Reinjection Mechanism Research Articles

Review of Chaotic Intermittency

Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of chaotic phases also increases. Therefore, intermittency presents a continuous route from regular behavior to chaotic motion. In this paper, a review of different types of intermittency is carried out. In addition, the description of two recent formulations to evaluate the reinjection processes is developed. The new theoretical formulations have allowed us to explain several tests previously called pathological. The theoretical background also includes the noise effects in the reinjection mechanism.

Chaotic intermittency with non-differentiable M(x) function

One-dimensional maps showing chaotic intermittency with discontinuous reinjection probability density functions are studied. For these maps, the reinjection mechanism possesses two different processes. The M function methodology is applied to analyze the complete reinjection mechanism and to determine the discontinuous reinjection probability density function. In these maps, the function M(x) is continuous and non-differentiable. Theoretical equations are found for the function M(x) and for the reinjection probability density function. Finally, the theoretical results are compared with numerical data finding high accuracy.

Intermittency Reinjection in the Logistic Map

Just below a Period-3 window, the logistic map exhibits intermittency. Then, the third iterate of this map has been widely used to explain the chaotic intermittency concept. Much attention has been paid to describing the behavior around the vanished fixed points, the tangent bifurcation, and the formation of the characteristic channel between the map and the diagonal for type-I intermittency. However, the reinjection mechanism has not been deeply analyzed. In this paper, we studied the reinjection processes for the three fixed points around which intermittency is generated. We calculated the reinjection probability density function, the probability density of the laminar lengths, and the characteristic relation. We found that the reinjection mechanisms have broader behavior than the usually used uniform reinjection. Furthermore, the reinjection processes depend on the fixed point.

Type III intermittency without characteristic relation.

Chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories from the chaotic region into the laminar one. The RPD classically was taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized non-uniform RPD has been observed in a wide class of 1D maps; hence, the intermittency theory has been generalized. Consequently, the characteristic relations were also generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. We extend the previous analytical results to deal with the mentioned class of maps. We found that in the mentioned maps, there is not a well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation classically associated to every type of intermittency is not well defined and, in general, cannot be determined experimentally. We illustrate the results with a 1D map and we develop the analytical expressions for every RPD and its characteristic relations. Moreover, we found a characteristic relation going to a constant value, instead of increasing to infinity. We found a good agreement with the numerical simulation.

Experimental Results Versus Computer Simulations of Noisy Poincaré Maps in an Intermittency Scenario

Intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories of the system from the chaotic region into the laminar one. The main results on chaotic intermittency strongly depend on the RPD. Recently a generalized power law RPD has been observed in a wide class of 1D maps. Noise has an impact on the intermittency phenomena and the generalized RPD introduces a novel scenario because it is affected by the noise. An analytical approach was introduced to estimate the noisy RPD (NRPD). In this work we investigate the noisy RPD in two cases: an experimental continuous system, by means of a Poincare map associated to it, and a numerical map chosen close to the experimental one. In the experimental map we use the internal noise of the circuit, whereas in the numerical map we introduce the noise in the usual way. We have compared both noisy dynamics and found important differences between them, concerning the propagation of the noise effect from the maximum of the map (where the power law is generated) into the laminar region. To mimic the numerical map by the experiment, we introduced an external noise during a short window of time, obtaining similar results to the ones obtained in the internal natural noise case. We found that our methodology developed to study the noise intermittency can be used to investigate which class of noise is present in continuous systems.

One-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator

A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass m, confined to bounce elastically between two rigid walls where one is described by a nonlinear van der Pol type oscillator while the other one is fixed, working as a reinjection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional nonlinear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; and (ii) the case where collisions of the particle do affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter (χ) controlling the nonlinearity of the moving wall. For large χ, a diffusion on the velocity is observed leading to the conclusion that Fermi acceleration is taking place. On the other hand, for case (ii), the motion of the moving wall is affected by collisions with the particle. However, due to the properties of the van der Pol oscillator, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicated organization.

