Noisy Attractors Research Articles

Stochastic chaos in chemical Lorenz system: Interplay of intrinsic noise and nonlinearity

In this paper, we address the question of whether the phenomenon of chaos can occur in a purely stochastic system or not. We show that chaos can arise in a purely stochastic chemical Lorenz system. A robust regime of stochastic chaos develops like the onset of chaos in a deterministic system. Stochastic trajectories, which are initially very close, show sensitivity to initial states where they diverge exponentially as time progress. The interplay of nonlinearity and intrinsic noise in the chemical Lorenz system is studied to understand the effect of intrinsic noise. We observe that intrinsic noise can destabilize the fixed points and limit cycle attractors where stochastic trajectories make excursions along the unstable manifold, giving rise to the noisy chaotic attractors. This study uses quantitative measures like power spectrums and invariant measures to characterize the strange noisy attractors. Our study establishes that the interplay of intrinsic noise and nonlinearity gives rise to chaos in the stochastic Lorenz system.

Shadowing in hidden attractors

Hidden attractors found in physical systems are different from self-exited attractors and may have a small basin of attraction. The issue of shadowing in these attractors using dynamical noise is discussed. We have particularly considered two classes of dynamical systems which have hidden attractors in their state space. In one of the systems, there is no fixed point but only a hidden attractor in the state space, while in the other, the system has one unstable fixed point along with a hidden attractor in the state space. The effect of dynamical noise on these dynamical systems is studied by using the Hausdorff distance between the noisy and deterministic attractors. It appears that, up to some threshold value of noise, the noisy trajectory completely shadows the noiseless trajectory in these attractors which is quite different from the results of self-exited attractors. We compare the results of hidden chaotic attractors with the self-exited chaotic attractors.

Quantifying Noisy Attractors: From Heteroclinic to Excitable Networks

Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between “nodes''). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related---one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored and can result in “macroscopic” random motion around the network. We perform an asymptotic analysis of local behavior of an escape model near heteroclinic/excitable nodes in the limit of noise $\eta\rightarrow 0^+$ as a model for the mean residence time $T$ near equilibria. The heteroclinic network case has $T$ proportional to $-\ln\eta$ while the excitable network has $T$ given by a Kramers' law, proportional to $\exp (B/\eta^2)$. There is singular scaling behavior (where $T$ is proportional to $1/\eta$) at the bifurcation between the two types of network. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realize any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modeled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.

Stochastic sensitivity analysis of the attractors for the randomly forced Ricker model with delay

Abstract Stochastically forced regular attractors (equilibria, cycles, closed invariant curves) of the discrete-time nonlinear systems are studied. For the analysis of noisy attractors, a unified approach based on the stochastic sensitivity function technique is suggested and discussed. Potentialities of the elaborated theory are demonstrated in the parametric analysis of the stochastic Ricker model with delay nearby Neimark–Sacker bifurcation.

Bistability in a stochastic RNA-mediated gene network

Small regulatory RNAs (srRNAs) are important regulators of gene expression in eukaryotes and prokaryotes. A common motif containing srRNA is a bistable two-gene motif where one gene codes for a transcription factor (TF) which represses the transcription of the second gene, whose transcript is a srRNA which targets the first gene's transcript. Here, we investigate the properties of this motif in a stochastic model which takes the low copy numbers of the RNA components into account. First, we examine the conditions for stability of the two "noisy attractors." We find that for realistic low copy numbers, extreme, but within realistic intervals, mutual repression strengths are required to compensate for the variability of the RNA numbers and thus, achieve long-term bistability. Second, the promoter initiation kinetics is found to strongly influence the bistability of the switch. Super-Poissonian RNA production disrupts the ability of the srRNA to silence its target, though sub-Poissonian RNA production does not rule out the need for strong mutual repression. Finally, we show that asymmetry between the two interactions forming the switch allows an external input to induce the transition from "high srRNA" to "'high TF" more easily (i.e., with a shorter input) than in the opposite direction. We hypothesize that this asymmetric switching property allows these circuits to be more sensitive to one external input, without sacrificing the stability of one of the noisy attractors.

