Fixed Point Behavior Research Articles

Experimental evidence of coexisting periodic stochastic resonance and coherence resonance phenomena.

We report evidence of coexisting period stochastic resonance (PSR) and coherence resonance (CR) phenomena in an electrochemical cell. The anodic voltage (V) in the cell is chosen such that the anodic current (I) exhibits excitable fixed point behavior. Subsequently, the anodic voltage is modulated by an external perturbation that is a composite of a subthreshold periodic pulse signal and Gaussian white noise (GWN). As the amplitude of the GWN is increased, the regularity of the invoked dynamics is analyzed using normalized variance curve. The calculated resonance curve shows a double minima, implying the existence of two optimum noise levels where enhanced regularity of the induced spike sequence is detected. Numerical simulations corroborate experimental findings.

Complex dynamics is abolished in delayed recurrent systems with distributed feedback times

AbstractFeedback systems with a single delay time—as described by delay‐differential equations—are known to exhibit various dynamical behaviors including complex oscillations and chaos. Here we show that the consideration of a broad distribution of delay times instead of a single delay results in a shift of the dynamical bifurcations toward higher parameter values, yielding a larger set of parameters with fixed point behavior or simple oscillatory behavior. We demonstrate similar phenomena in three different systems: neuronal feedback in the hippocampus, white blood cell production, i.e., the Mackey‐Glass equation, and population dynamics in theoretical ecology. Our results suggest that the observed simplification of the dynamics is independent of the shape of the delay distribution and the precise nature of the feedback. The existence of distributed delay times may yield a mechanism to avoid irregular fluctuations in biological feedback systems. © 2003 Wiley Periodicals, Inc.

Nonminimal supersymmetric standard model with baryon and lepton number violation

We carry out a comprehensive analysis of the nonminimal supersymmetric standard model (NMSSM) with baryon and lepton number violation. We catalogue the baryon and lepton number violating dimension four and five operators of the model. We then study the renormalization group evolution and infrared stable fixed points of the Yukawa couplings and the soft supersymmetry breaking trilinear couplings of this model with baryon and lepton number (and R-parity) violation involving the heaviest generations. We show analytically that in the Yukawa sector of the NMSSM there is only one infrared stable fixed point. This corresponds to a non-trivial fixed point for the top-, bottom-quark Yukawa couplings and the $B$ violating coupling $\lambda_{233}''$, and a trivial one for all other couplings. All other possible fixed points are either unphysical or unstable in the infrared region. We also carry out an analysis of the renormalization group equations for the soft supersymmetry breaking trilinear couplings, and determine the corresponding fixed points for these couplings. We then study the quasi-fixed point behaviour, both of the third generation Yukawa couplings and the baryon number violating coupling, and those of the soft supersymmetry breaking trilinear couplings. From the analysis of the fixed point behaviour, we obtain upper and lower bounds on the baryon number violating coupling $\lambda_{233}''$, as well as on the soft supersymmetry breaking trilinear couplings. Our analysis shows that the infrared fixed point behavior of NMSSM with baryon and lepton number violation is similar to that of MSSM.

Stationary BPS solutions in N = 2 supergravity with R2-interactions

We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimensional N = 2 supergravity theories with terms quadratic in the Weyl tensor. These terms are encoded in a holomorphic function, which determines the most relevant part of the action and which plays a central role in our analysis. The solutions include extremal black holes and rotating field configurations, and may have multiple centers. We prove that they are expressed in terms of harmonic functions associated with the electric and magnetic charges carried by the solutions by a proper generalization of the so-called stabilization equations. Electric/magnetic duality is manifest throughout the analysis. We also prove that spacetimes with unbroken supersymmetry are fully determined by electric and magnetic charges. This result establishes the so-called fixed-point behavior according to which the moduli fields must flow towards certain prescribed values on a fully supersymmetric horizon, but now in a more general context with higher-order curvature interactions. We briefly comment on the implications of our results for the metric on the moduli space of extremal black hole solutions.