The geometry of mixed-mode oscillations in the Olsen model for the Peroxidase-Oxidase reaction

We study the organization of mixed-mode oscillations (MMOs) in the Olsen model for the peroxidase-oxidase reaction, which is a four-dimensional system with multiple time scales. A numerical continuation study shows that the MMOs appear as families in a complicated bifurcation structure that involves many regions of multistability. We show that the small-amplitude oscillations of the MMOs arise from the slow passage through a (delayed) Hopf bifurcation of a three-dimensional fast subsystem, while large-amplitude excursions are associated with a global reinjection mechanism. To characterize these two key components of MMO dynamics geometrically we consider attracting and repelling slow manifolds in phase space. More specifically, these objects are surfaces that are defined and computed as one-parameter families of stable and unstable manifolds of saddle equilibria of the fast subsystem. The attracting and repelling slow manifolds interact near the Hopf bifurcation, but also explain the geometry of the global reinjection mechanism. Their intersection gives rise to canard-like orbits that organize the spiralling nature of the MMOs.

Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

The saddle-node Hopf bifurcation (SNH) is a generic codimension-two bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of the equilibrium. Such a SNH bifurcation with global reinjection occurs naturally in applications, most notably in models of semiconductor lasers.We construct a three-dimensional model vector field that allows us to study the possible dynamics near a SNH bifurcation with global reinjection. This model follows on from our earlier results on a planar (averaged) vector field model, and it allows us to find periodic and homoclinic orbits with global excursions out of and back into a neighbourhood of the SNH point. Specifically, we use numerical continuation techniques to find a two-parameter bifurcation diagram for a well-known and complicated case of a SNH bifurcation that involves the break-up of an invariant sphere. As a particular feature we find a concrete example of a phenomenon that was studied theoretically by Rademacher: a curve of homoclinic orbits that accumulates on a segment in parameter space while the homoclinic orbit itself approaches a saddle-periodic orbit.

Viscous Shocks in the Destabilized Kuramoto-Sivashinsky Equation

Abstract We study stationary periodic solutions of the Kuramoto-Sivashinsky (KS) model for complex spatio-temporal dynamics in the presence of an additional linear destabilizing term. In particular, we show the phase space origins of the previously observed stationary “viscous shocks” and related solutions. These arise in a reversible four-dimensional dynamical system as perturbed heteroclinic connections whose tails are joined through a reinjection mechanism due to the linear term. We present numerical evidence that the transition to the KS limit contains a rich bifurcation structure even within the class of stationary reversible solutions.

Non-linear analysis and modelling of the cellular mechanisms that regulate arterial vasomotion

Non-linearity intrinsic to the ion transport systems that regulate intracellular [Ca2+] and smooth muscle tone allows the emergence of fluctuations in vascular diameter and resistance in isolated arteries (vasomotion) that can be classified as chaotic. Correlation analysis suggests an underlying low dimensional system and a four-dimensional model of vasomotion has been formulated to simulate the effects of pharmacological manipulation of smooth muscle tone or nitric oxide (NO) synthesis by the vascular endothelium. The oscillatory patterns observed experimentally and in modelling studies may be considered ‘universal’ in the sense that they also occur in many physico-chemical systems and include: (a) period-doubling, a feature of the Feigenbaum route to chaos; (b) mode-locking and quasiperiodicity, which reflect the interaction of two nonlinear oscillatory subsystems; and (c) intermittency, in which segments of nearly periodic oscillations of variable duration are interrupted by short chaotic bursts. Low dimensionality allows the construction of iterative maps that confirm the existence of types I and III Pomeau-Manneville intermittency in vascular dynamics, with attractor reconstructions indicating that the reinjection mechanism underlying the type III scenario involves a Shil'nikov-type homoclinic trajectory. Dimensional analysis of experimental data and corresponding surrogate time series generated by randomization of Fourier phase also provide evidence for underlying nonlinear structure in fluctuations in red cell velocity and arteriolar calibre in vivo.

