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PhysRevA78 , where only a case of [MATH] was investigated. Combining all above suppositions into the one model for a single-mode laser system, we will get the generalized system of nonlinear equations type of |
[EQUATION] Using two type of additional medium in the cavity, one can expect that some combinations of parameters for both modulator and absorber should exist to provide the stable periodic radiation of the laser. |
III Main equations To find mechanisms which takes care of the stable dissipative structures formation we will use the standard procedure to analyze conditions where bifurcation into limit cycle occurs hassard . To this end we rewrite the system ( ) in a most general form |
[EQUATION] where effective forces are as follows: [EQUATION] here constructions ( ), ( ) are used. We deal with a problem of nonlinear dynamics and present a behaviour of the system in the phase plane [MATH] . Firstly, we consider steady states [MATH] and |
[MATH] , defined as coordinates of fixed points in the phase plane. Setting [MATH] and [MATH] , one can find steady states as solutions of stationary equations |
[EQUATION] A behaviour of phase trajectories in the vicinity of these fixed points can be analyzed with a help of the Lyapunov exponents approach. Here time dependent solutions of above system are assumed to be in the form [MATH] , where [MATH] controls the stability of the phase trajectories, [MATH] determines pulse f... |
[EQUATION] where subscript 0 relates to steady states. Inserting ( ) into definition ( 10 ), we get matrix elements [EQUATION] Then, an equation for eigenvalues and eigenvectors |
[EQUATION] gives expressions for [MATH] and [MATH] as follows: [EQUATION] If the real part of the Lyapunov exponent [MATH] then a fixed point |
[MATH] is addressed to a center of a limit cycle. It leads to relation [EQUATION] and yields a condition for the frequency of oscillations |
[EQUATION] To investigate a stability of such a limit cycle we analyze a behaviour of trajectories in the vicinity of the fixed point [MATH] . To this end we rewrite motion equations ( ) where variables [MATH] and [MATH] are count off from stationary magnitudes [MATH] . To do this one can use following transformation |
[EQUATION] where notations for pseudovectors are used: [EQUATION] The corresponding transformation matrix [MATH] is obtained with a help of eigenvector [MATH] components, i.e.: |
[EQUATION] Assuming [MATH] , for the second component [MATH] from Eq.( 13 one gets [EQUATION] Hence, the transformation matrix ( 19 ) takes the form |
[EQUATION] It leads to evolution equations for deviations written in a vector form [EQUATION] Here a pseudovector of the canonical force |
[EQUATION] satisfies conditions hassard Poincare Andronov Leontovich [EQUATION] and has following components: [EQUATION] [EQUATION] |
Above procedure allows to find the stability of the manifold formed by the fixed point [MATH] . Using the standard technique hassard , one can say that the limit cycle is stable only if a real part of the Floquet exponent |
[EQUATION] is negative in a bifurcation point. Structure constants in the definition 27 ) are described by derivatives with respect to [MATH] and [MATH] denoted with subscripts: |
[EQUATION] [EQUATION] [EQUATION] Using some algebra, the stability condition for the limit cycle can be written as follows [EQUATION] |
where notations [EQUATION] [EQUATION] are used. IV Analysis of Hopf bifurcations IV.1 Influence of nonlinear absorber To proceed let us consider steady states behaviour under supposition that action of the absorber is given by expression ( ), [MATH] . Setting |
[MATH] , one gets stationary values of the electric field amplitude [MATH] shown in Fig. . A steady states analysis allows to find that a bistable regime is realized only if [MATH] , here |
[MATH] . In such a case one gets the hysteresis loop in [MATH] dependence in the domain [MATH] (curve 1) which disappears when the threshold [MATH] is crossed, where |
[EQUATION] The behaviour of the amplitude [MATH] is the same as in the first order phase transitions where zero value of [MATH] below [MATH] corresponds to a disordered state, values [MATH] (solid line) relate to an ordered state, whereas intermediate magnitudes of [MATH] (dotted line) correspond to unstable state. The... |
The analysis of the Floquet exponent allows to find the phase diagram (Fig. ), which shows the stable periodic radiation (formation of limit cycle in the phase plane [MATH] ). |