Renormalization group evolution and infrared fixed points in the minimal supersymmetric standard model with baryon and lepton number violation

We study the renormalization group evolution and infrared stable fixed points of the Yukawa couplings and the soft supersymmetry breaking trilinear couplings of the minimal supersymmetric standard model with baryon and lepton number (and R-parity) violation involving the heaviest generations. We show analytically that in the Yukawa sector there is only one infrared stable fixed point. This corresponds to nontrivial fixed point for the top-, bottom-quark Yukawa couplings and the B violating coupling ${\ensuremath{\lambda}}_{233}^{\ensuremath{''}},$ and a trivial one for the $\ensuremath{\tau}$-Yukawa coupling. All other possible fixed points are either unphysical or unstable in the infrared region. We then carry out an analysis of the renormalization group equations for the soft supersymmetry breaking trilinear couplings, and determine the corresponding fixed points for these couplings. We also study the quasi-fixed-point behavior, both algebraically and with numerical solutions of the evolution equations, of the third generation Yukawa couplings and the baryon number violating coupling, and those of the soft supersymmetry breaking trilinear couplings. From the analysis of the fixed point behavior, we obtain upper and lower bounds on the baryon number violating coupling ${\ensuremath{\lambda}}_{233}^{\ensuremath{''}},$ as well as on the supersymmetry breaking trilinear couplings.

Approximate analytic solutions of the renormalization group equations for Yukawa and soft couplings in SUSY models

We present simple analytical formulae which describe solutions of the RG equations for Yukawa couplings in SUSY gauge theories with the accuracy of a few per cent. Performing the Grassmannian expansion in these solutions, one finds those for all the soft couplings and masses. The solutions clearly exhibit the fixed point behaviour which can be calculated analytically. A comparison with numerical solutions is made.

The stability of an <I>NPZ</I> model subject to realistic levels of vertical mixing

The linear stability of a vertically-distributed, Nutrient-Phytoplankton-Zooplankton (NPZ) ocean ecosystem model is analyzed to understand how vertical mixing influences biological dynamics. In the absence of vertical diffusion, the model generally exhibits both stable fixed point and limit cycle behavior, depending on the depth and choice of parameters. Diffusion couples the dynamics of nearby levels and can induce stable profiles as well as oscillatory dynamical trajectories that become vertically phase-locked for large mixing levels. Calculations of the Lyapunov exponent reveal that vertical diffusion can drive this model into a chaotic state, though this occurs only for levels of diffusion well below those found in nature. The dynamics of the model, assuming macrozooplankton are the dominant grazers in the ecosystem, are compared to those in which microzooplankton dominate, with a faster grazing rate and poor assimilation efficiency. While the coupled physical-macrozooplanton system has a stable profile, the coupled microzooplankton profile remains unstable, even at large mixing levels. Fluctuations occur on time scales varying between a few days and a few months, depending on the parameters and magnitude of diffusion.

Approximate analytic solutions of the renormalization group equations for Yukawa and soft couplings in SUSY models

We present simple analytical formulae which describe solutions of the RG equations for Yukawa couplings in SUSY gauge theories with the accuracy of a few per cent. Performing the Grassmannian expansion in these solutions, one finds those for all the soft couplings and masses. The solutions clearly exhibit the fixed point behaviour which can be calculated analytically. A comparison with numerical solutions is made.

Classical corrections in string cosmology

An important element in a model of non-singular string cosmology is a phase in which classical corrections saturate the growth of curvature in a deSitter-like phase with a linearly growing dilaton (an `algebraic fixed point'). As the form of the classical corrections is not well known, here we look for evidence, based on a suggested symmetry of the action, scale factor duality and on conformal field theory considerations, that they can produce this saturation. It has previously been observed that imposing scale factor duality on the $O(\alpha')$ corrections is not compatible with fixed point behavior. Here we present arguments that these problems persist to all orders in $\alpha'$. We also present evidence for the form of a solution to the equations of motion using conformal perturbation theory, examine its implications for the form of the effective action and find novel fixed point structure.

Fixed points and fermion mass structure from large extra dimensions

We examine the fixed point behaviour of Yukawa couplings in supersymmetric theories with varying numbers of dimensions. We show that Pendleton-Ross fixed point behaviour is greatly amplified in the MSSM with no extra dimensions and 4 extra $5+\bar{5}$ multiplets or the MSSM with one extra large dimension and 3 extra $5+\bar{5}$ multiplets. We also show that power law running in models with large extra dimensions can give a hierarchical set of quasi-fixed points for the Yukawa couplings in a manner which is similar to the Froggatt-Nielsen mechanism. However, we also point out the limited perturbative domain in models with power law running

An experimental study of the dynamics of the certainty neuron fuzzy cognitive maps

Certainty Neurons have been introduced as a new type of artificial neurons that use a two variable transfer function that provides them with memory capabilities and decay mechanism. They are used in fuzzy cognitive maps which is an artificial neural network structure that creates models as collections of concepts – neurons and the various causal relationships – weighted arcs that exist between them. An experimental study of the certainty neuron fuzzy cognitive maps (CNFCMs) dynamical behaviour is presented as this appears through simulations. Two control parameters are used: the symmetry of the system's weight matrix and the strength of the decay mechanism. The values of these two parameters can lead the system to exhibit stable fixed point behaviour, limit cycle behaviour or to collapse. The ways that the two control parameters cause the change of the system's dynamical behaviour from fixed point to limit cycle are also presented. The areas where the systems exhibit specific dynamical behaviour are identified.