A planar model system for the saddle–node Hopf bifurcation with global reinjection

A simultaneous saddle–node and Hopf bifurcation is a basic codimension-two local bifurcation of vector fields with a phase space of dimension at least three. Its local unfoldings are now well known and it has been found as an organizing centre of the dynamics in many vector fields arising in applications. Here we study this very bifurcation but in the presence of a reinjection mechanism that causes trajectories to return to the relevant local neighbourhood in phase space. This happens in applications, for example, in a semiconductor laser subject to optical injection.We propose and study a -symmetric planar vector field with an additional 2π periodicity as a global model for the planar vector field reduction near a saddle–node Hopf (SNH) bifurcation with global reinjection. The phase space of this model vector field is a half-cylinder and the SNH bifurcation reduces to a saddle–node pitchfork (SNP) bifurcation with global reinjection.Two-parameter unfoldings are presented for the different cases of the SNP bifurcation in the presence of global reinjection. New phenomena that we find are periodic, homoclinic and families of heteroclinic orbits that wind around the cylinder. The topological unfoldings are developed hand-in-hand with careful numerical investigations of these global bifurcations. We demonstrate how our results can be applied to a planar model of a semiconductor laser with optical injection. This is followed by a discussion of how to interpret the presented unfoldings in terms of the full three-dimensional dynamics near a SNH bifurcation with global reinjection.

Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos

Imitative and contrarian behaviours are the two typical opposite attitudes of investors in stock markets. We introduce a simple model to investigate their interplay in a stock market where agents can take only two states, bullish or bearish. Each bullish (bearish) agent polls m ‘friends’ and changes her opinion to bearish (bullish) if (i) at least mρ hb (mρ bh ) among the m agents inspected are bearish (bullish) or (ii) at least mρ hh >mρ hb (mρ bb >mρ bh ) among the m agents inspected are bullish (bearish). The condition (i) ((ii)) corresponds to imitative (antagonistic) behaviour. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in a significant domain of the parameter space {ρ hb ,ρ bh ,ρ hh ,ρ bb ,m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behaviour and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power-law growth rate leading to a finite-time singularity. The reinjection mechanism provided by the contrarian behaviour introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the ‘thermodynamic’ limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics.

Torus fractalization and intermittency

The bifurcation transition is studied for the onset of intermittency analogous to the Pomeau-Manneville mechanism of type I, but generalized for the presence of a quasiperiodic external force. The analysis is concentrated on the torus-fractalization (TF) critical point that occurs at some critical amplitude of driving. (At smaller amplitudes the bifurcation corresponds to a collision and subsequent disappearance of two smooth invariant curves, and at larger amplitudes it is a touch of attractor and repeller at some fractal set without coincidence.) For the TF critical point, renormalization group (RG) analysis is developed. For the golden mean rotation number a nontrivial fixed-point solution of the RG equation is found in a class of fractional-linear functions with coefficients depending on the phase variable. Universal constants are computed that are responsible for scaling in phase space (alpha=2.890 053... and beta= -1.618 034...) and in parameter space (delta(1)=3.134 272... and delta(2)=1.618 034...). An analogy with the Harper equation is outlined, which reveals important peculiarities of the transition. For amplitudes of driving less than the critical value the transition leads (in the presence of an appropriate reinjection mechanism) to intermittent chaotic regimes; in the supercritical case it gives rise to a strange nonchaotic attractor.