Effects of gene length on the dynamics of gene expression

In Escherichia coli, the nucleotide length of a gene is bound to affect its expression dynamics. From simulations of a stochastic model of gene expression at the nucleotide and codon levels we show that, within realistic parameter values, the nucleotide length affects RNA and protein mean levels, as well as the expected transient time for RNA and protein numbers to change, following a signal. Fluctuations in RNA and protein numbers are found to be minimized for a small range of lengths, which matches the means of the distributions of lengths found in E. coli of both essential and non-essential genes. The variance of the length distribution for essential genes is found to be smaller than for non-essential genes, implying that these distributions are far from random. Finally, gene lengths are shown to affect the kinetics of a genetic switch, namely, the correlation between temporal proteins numbers, the stability of the two noisy attractors of the switch, and how biased is the choice of noisy attractor. The stability increases with gene length due to increased ‘memory’ about the previous states of the switch. We argue that, by affecting the dynamics of gene expression and of genetic circuits, gene lengths are subject to selection.

Effects of codon sequence on the dynamics of genetic networks

In prokaryotes, the rate at which codons are translated varies from one codon to the next. Using a stochastic model of transcription and translation at the nucleotide and codon levels, we investigate the effects of the codon sequence on the dynamics of protein numbers. For sequences generated according to the codon frequencies in Escherichia coli, we find that mean protein numbers at near equilibrium differ with the codon sequence, due to the mean codon translation efficiencies, in particular of the codons at the ribosome binding site region. We find close agreement between these predictions and measurements of protein expression levels as a function of the codon sequence. Next, we investigate the effects of short codon sequences at the start/end of the RNA sequence with linearly increasing/decreasing translation efficiencies, known as slow ramps. The ramps affect the mean, but not the fluctuations, in proteins numbers by affecting the rate of translation initiation. Finally, we show that slow ramps affect the dynamics of small genetic circuits, namely, switches and clocks. In switches, ramps affect the frequency of switching and bias the robustness of the noisy attractors. In repressilators, ramps alter the robustness of periodicity. We conclude that codon sequences affect the dynamics of gene expression and genetic circuits and, thus, are likely to be under selection regarding both mean codon frequency as well as spatial arrangement along the sequence.

Dynamics of a genetic toggle switch at the nucleotide and codon levels

We study the dynamics of a model stochastic two-gene switch at the nucleotide and codon levels. First, we show that its stability, the mean lifetime of the noisy attractors, differs from that of a model where transcription and translation elongation are modeled as single-step delayed events, indicating the need of detailed models to study the dynamics of switches. Next, we vary the coupling between the two genes by varying the affinity of repressor proteins to the promoters and measure the mutual information between the two proteins times series. We find that there is a degree of coupling that maximizes information propagation between the two genes. This is explained by the effects of the coupling on mean and entropy of RNA and protein numbers of each gene, as well as correlation, 2-tuple entropy between the two proteins numbers, and, finally, the stability of the noisy attractors. We also find that increasing the rate of translation initiation increases the correlation between RNA and protein numbers and between the two proteins, due to increased stability of the noisy attractors. Increasing the rate of transcription or decreasing RNA degradation causes opposite effects to the correlation between RNA and proteins of each gene and the stability of the noisy attractors. Finally, we add a sequence-dependent transcription pause site and show that both its probability of occurrence, as well as its mean time length, affects the dynamics of the switch, further demonstrating the dependence of the dynamics of this circuit on sequence level events.

Determining noisy attractors of delayed stochastic gene regulatory networks from multiple data sources

Gene regulatory networks (GRNs) are stochastic, thus, do not have attractors, but can remain in confined regions of the state space, i.e. the 'noisy attractors', which define the cell type and phenotype. We propose a gamma-Bernoulli mixture model clustering algorithm (GammaBMM), tailored for quantizing states from gamma and Bernoulli distributed data, to determine the noisy attractors of stochastic GRN. GammaBMM uses multiple data sources, naturally selects the number of states and can be extended to other parametric distributions according to the number and type of data sources available. We apply it to protein and RNA levels, and promoter occupancy state of a toggle switch and show that it can be bistable, tristable or monostable depending on its internal noise level. We show that these results are in agreement with the patterns of differentiation of model cells whose pathway choice is driven by the switch. We further apply GammaBMM to a model of the MeKS module of Bacillus subtilis, and the results match experimental data, demonstrating the usability of GammaBMM. Implementation software is available upon request.

