Infinite Cascades Research Articles

Infinite cascades of phase transitions in the classical Ising chain.

We report exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as disorder lines or geometric phase transitions). The phase transition is signaled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonic decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied (m-periodic) strings with arbitrary odd m. We propose a duality transformation which maps the Ising chain onto the m-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the m-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the m-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in m-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.

Features of light scattering by particles with a high absorption coefficient

This paper presents a brief review of resonance scattering of light by subwavelength particles with a large refraction index and small dissipation. It shows that each partial mode can be represented outside the particles as a superposition of infinite cascades of Fano resonances. The nature of these resonances is explained. Simple relationships are given that directly express the parameters of an asymmetric Fano line in terms of the particle size, the wave number of the incident radiation, and the complex refractive index. The resonances are characterized inside the particles by a traditional Lorentzian line shape; however, the amplitude of the resonances can be very large, while resonances of various orders can substantially overlap. This creates conditions for giant (by orders of magnitude) amplification of the field inside such a particle, as well as for controlling the contrast of this field. The indicated properties open up new possibilities for creating nonlinear heterogeneous nanostructures and other metamaterials.

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincare) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map \begin{document}$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$\end{document} which, as known [ 14 ], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.

Exploring percolative landscapes: Infinite cascades of geometric phase transitions.

The evolution of many kinetic processes in 1+1 (space-time) dimensions results in 2D directed percolative landscapes. The active phases of these models possess numerous hidden geometric orders characterized by various types of large-scale and/or coarse-grained percolative backbones that we define. For the patterns originated in the classical directed percolation (DP) and contact process we show from the Monte Carlo simulation data that these percolative backbones emerge at specific critical points as a result of continuous phase transitions. These geometric transitions belong to the DP universality class and their nonlocal order parameters are the capacities of corresponding backbones. The multitude of conceivable percolative backbones implies the existence of infinite cascades of such geometric transitions in the kinetic processes considered. We present simple arguments to support the conjecture that such cascades of transitions are a generic feature of percolation as well as of many other transitions with nonlocal order parameters.

Self-organization of antiperiodic oscillations

Antiperiodic oscillations forming infinite cascades of spirals were recently found experimentally and numerically in the control parameter space of an autonomous electronic circuit. They were discovered while recording one specific voltage of the circuit. Here, we show that such regular self-organization may be measured in any of the four variables of the circuit. Although the relative size of individual phases, their boundaries and the number of peaks of each characteristic oscillation depends on the physical quantity used to record them, the global structural organization of the complex phase diagrams is an invariant of the circuit. Tunable families of antiperiodic oscillations cast fresh light on new intricate behavior of nonlinear systems and open the possibility of studying hitherto unobserved phenomena.

The application of sinusoidal blade-leading edges in a fan-design methodology to improve stall resistance

Taking inspiration from previous biomimetic studies on the performance of humpback whale flippers, this paper reports a programme of work to design a ‘whale-fan’ that incorporates a sinusoidal leading-edge blade profile that mimics the tubercles on humpback whales flippers. Previous researchers have used two-dimensional cascades of aerofoils to study the effects of a sinusoidal profile on aerofoil lift and drag performance. The research was primarily concerned with elucidating the fluid-flow mechanisms induced by the sinusoidal profile and the impact of those mechanisms on aerofoil performance. The results indicate that a sinusoidal leading-edge profile has improved lift recovery post-stall and, thus, is inherently more aerodynamically resistant to the effect of stall. The reported research focuses on the application of previous research conducted with infinite span cascades of aerofoils to the design and optimisation of a finite span aerofoil. The paper presents the assumptions when developing a three-dimensional aerofoil-design methodology that correlates the sinusoidal profile of the blade-leading edge with the desired vorticity distribution at the trailing edge. The authors apply the developed methodology to the design of a fan blade’s tip region to control separation at the trailing edge. The paper presents numerically derived whale-fan performance characteristics and compares them with both numerically and experimentally derived performance characteristics of the baseline fan.

Poincaré analyticity and the complete variational equations

According to a theorem of Poincaré, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters provided that the right sides of the differential equations are analytic in the variables, the time, and the parameters. We describe how these Taylor expansions may be obtained, to any desired order, by integration of what we call the complete variational equations. As illustrated in a Duffing equation stroboscopic map example, these Taylor expansions, truncated at an appropriate order thereby providing polynomial approximations, can well reproduce the behavior (including infinite period doubling cascades and strange attractors) of the solutions of the underlying differential equations.

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Rössler oscillators.

Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows

Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow, namely a Rössler's oscillator. We show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor.

Shear-Induced Bifurcations and Chaos in Models of Three Coupled Lasers

We study nonlinear dynamics in a linear array of three coupled laser oscillators with rotational $\mathbb{S}^1$ and reflectional $\mathbb{Z}_2$ symmetry. The focus is on a coupled-laser model with dependence on three parameters: laser coupling strength, $\kappa$, laser frequency detuning, $\Delta$, and degree of coupling between the amplitude and phase of the laser, $\alpha$, also known as shear or nonisochronicity. Numerical bifurcation analysis is used in conjunction with Lyapunov exponent calculations to study the different aspects of the system dynamics. First, the shape and extent of regions with stable phase locking in the $(\kappa,\Delta)$ plane change drastically with $\alpha$. We explain these changes in terms of codimension-two and -three bifurcations of (relative) equilibria. Furthermore, we identify locking-unlocking transitions due to global homoclinic and heteroclinic bifurcations and the associated infinite cascades of local bifurcations. Second, vast regions of deterministic chaos emerge in the $(\kappa,\Delta)$ plane for nonzero $\alpha$. We give an intuitive explanation of this effect in terms of $\alpha$-induced stretch-and-fold action that creates horseshoes and discuss chaotic attractors with different topologies. Similar analysis of a more accurate composite-cavity mode model reveals good agreement with the coupled-laser model on the level of local and global bifurcations as well as chaotic dynamics, provided that coupling between lasers is not too strong. The results give new insight into modeling approaches and methodologies for studying nonlinear behavior of laser arrays.