Right-handed neutrino mass and bottom-tau ratio in strong coupling unification

The recent results of neutrino experiments indicate the existence of the right-handed neutrinos with masses around intermediate scale, which affects the prediction of the bottom-tau mass ratio. In the minimal supersymmetric standard model, this effect largely depends on the right-handed neutrino mass scale and then may severely limit its lower bound. In this letter, we show that in the case of strong coupling unification the bottom-tau mass ratio is little affected by the presence of the right-handed neutrinos. This is because of the infrared fixed-point behavior of Yukawa couplings, which is a common feature of this kind of model.

Higgs limit and b→ sγ constraints in minimal supersymmetry

New limits on the Higgs mass from LEP and new calculations on the radiative (penguin) decay of the b-quark into sγ restrict the parameter space of the Constrained Minimal Supersymmetric Standard Model (CMSSM). We find that for the low tan β scenario only one sign of the Higgs mixing parameter is allowed, while the high tan β scenario is practically excluded, if one requires all sparticles to be below 1 TeV and imposes radiative electroweak symmetry breaking as well as gauge and Yukawa coupling unification. For squarks between 1 and 2 TeV high tan β scenarios are allowed. We consider especially a new high tan β=64 scenario with triple unification of all Yukawa couplings of the third generation, which shows an infrared fixed point behaviour. The upper limit on the mass of the lightest Higgs in the low (high) tan β scenarios is 97±6 (120±2) GeV, where the largest error originates predominantly from the uncertainty in the top mass.

Phase structures of magnetic impurity models with two-body hybridization

The most general model with a magnetic impurity coupled to hybridizing and screening channels of a conduction band is considered. The partition function of the system is asymptotically equivalent to that of the multicomponent kink plasma with a weak external field. The scaling properties of the models for finite $U$ are sketched by using the Anderson-Yuval-Hamann-Cardy poor-man's scaling theory. We point out that it is proper to include a two-body hybridization in order to obtain correct renormalization flows. The phase structures are studied graphically for the general model and various reduced models. A Fermi-liquid---non-Fermi-liquid phase transition is found for all the models. We also show all possible phases with different finite-temperature behaviors though they have the same Fermi-liquid fixed point at low temperature. We also discuss the fixed-point behaviors in the mixed-valence-state regime.

Deconfinement from action restriction

The effect of restricting the plaquette to be greater than a certain cutoff value is studied. The action considered is the standard Wilson action with the addition of a plaquette restriction, which should not affect the continuum limit of the theory. In this investigation, the strong coupling limit is also taken. It is found that a deconfining phase transition occurs as the cutoff is increased, on all lattices studied (up to 20 4). The critical cutoff on the infinite lattice appears to be around 0.55. For cutoffs above this, a fixed point behavior is observed in the normalized fourth cumulant of the Polyakov loop, suggesting the existence of a line of critical points corresponding to a massless gluon phase, not unlike the situation in compact U(1). The Polyakov loop susceptibility also appears to be diverging with lattice size at these cutoffs. A strong finite volume behavior is observed in the pseudo-specific heat. It is discussed whether these results could still be consistent with the standard crossover picture which precludes the existence of a deconfining phase transition on an infinite symmetric lattice.

