinic orbits; these orbits can be of Sil'nikov type with many coexisting periodic solutions. We may use the bifurcation calculations to divide the two-parameter plane into a number of regions, for which there are qualitatively different dynamics. We classify and further investigate the dynamical behavior in each of these regions, using a Monte Carlo method to investigate basins of attraction of various stable states. We then show how this classification can be used to denote each regions as either safe or unsafe with respect to the likelihood of voltage collapse.

Global bifurcations in a laser with injected signal: Beyond Adler's approximation.

We discuss the dynamics in the laser with an injected signal from a perturbative point of view showing how different aspects of the dynamics get their definitive character at different orders in the perturbation scheme. At the lowest order Adler's equation [Proc. IRE 34, 351 (1946)] is recovered. More features emerge at first order including some bifurcations sets and the global reinjection conjectured in Physica D 109, 293 (1997). The type of codimension-2 bifurcations present can only be resolved at second order. We show that of the two averaging approximations proposed [Opt. Commun. 111, 173 (1994); Quantum Semiclassic. Opt. 9, 797 (1997); Quantum Semiclassic. Opt. 8, 805 (1996)] differing in the second order terms, only one is accurate to the order required, hence, solving the apparent contradiction among these results. We also show in numerical studies how a homoclinic orbit of the Sil'nikov type, bifurcates into a homoclinic tangency of a periodic orbit of vanishing amplitude. The local vector field at the transition point contains a Hopf-saddle-node singularity, which becomes degenerate and changes type. The overall global bifurcation is of codimension-3. The parameter governing this transition is theta, the cavity detuning (with respect to the atomic frequency) of the laser. (c) 2001 American Institute of Physics.

Mel’nikov analysis of a symmetry-breaking perturbation of the NLS equation

The effects of loss of symmetry due to noneven initial conditions on the chaotic dynamics of a Hamiltonian perturbation of the nonlinear Schrödinger equation (NLS) were first numerically studied in [Phys. A 228 (1996)], where it was observed that temporally irregular evolutions can occur even in the absence of homoclinic crossings. In this article, we introduce a symmetry-breaking damped-driven perturbation of the NLS equation in order to develop a Mel’nikov analysis of the noneven chaotic regime. We obtain the following results: (1) spatial symmetry breaking within the chaotic regime causes the wave form to exhibit a more complex dynamics than in [Phys. A 228 (1996)]: center-wing jumping (which characterizes the even chaotic dynamics) about a shifted lattice site alternates (at random times) with the occurrence of modulated traveling wave solutions whose velocity changes sign in a temporally random fashion; (2) we give a heuristic description of the geometry of the full phase space (with no evenness imposed) and compute Mel’nikov type measurements in terms of the complex gradient of the Floquet discriminant. The Mel’nikov analysis yields explicit conditions for the onset of chaotic dynamics which are consistent with the numerical observations; in particular, the imaginary part of the Mel’nikov integral appears to be correlated with purely noneven features of the chaotic wave form.

Homoclinic bifurcations in Chua's circuit

In this paper we study the possible relationship between the Birth of the Double Scroll [L.O. Chua et al., IEEE-CAS 33(11) (1986) 1073] and the homoclinic bifurcations in the traditional Chua's equations. Using a one-dimensional Poincaré map we determine the existence of secondary symmetric homoclinic orbits of Shil'nikov type, born with the Chua's attractor, connecting unstable and stable manifolds of the trivial equilibrium point. In addition, taking into account the presence of other homoclinic orbits for the asymmetric attractor and heteroclinic orbits for the symmetric attractor (connecting unstable and stable manifold of the non-trivial equilibrium points), we suggest a hypothesis about the Birth of Double Scroll structure on the ( α, β) plane.

Controlling the onset of homoclinic chaos due to parametric noise

The effect of parametric noise on the properties of a periodically forced, damped nonlinear oscillator is investigated at the onset of chaos, at parameter values of homoclinic tangency. A Mel'nikov type analysis shows that the Fourier spectrum of the noise controls the width of the chaotic layer thus produced.

The dynamics of coupled perturbed discretized NLS equations

We study the dynamics of two coupled perturbed discretized NLS equations with periodic boundary conditions. The unperturbed system consists of two integrable discrete NLS equations, for which the corresponding Lax pairs are known. We describe here the homoclinic orbits of the coupled system and give new results for a class of non-integrable perturbations through a Mel'nikov type approach, using the spectral analysis of Lax's operators and Fenichel's theory.

DYNAMICS OF CHUA'S CIRCUIT IN A BANACH SPACE

In this paper we present generalized theorems of the Shil'nikov type for evolution equations in Banach spaces of infinite dimension, which describe the behaviour of subsystems of solutions in a neighborhood of a double homoclinic orbits to the same saddle-focus point.

Numerical Chaos, Symplectic Integrators, and Exponentially Small Splitting Distances

Two discrete versions of Duffing's equation are derived as reductions of discretizations of the nonlinear Schrödinger (NLS) equation. Inheriting the properties of the NSL discretizations, one is shown to be integrable and, using a Mel'nikov type analysis, the other discretization is shown to be nonintegrable. Arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation, as h → 0. This suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous Hamiltonian systems.

Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel'nikov type

One considers V. K. Mel'nikov's new class of nonlinear dynamical systems, which is a generalization of the Korteweg-de Vries dynamical system. One investigates the differential-geometric and spectral properties of dynamical systems of Mel'nikov type, one gives their Hamiltonian form, one establishes the so-called gradient identity. The class of finite-zone potentials of a Sturm-Liouville operator, satisfying the given dynamical systems, is described.