In Fig. the domain I defines configuration of the phase space with both a stable focus (ordered state) and a saddle point (disordered state); in the domain II only disordered state is realized (node); the domain III is characterized by the hysteresis loop, where ordered state corresponds to unstable focus, unstable sta... |
An influence of the parameters of the absorber on a topology of phase plane is shown in Fig. Here an increase in the absorption coefficient [MATH] at small [MATH] leads to transformation of unstable focus into a stable one with additional node and saddle points appearing. At values [MATH] and [MATH] , corresponding to ... |
[MATH] transforms this focus into a node. The frequency of pulse radiation regime appears at non zero value, that correspond to the first bifurcation point [MATH] a further increase in the pump intensity, leads to the growth of [MATH] till the second critical point |
[MATH] is achieved. We have analyzed behavior of pulse radiation frequency at different values of the absorption coefficient [MATH] . According to Fig. an increase in [MATH] at fixed saturation amplitude magnitudes leads to the shift of minimal and maximal values of [MATH] despite a topology of the dependence [MATH] is... |
casperson Therefore, the dispersion in the relaxation time of the electric field amplitude [MATH] , promoting by the absorbing influence, leads to formation of the stable periodic radiation at saturation amplitude [MATH] |
IV.2 Influence of external modulator Let us consider an influence of the external source [MATH] at [MATH] . It is principally important that the periodic radiation is possible only if parameter that controls nonlinear effects [MATH] . Here stationary behavior of the field [MATH] versus pump intensity [MATH] is shown in... |
Analysis of the Floquet exponent shows that limit cycles can be formed only if a stable focus is transformed into an unstable one and vice versa (see Fig. ). Here at [MATH] and [MATH] the phase portrait is characterized by single saddle point [MATH] or [MATH] , respectively. In the domain [MATH] one gets two saddles [M... |
[MATH] . It means a formation of nested loops of neutral stability (Fig. ). Therefore, external force suppresses processes of dissipative structure formation. |
IV.3 Combined effect of external modulator and nonlinear absorber Now we consider an influence of both external modulator and nonlinear absorber on the processes of dissipative structure formation. Setting [MATH] in the system ( ), one gets stationary values of the electric field amplitude [MATH] shown in Fig. |
As it is seen, if the modulator is turned off ( [MATH] ) then we have a single stable state with no radiation at small values of the pump parameter [MATH] . If the threshold given by expression [MATH] is crossed, then a new solution of the steady state equation appears and we have a stationary radiation with an amplitu... |
Next, we investigate conditions where stable periodic radiation can be realized. To this end we need to determine a domain defined by conditions |
[MATH] and [MATH] where periodic solutions of the system ) are exist. Corresponding solutions of the Eq.( 31 are shown in Fig. . It illustrates domains of the absorption coefficient [MATH] and pump intensity [MATH] magnitudes at different intensities [MATH] [MATH] where the stable radiation process is realized. |
As Fig. shows, if we set an absorber inside the cavity only, then a semi-limited domain of [MATH] and [MATH] magnitudes is formed; inside of this domain the stable periodic radiation is possible. Introducing a modulator with |
[MATH] [MATH] (see Fig. a), such a domain becomes totally limited. Moreover, an increase in the parameter [MATH] leads to restriction of the values for the collective parameter [MATH] and pump intensity [MATH] , at which one has stable periodic radiation. At large values [MATH] such domain is degenerated into the line.... |
[MATH] leads to extension of the domain of stable periodic radiation that occurs at large magnitudes of pump intensity parameter. |
An influence of nonlinear processes in the modulator on a picture of the stable periodic radiation formation is presented in Fig. 10 |
It is seen, if [MATH] then there is only stable stationary state (see Fig. ) which is a focus ( [MATH] [MATH] ) on a phase plane [MATH] . Such a fixed point is transformed into a manifold if control parameters are in the domain including its border shown in Fig. 10 a. Such a manifold is a limit cycle ( [MATH] [MATH] ) ... |