String gauge unification and infrared fixed points in MSSM+X models

In order to achieve gauge unification at the string scale M X ∼ 5 × 10 17 GeV in the minimal supersymmetric standard model (MSSM), it is necessary to add extra gauge nonsinglet representations at an intermediate scale M I < M X , leading to a class of models which we refer to as MSSM+X models. We perform a detailed analysis of a large class of MSSM+X models and find that the number of (3,1) representations added must be greater than the total of the number of (3,2) and (1,2) representations. Predictions of M I , M X and α ( M x ) are obtained for models with up to 5 extra vector representations than the MSSM. Upper bounds on the U(1) string gauge normalisation k 1 and the sum of the squares of the hypercharge assignments of the extra matter are also obtained for the models. We also study the infrared fixed point behaviour of the top quark Yukawa coupling in a large class of MSSM+X models and find that the low energy MSSM quasi-fixed point prediction of the top quark mass is more likely to be realised in these theories than in the MSSM. In other words, the top quark tends to be heavier in MSSM+X models than in the MSSM. The implementation of a U(1) X family symmetry into MSSM+X models to account for the Standard Model fermion masses is discussed and a particular viable model is presented.

Renormalization group evolution of R-parity-violating Yukawa couplings.

We study the evolution of $R$-parity-violating (RPV) couplings in the minimum supersymmetric standard model, between the electroweak and grand unification scales, assuming a family hierarchy for these coupling strengths. Particular attention is given to solutions where both the $R$-conserving and $R$-violating top quark Yukawa couplings simultaneously approach infrared fixed points; these we analyse both algebraically and with numerical solutions of the evolution equations at one-loop level. We identify constraints on these couplings at the GUT scale, arising from lower limits on the top quark mass. We show that fixed points offer a new source of bounds on RPV couplings at the electroweak scale. We derive evolution equations for the CKM matrix, and show that RPV couplings affect the scaling of the unitarity triangle. The fixed-point behaviour is compatible with all present experimental constraints. However, fixed-point values of RPV top-quark couplings would require the corresponding sleptons or squarks to have mass $\agt m_t$ to suppress strong new top decays to sparticles.

Quark singlets: Implications and constraints.

Quarks whose left- and right-handed chiral components are both singlets with respect to the SU(2) weak-isospin gauge group, offer interesting physics possibilities beyond the Standard Model (SM) already studied in many contexts. We here address some further aspects. We first collect and update the constraints from present data on their masses and mixings with conventional quarks. We discuss possible effects on $b\to s\gamma$ and $Z\to b\bar b$ decays and give fresh illustrations of CP asymmetries in $B^0$ decays differing dramatically from SM expectations. We analyse singlet effects in grand unification scenarios: $d$-type singlets are most economically introduced in ${\bf 5+5^*}$ multiplets of $SU(5)$, with up to three generations, preserving gauge coupling unification with perturbative values up to the GUT scale; $u$-type singlets can arise in ${\bf 10+10^*}$ multiplets of $SU(5)$ with at most one light generation. With extra matter multiplets the gauge couplings are bigger; we give the two-loop evolution equations including exotic multiplets and a possible extra $U(1)$ symmetry. Two-loop effects can become important, threatening unification (modulo threshold effects), perturbativity and asymptotic freedom of $\alpha_3$. In the Yukawa sector, top-quark fixed-point behaviour is preserved and singlet-quark couplings have infrared fixed points too, but unification of $b$ and $\tau$ couplings is not possible in a three-generation $E_6$ model.

Dynamics of damped coupled oscillators near resonance.

We study the dynamics of two conservative oscillators with perturbations from a linear displacement coupling and non-Hamiltonian forces such as damping. We examine the dynamics of these systems when they are near the primary resonance using secular perturbation theory. We show that near resonance a large class of driven oscillators and two coupled oscillators can be transformed to the same ordinary differential equations (ODEs). This common type of dynamics near the resonance is a generalization of the standard Hamiltonian dynamics of two coupled conservative oscillators. We derive expressions for the parameters in these ODEs. From these parameters, we derive analytical expressions for the parameters in these ODEs. From these parameters, we derive analytical expressions for the linear fixed point behavior of these oscillators near resonance. We find a relation between the amplitude frequency coupling of the oscillators and their phase-locking behavior. In particular, we show that two hard oscillators lock in phase and two soft oscillators lock out of phase. We compare our theoretical predictions with computer simulations of two examples: a sinusoidally driven ${\mathit{X}}^{3}$ force oscillator and two coupled van der Pol oscillators with ${\mathit{X}}^{3}$ force.

Controlling chaos in the brain.

In a spontaneously bursting neuronal network in vitro, chaos can be demonstrated by the presence of unstable fixed-point behaviour. Chaos control techniques can increase the periodicity of such neuronal population bursting behaviour. Periodic pacing is also effective in entraining such systems, although in a qualitatively different fashion. Using a strategy of anticontrol such systems can be made less periodic. These techniques may be applicable to in vivo epileptic foci.