How similar is the performance of the cubic and the piecewise-linear circuits of Chua?

This Letter reports phase diagrams quantifying and contrasting the dynamical performance of the paradigmatic piecewise-linear and cubic circuits of Chua. Although both circuits may be regarded as macroscopically isomorphic over wide regions in control parameter space, we show that their microscopic structure displays a myriad of rather distinctive intrinsic features making them unique. Inhomogeneities embedded in periodic and chaotic phases complicate some applications of the circuits but may also adequately act as realistic noise proxies in synchronization problems. In addition, infinite cascades of spirals and hubs observed experimentally very recently in a related dissipative flow are shown to be also present in both circuits of Chua, emerging however in a rather distinctive asymmetric way. Thus Chua's circuits may be used to study experimentally elusive and theoretically intricate phenomena generating periodicity hubs.

THE STRUCTURE OF INFINITE PERIODIC AND CHAOTIC HUB CASCADES IN PHASE DIAGRAMS OF SIMPLE AUTONOMOUS FLOWS

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.

On Capacity of Line Networks

We consider communication through a cascade of discrete memoryless channels (DMCs). The source and destination node of this cascade are allowed to use coding schemes of arbitrary complexity, but the intermediate relay nodes are restricted to process only blocks of a fixed length. We investigate how the processing at the relays must be chosen in order to maximize the capacity of the cascade, that is, the maximum achievable end-to-end rate between the source and the destination. For infinite cascades with fixed intermediate processing length at the relays, we prove that this intermediate processing can be chosen to be identical without loss of optimality, and that the capacity of the cascade coincides with the rate of the best zero-error code of length equal to the block length of the intermediate processing. We further show that for fixed and identical intermediate processing at all relays, convergence of capacity as the length of the cascade goes to infinity is exponentially fast. Finally, we characterize how the block length of the intermediate processing must scale with the length of the cascade to guarantee a constant end-to-end rate. We prove that it is sufficient that the block length scales logarithmically with the network length in order to achieve any rate above the zero-error capacity. We show that in many cases of interest logarithmic growth is also necessary.

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincare) maps and show that the rescaled maps can be brought to so-called generalized Henon maps which have non-degenerate bifurcations of fixed points including those with multipliers e ±iϕ . On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.

Accumulation horizons and period adding in optically injected semiconductor lasers

We study the hierarchical structuring of islands of stable periodic oscillations inside chaotic regions in phase diagrams of single-mode semiconductor lasers with optical injection. Phase diagrams display remarkable accumulation horizons: boundaries formed by the accumulation of infinite cascades of self-similar islands of periodic solutions of ever-increasing period. Each cascade follows a specific period-adding route. The riddling of chaotic laser phases by such networks of periodic solutions may compromise applications operating with chaotic signals such as, e.g., secure communications.

Resonant islands without separatrix chaos.

An alternative type of Hamiltonian nonlinear resonant island is analyzed. In the usual case where the resonant island is pendulumlike, chains bifurcated out of the central elliptic point undergo infinite cascades of period-doubling bifurcations as they approach the island boundary. In the present case we find that those chains undergo either period doubling or inverse saddle-node bifurcations, depending on the strength of perturbing terms. In the saddle-node case we show that just after a reconnection process, external chains cross the island boundary to collapse against the bifurcated internal chains.

Bifurcation phenomena of the non-associative octonionic quadratic

In this work we have performed an investigation into the limiting dynamics and bifurcation phenomena of the non-associative octonionic quadratic map. It displayed a wealth of nonlinear structure including fixed points, Hopf bifurcations, phase locking, periodic cycles, tori, nontrivial knots, loop doubling and tripling, infinite period doubling cascades and hyperchaos. The evolution of the limiting structure was characterized by the recursive interplay of these various bifurcation mechanisms, which led to the appearance of complex attracting structures. Connections from this behaviour to the theory of the Mandelbrot set were established.

Infinite cascades of braids and smooth dynamical systems

Infinite cascades of braids and smooth dynamical systems

Theory of field-induced spin-density wave states in Bechgaard salts: (TMTSF) 2X

The field-induced SDW or CDW is shown to be characterized by a set of countably many order parameters (OP) of DW modulation due to interference between the lattice commensurability and magnetic length. The OP-distribution differs for every rational field, resulting in a phase diagram with devil's-staircase-like infinite cascades. Recent experiments on (TMTSF) 2X are investigated.

Theory of Field-Induced Spin Density Wave States in Anisotropic Two-Dimensional Electron Systems

In order to explain recursive cascades of the field-induced spin density wave phase in organic conductors (TMTSF) 2 X discovered recently, a standard model or two-dimensional anisotropic Hubbard model under a perpendicular field is investigated theoretically: The field-induced SDW ground state is in general to be characterized by a set of countable many order parameters (OP) of the Fourier components of the SDW modulation or by a distribution of these OP's. Due to interference between the lattice commensurability and magnetic length the OP distribution differs for every rational field, resulting in a phase diagram with devil's staircase like infinite cascades. The band structure and spatial profile of the FISDW are analyzed and compared with ordinary density wave states.