Universal scaling properties of type-I intermittent chaos in isolated resistance arteries are unaffected by endogenous nitric oxide synthesis

Spontaneous fluctuations in flow in isolated rabbit ear resistance arteries may exhibit almost-periodic behavior interrupted by chaotic bursts that can be classified as type-I Pomeau-Manneville intermittency. This conclusion was supported by the construction of parabolic return maps and identification of the characteristic probability distributions for the number of oscillations per laminar segment (n) associated with the type-I scenario. Pharmacological inhibition of nitric oxide (NO) synthesis by the vascular endothelium modulated the dynamics of the reinjection mechanism, and thus the generic shape of the probability distribution for n. Nevertheless, average laminar length was related to a derived bifurcation parameter epsilon according to power-law scaling of the form <n> approximately epsilon(beta), where the estimated critical exponent beta was close to the theoretical value of -0.5 both in the presence and absence of NO synthesis.

Power law periodicity in the tangent bifurcations of the logistic map

Numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows. A critical exponent of 1 2 is found on the average amplitude as one approaches the transition. Additionally, the averages oscillate with a period that decreases with the same exponent. This Power Law Periodicity is related to the reinjection mechanism of the map. The undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows significant changes in the average amplitude.

On the scaling properties of the period-increment scenario in dynamical systems

Two one-dimensional dynamical systems discrete in time are presented, where the variation of one parameter causes a sequence of global bifurcations; at each bifurcation the period increases by a constant value (period-increment scenario, usually denoted as a period-adding scenario). We determine all the bifurcation points and the scaling constants of the period-increment scenario analytically. A re-injection mechanism, leading to the period-increment scenario, is discussed. It will be shown, that in systems with more than one parameter the scaling constants can depend on the values of the parameters.

Wave-induced chemical chaos.

Spatiotemporal chaos in a two-variable, cubic autocatalator model with equal diffusivities of thespecies is described. The interplay between an unstable homogeneous state and propagating frontswhich return the system to that state gives rise to a reinjection mechanism for chaotic behavior. Extremesensitivity to initial conditions in both space and time and a rapid falloff of the spatial correlationfunction are exhibited in the chaotic regime.

Laser with injected signal: perturbation of an invariant circle

We study the locked-unloked transition for a class of lasers with injected signal. The transition is produced by a parametric breaking of the invariant circle that represents the free running laser. A Hopf-saddle-node codimension two bifurcation coupled with a phase-drift re-injection mechanism organizes the flow. Fixed points (locked states), periodic orbits and tori, T 2, of two inequivalent types as well as hetero-homoclinic loops are found by using methods of bifurcation theory and are illustrated with computer simulations. We discuss the dependence of the flow patterns with respect to the laser parameters and, in particular, we show that the detuning between atomic and cavity frequencies plays a fundamental role for the dynamics.

Šil'nikov-Hopf bifurcation

We derive a 2-parameter Poincaré map modeling a 3-dimensional vector flow near the codimension-2 point defined by a Šil'nikov homoclinic orbit to a fixed point undergoing a Hopf bifurcation. In one parameter regime the map reproduces Glendinning and Sparrow's results for the Šil'nikov bifurcation while in another regime it provides new results for a small Hopf cycle with a global reinjection mechanism. The map is analyzed finding primary homoclinic tangencies to the small Hopf cycle. Parameter dependence of periodic orbits associated with these tangencies is described. Global behavior of the tube-like unstable manifold of the Hopf cycle is studied in depth, analytically and numerically, locating its simplest subsidiary tangencies.

Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle focus in the indium/thiocyanate electrochemical oscillator

The complex mixed-mode and chaotic oscillations in the reduction of indium(III) at a hanging mercury drop electrode in thiocyanate solution are studied in experimental detail in the parameter plane. With the aid of a simple model developed previously, we critically discuss the relation of the experimentally observed periodic–chaotic sequences with those predicted by the Shil’nikov theory of homoclinic orbits. It appears helpful to distinguish between two fundamentally different types of mixed-mode behavior: ‘‘type 1’’ related to the incomplete homoclinic scenarios observed in the present system and its model; ‘‘type 2’’ related more directly to the Shil’nikov behavior and the classical Rössler type of reinjection mechanism. The definitions and properties of both types of mixed-mode oscillation are worked out in some detail.