[MATH] induces formation of stable periodic radiation at magnitude [MATH] and destroys it at [MATH] . In other words, one gets the situation where the only one reason serves as stimulus for both self-organization and desorganization. |
A picture became more complicated at [MATH] . At first let us discuss the phase diagram shown in Fig. 10 b. At pump limited by the dashed curve in Fig. 10 b there are no stationary solutions and, hence, no stable regimes of radiation. Next, processes of spontaneous photon annihilation reduce a domain of stable periodic... |
[MATH] such a stationary regime is defined by the corresponding stationary solution which is an unstable focus ( [MATH] [MATH] ). |
The related fixed point is defined as an upper branch of the dashed curve in Fig. . At large values [MATH] one has a picture similar to discussed above. At intermediate values [MATH] one can get a very complicated picture of self-organization. Here with an increase in [MATH] we have following picture of transformations... |
Let us consider more closely properties of phase diagram (Fig. ), which shows magnitudes of the absorption coefficient [MATH] and the control parameter [MATH] |
Here the thin solid curve (bifurcation line) defines critical magnitudes for [MATH] and [MATH] where stationary states appeared. In the domain with |
[MATH] one has an unstable focus (see Fig. 13 a). The stable limit cycle is realized inside the bounded domain with [MATH] (Fig. 13 b). At small [MATH] and large [MATH] one has a stable focus (Fig. 13 c). Therefore, one gets a transformation of topology of attractors in the phase plane if parameters [MATH] or [MATH] ar... |
(transition from the limit cycle into repeller — unstable focus) and a further increase in [MATH] leads to the absence of any stationary regime at all. However, the stable periodic solution is observed not on a whole border of the indicated domain. Figure 12 shows that a stable dissipative structure is formed inside th... |
[MATH] and periodic solution changes its stability. In this point the phase portrait of the system is characterized by a set of nested loops. |
Conclusions In this Paper we have analyzed properties of self-organization processes in the two-level class-B laser systems in the presence of absorption effects and influence of the external force. We have shown that due to the nonlinear damping the domain of control parameters of the cavity with the stable pulse radi... |
In our investigation we have considered the simplest case, where relaxation velocities of the electric field and population inversion are of the same order. In real systems of the solid-state class-B lasers |
[MATH] , in gas lasers of such class [MATH] . As was shown theoretically and experimentally GETF71 a difference between above relaxation velocities will not change the picture of stable pulse regime qualitatively. Experimental investigation shows quantitative changes only. |
In our consideration the construction for the external force can be applied to describe influence of the nonlinear processes: in the nonlinear medium with the nonlinear dependence of refractive index (a variation of the parameter [MATH] ); introducing an external incident field with amplitude [MATH] ; more complicated ... |
# Source: arxiv 0804.4714 # Title: Network Structure and Dynamics, and Emergence of Robustness by Stabilizing Selection in an Artificial Genome # Sections: all # Downloaded: 2026-03-03T01:55:59.263932+00:00 |
NETWORK STRUCTURE AND DYNAMICS, AND EMERGENCE OF ROBUSTNESS BY STABILIZING SELECTION IN AN ARTIFICIAL GENOME Abstract Genetic regulation is a key component in development, but a clear understanding of the structure and dynamics of genetic networks is not yet at hand. In this work we investigate these properties within ... |
Introduction The transcription of DNA into mRNA and subsequent translation into protein is the fundamental genetic process; it is the crucial first step by which genetic information gives rise to an organism. Development is not such a linear process, however. By binding to specific regions of the genome, the protein pr... |
Today, the available amount of data for regulatory interactions in a number of model organisms, as, for example, Yeast Wagner2000 |
is steadily increasing. A number of distinguishing structural properties have been identified, namely scale-free degree distributions Jordan2004 , motifs Dobrin2004 and modular organization Thieffry1999 |
Still, there is not enough information to suggest a comprehensive theory of how genetic regulatory networks attain a particular structure, how genes in such networks interact and respond to perturbation, and how evolution has shaped these factors. This study is an attempt to explore these questions in the context of on... |