RENORMALIZATION GROUP APPROACH TO THE EFFECT OF NOISE AT THE BIRTH OF A STRANGE NONCHAOTIC ATTRACTOR

Scaling regularities associated with additive noise are examined in a model of the pitch-fork bifurcation map with multiplicative quasiperiodic driving (Grebogi et al., Physica D13, 261) with the golden-mean frequency ratio at the birth of a strange nonchaotic attractor (SNA). This case of the onset of SNA termed as the blowout bifurcation route was discussed in the context of realistic systems governed by non-autonomous differential equations (Yalçynkaya and Lai, Phys. Rev. Lett., 77, 5039). Our method taking into the account of noise is based on renormalization group (RG) analysis of the birth of SNA (Kuznetsov et al., Phys. Rev. E51, 1629) with application of an appropriate generalization of the approach of Crutchfield et al. (Phys. Rev. Lett., 46, 933) and Shraiman et al. (Phys. Rev. Lett., 46, 935) originally developed for the period doubling transition to chaos. A constant γ=7.4246 is evaluated that determines the scaling law regarding the intensity of noise: A decrease of the noise amplitude by this factor allows resolving one more level of the fractal-like structure associated with the characteristic time scale which is increased by a factor of [Formula: see text]. Numeric results demonstrating evidence of the expected regularities are presented, e.g. portraits of the noisy attractors in different scales.

Noisy attractors and ergodic sets in models of gene regulatory networks

We investigate the hypothesis that cell types are attractors. This hypothesis was criticized with the fact that real gene networks are noisy systems and, thus, do not have attractors [Kadanoff, L., Coppersmith, S., Aldana, M., 2002. Boolean Dynamics with Random Couplings. 〈 http://www.citebase.org/abstract?id=oai:arXiv.org:nlin/0204062 〉 ]. Given the concept of “ergodic set” as a set of states from which the system, once entering, does not leave when subject to internal noise, first, using the Boolean network model, we show that if all nodes of states on attractors are subject to internal state change with a probability p due to noise, multiple ergodic sets are very unlikely. Thereafter, we show that if a fraction of those nodes are “locked” (not subject to state fluctuations caused by internal noise), multiple ergodic sets emerge. Finally, we present an example of a gene network, modelled with a realistic model of transcription and translation and gene–gene interaction, driven by a stochastic simulation algorithm with multiple time-delayed reactions, which has internal noise and that we also subject to external perturbations. We show that, in this case, two distinct ergodic sets exist and are stable within a wide range of parameters variations and, to some extent, to external perturbations.

Effect of noise on the dynamics at the torus-doubling terminal point in a quadratic map under quasiperiodic driving

Scaling regularities are examined associated with the effect of additive noise on a system driven by an external quasiperiodic force with the golden-mean frequency ratio near the terminal point of the torus-doubling bifurcation curve (TDT point). This point was studied in the context of the problem of the onset of a strange nonchaotic attractor on the basis of renormalization group (RG) analysis [Kuznetsov, Phys. Rev. E 57, 1585 (1998)] and observed in experiments with a quasiperiodically driven resistor-inductor-diode circuit [Bezruchko, Phys. Rev. E 62, 7828 (2000)]. The method implemented in the present paper is based on a generalization of the RG approach of Crutchfield [Phys. Rev. Lett. 46, 933 (1981)] and Shraiman [Phys. Rev. Lett., 46, 935 (1981)], originally developed for the period-doubling transition to chaos in the presence of noise. At the TDT point, a constant determining the rescaling rule for the intensity of noise is found to be gamma = 20.048 637 7 . It means that a decrease of the noise amplitude by this factor ensures the possibility of observing one more level of the fractal-like structure of the dynamics, with increase of the characteristic time scale by [(square root (5) + 1)/2]3. Numeric results demonstrating evidence of the expected scaling are presented, e.g., portraits of the noisy attractors and Lyapunov charts on the parameter plane in different scales.

