Narrow Results

We present a detailed study of the external-cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback. Mathematically, lasers with feedback are modeled by delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical bifurcation tools for DDEs to continue steady states and periodic orbits, irrespective of their stability. In this way, we show that the periodic orbits corresponding to the ECMs are connected to the steady state solution associated with the locking range of the laser. We also identify symmetric and nonsymmetric homoclinic orbits and hysteresis in the system.

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

This presentation investigates the dynamics of discrete-time cellular neural networks (DT-CNN). In contrast to classical neural networks that are mostly gradient-like systems, DT-CNN possesses both complete stability and chaotic behaviors as different parameters are considered. An energy-like function which decreases along orbits of DT-CNN as well as the existence of a globally attracting set are derived. Complete stability can then be concluded, with further analysis on the sets on which the energy function is constant. The formations of saturated stationary patterns for DT-CNN are shown to be analogous to the ones in continuous-time CNN. Thus, DT-CNN shares similar properties with continuous-time CNN. By confirming the existence of snap-back repellers, hence transversal homoclinic orbits, we also conclude that DT-CNN with certain parameters exhibits chaotic dynamics, according to the theorem by Marotto.

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films presented by G. Cipparrone, G. Russo, C. Versace, G. Strangi and V. Carbone allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, relying on the original model for chaos by cascade of gluing bifurcations proposed by A. Arneodo, P. Coullet and C. Tresser about twenty years before, does not have the central symmetry which one would expect for minimal dimensional model for chaos in nematics in view of the time series near the gluing bifurcation. What we show here is that the simplest truncated normal forms for gluing with the appropriate symmetry and minimal dimension do exhibit time signals that are embarrassingly similar to the ones that could be found using the above-mentioned experimental settings. It so happens that the gluing bifurcation scenario itself is only visible in limited parameter ranges, and that a substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion which easily produces chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach to attracting homoclinic pairs contains a rich behavior, minute noise can transform the complex damping into sustained chaos. As Leonid Shilnikov has taught us, combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations in dissipative systems. We see here on an example how this helps construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for suitable parameters. It is verified that the model has a saddle-node bifurcation, Hopf bifurcations and a cusp of codimension 2 or higher codimension. Stable limit cycles and unstable limit cycles are examined as the infection-reduced reproduction rate varies. It is shown that the model goes through the change of stages of infection extinction, infection persistence, infection extinction, and the extinction of both parasites and the population as the contact coefficient increases.

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L₁ between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L₁ and the other collinear libration points L₂, L₃ is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit.

This manuscript reports numerical investigations about the relative abundance and structure of chaotic phases in autonomous dissipative flows, i.e. in continuous-time dynamical systems described by sets of ordinary differential equations. In the first half, we consider flows containing "periodicity hubs", which are remarkable points responsible for organizing the dynamics regularly over wide parameter regions around them. We describe isolated hubs found in two forms of Rössler's equations and in Chua's circuit, as well as surprising infinite hub cascades that we found in a polynomial chemical flow with a cubic nonlinearity. Hub cascades converge orderly to accumulation points lying on specific parameter paths. In sharp contrast with familiar phenomena associated with unstable orbits, hubs and infinite hub cascades always involve stable periodic and chaotic orbits which are, therefore, directly measurable in experiments. In the last part, we consider flows having no hubs but unusual phase diagrams: a cubic polynomial model containing T-points and wide regions of dense chaos, a nonpolynomial model of the Belousov–Zhabotinsky reaction and the Hindmarsh–Rose model of neuronal bursting, both having chaotic phases with "fountains of chaos". The chaotic regions for the flows discussed here are different from what is known for discrete-time maps. This forcefully shows that knowledge about phase diagrams is quite fragmentary and that much work is still needed to classify and to understand them.

We construct a theory-based numerical method for starting the continuation of homoclinic tangencies near 1 : 1 resonances, for systems with arbitrary dimension ≥ 2. The core of the method is numerical center manifold reduction and flow approximation. The reduction is implemented by means of the homological equation. The starting procedure is applied in numerical examples.

For a class of 4-dim piecewise linear dynamical systems, some sufficient conditions for the existence of homoclinic orbit and chaotic invariant set are provided. Especially, the existence of chaotic invariant set is proved in terms of the constructed Poincaré map with a topological horseshoe. A numerical example with chaotic attractor is provided to illustrate our main result.

Based on mathematical analysis, this paper provides a methodology to ensure the existence of homoclinic orbits in a class of four-dimensional piecewise affine systems. In addition, an example is provided to illustrate the effectiveness of the method.

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

The ionization of hydrogen in a circularly polarized electric field consists of the electron escaping over a rotating Stark saddle. The escape is mediated by the stable and unstable manifolds of a normally hyperbolic invariant manifold associated with the saddle. General principles guarantee the existence of this normally hyperbolic invariant manifold which is referred to as a transition state for energies close to the saddle. We use a numerical procedure to continue the transition state to high energies and study the dynamics restricted to the transition state. We find that for the strength of the electric field under consideration, the transition state persists to very high energies and the restricted dynamics remains amazingly regular. Our analysis moreover includes a study of homoclinic orbits connecting the transition state to itself.

We characterize the families of periodic orbits of two discontinuous piecewise differential systems in ${ℝ}^3$ separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the $X$, $Y$, and $Z$ state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The $X$ and $Z$ solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ($\mathrm{\text{sech}}$) and a hyperbolic secant squared function (${\mathrm{\text{sech}}}^2$), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the $Z$ component and the KdV equation were obtained. Additionally, the ODE for the $X$ component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

In this paper, a 5D Lorenz system is discussed. The discrete mappings are developed to solve the periodic motions in the 5D Lorenz system. Then the stability and bifurcations are determined by eigenvalue analysis. A bifurcation tree is presented to demonstrate that the discrete mapping method can provide not only stable orbits but also unstable motions. Finally, trajectory illustrations are given to show bifurcation influences on periodic orbits and homoclinic orbits in the 5D Lorenz system.

Some dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system from the Poisson geometry point of view are described. The integrability of such system are also studied.

Through nonlinear normal form analysis of the equilibrium equations developed in a neighbourhood of the straight rod solution we show the existence of a ‘phase transition’ from mildly anisotropic to full ‘tape-like’ buckling in non-symmetric rods subject to terminal loads, provided the material has a sufficiently low Poisson's ratio. For a solid elliptical rod with a Poisson's ratio of 0.2 the critical point is found to occur at a cross-sectional aspect ratio of just over 3. For larger aspect ratios the rod locks on to a one-twist-per-wave buckling mode with no internal twist.

In the mildly anisotropic regime there turn out to be two physically distinct pairs of symmetric so-called primary localised buckling modes, differing in the phase of the internal twist. These four modes (homoclinic orbits) remain of the full circle of localised buckling modes existing in the symmetric case after breaking of the circular symmetry of the rod's cross-section. In the strongly anisotropic regime only the energetically favourable pair of ‘flat’ buckling modes survives.