Effect of noise on the critical golden-mean quasiperiodic dynamics in the circle map

Scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden-mean rotation number. We present an improved numerical estimate for the scaling constant of Hamm and Graham (Phys. Rev. A46, (1992) 6323) responsible for the effect of noise, γ = 2.3061852653 . Decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal-like structure, associated with increase of characteristic time scale by the golden mean factor ( 5 + 1 ) / 2 . Numeric results demonstrating evidence of the expected scaling are presented, e.g. portraits of the noisy attractors, devil's staircase plots, Lyapunov charts on the parameter plane in different scales.

Global analysis of crisis in twin-well Duffing system under harmonic excitation in presence of noise

Abstract Evolution of a crisis in a twin-well Duffing system under a harmonic excitation in presence of noise is explored in detail by the generalized cell mapping with digraph (GCMD in short) method. System parameters are chosen in the range that there co-exist chaotic attractors and/or chaotic saddles, together with their evolution. Due to noise effects, chaotic attractors and chaotic saddles here are all noisy (random or stochastic) ones, so is the crisis. Thus, noisy crisis happens whenever a noisy chaotic attractor collides with a noisy saddle, whether the latter is chaotic or not. A crisis, which results in sudden appear (or dismissal) of a chaotic attractor, together with its attractive basin, is called a catastrophic one. In addition, a crisis, which just results in sudden change of the size of a chaotic attractor and its attractive basin, is called an explosive one. Our study reveals that noisy catastrophic crisis and noisy explosive crisis often occur alternatively during the evolutionary long run of noisy crisis. Our study also reveals that the generalized cell mapping with digraph method is a powerful tool for global analysis of crisis, capable of providing clear and vivid scenarios of the mechanism of development, occurrence, and evolution of a noisy crisis.

ON THE EVALUATION OF NOISE LEVELS FOR QUANTIZED DATA

In recent years, quantitative methods for evaluating chaotic properties have been developed in the field of nonlinear time-series analysis. The embedding theorem, which is a mathematical background for the methods, assumes an ideal situation in which noiseless time series are observed with infinite resolution and an infinite amount of data points. However, under real situations we cannot ignore two classes of noises which are included in really observed data. The first one is observational and dynamical noises which depend on internal nonlinear systems and performance of observational instruments. The second one is quantization error included in the time series since we usually use digital computers for applying the methods. In the present paper, we derive formulae for evaluating the levels of observational and quantization noises in the case of embedding observed time series in reconstructed state spaces. By measuring a distance between noiseless and noisy attractors, we also confirm that the derived formulae are appropriate for quantifying the noise included in the reconstructed attractor.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems.

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Noise-induced chaos in an optically injected semiconductor laser model

The chaos induced by an intrinsic spontaneous-emission noise in an optically injected semiconductor laser is investigated through a single-mode injection model. A method is developed to quantitatively study the scale-dependent noise effect in general, and the noise-induced chaotic feature in particular. We find that noise at an experimentally measured level can induce chaos in the system. This suggests that noise-induced chaos may indeed exist in real systems. Certain required characteristics for noise to induce chaos are identified: the periodic state itself, when subject to weak noise, should undergo a process that is much more diffusive than the Brownian motion, and the adjacent chaotic states should still behave chaotically on certain finite scales when subject to noise. We believe they are generic features for noise to induce chaos. The correlation dimension of the clean and noisy attractors is also calculated to study noise-induced changes in the geometrical structure of the attractors.

Estimation of the dimension of a noisy attractor.

A simple method is proposed to estimate the correlation dimension of a noisy chaotic attractor. The method is based on the observation that the noise induces a bias in the observed distances of trajectories, which tend to appear farther apart than they are. Under the assumption of noise being strictly bounded in amplitude, this leads to a rescaling of interpoint distances on the attractor. A correlation integral function is obtained that accounts for this effect of noise. The applicability of the method is illustrated with two examples, viz., the Lorenz attractor with additive noise and experimental time series of pressure fluctuation data measured in gas-solid fluidized beds.