Traditionally, attempts to understand the characteristics of regulatory networks have focused on dynamical properties. That is, a network topology is specified and rules are applied to describe how each gene in the network responds to inputs. Some initial state is then assigned and the time evolution of gene activity i... |
Kauffman1969 , thresholds Kurten1988b Rohlf2002 , and differential equations Glass1973 . Much less work has been done in understanding how the machinery of transcription, translation, and binding might act throughout the genome to produce the topology of a genetic network. In fact, most studies of genetic networks igno... |
A description of the method we will use for building genetic regulatory networks follows, along with comparisons to published and publically available experimental data. Statistical properties of random realizations of artificial genomes are derived, and related to network structure. Next, we investigate the dynamics o... |
Bornholdt1998 Bornholdt2000a Ciliberti2007 . Finally, we are interested in understanding the role evolution might play in selecting particular network topologies. This is explored by asking how genome structure changes when those networks with certain dynamical properties are preferentially selected. Similar questions ... |
II Model Details II.1 Regulatory network construction from random sequences An artificial genome can be constructed as follows (also see Fig. ). Randomly string together [MATH] integers drawn uniformly between 0 and 3. The use of 4 digits need not be the case, but does provide correspondence with the ATGC alphabet of r... |
Clearly this model greatly simplifies the true transcription, translation, and binding processes. The binding of a transcription factor to a cis-site, for example, depends on the protein’s structure, shape, and environment, rather than a simple template matching approach. Moreover, there is a stochastic element to all ... |
Although it represents a strong simplification, the model does have biological justification Reil1999 . The use of a base promoter sequence is reminiscent of the TATA box frequently found in eukaryotic organisms. Binding is modeled in a DNA-specific way, just as in real organisms. Additionally, the model has the potent... |
Leier2007 . In this paper, we will restrict ourselves to single base pair mutations, and keep the genome size constant. II.1.1 Regulatory dynamics |
Dynamics of state changes (activity or inhibition of genes) on the constructed networks can be defined in various ways. In our study, we apply random threshold network (RTN) dynamics: An RTN consists of [MATH] randomly interconnected binary sites (spins) with states [MATH] For each site [MATH] , its state at time [MATH... |
[EQUATION] with [EQUATION] The [MATH] network sites are updated synchronously. In the following discussion the threshold parameter [MATH] is set to zero. The interaction weights |
[MATH] take discrete values [MATH] or [MATH] with equal probability. If [MATH] does not receive signals from [MATH] , one has [MATH] |
For a finite system size [MATH] , the dynamics of RTN, which are closely related to Boolean networks Kauffman1969 converge to periodic attractors (limit cycles) after a finite number of updates. It has been suggested that different limit cycles may correspond to different gene expression states (cell types) Kauffman196... |
III Statistical properties of the artificial genome In the following, [MATH] denotes the number of genes in the artificial genome. |
[MATH] is directly related to [MATH] , the number of bases, via the combinatorial construction of the artificial genome. III.1 Genome size scaling |
Under the assumption that [MATH] , one derives easily the following two relations: [EQUATION] and [EQUATION] III.2 Length of binding regions |
III.2.1 Average length Constructing the artificial genome by random sampling from an alphabet of length [MATH] the probability [MATH] to get a promoter sequence after a given point of time is [MATH] Thus the expectation value for the length of the binding regions in Reil’s artificial genome is given by |
[EQUATION] III.2.2 Statistical distribution of [MATH] To derive the exact statistical distribution of the lengths [MATH] of the binding regions in the A.G., we first remark that the random production of promoter sequences during the process of genome construction is a Bernoulli chain of length [MATH] with the two possi... |
random sampling steps is given by [EQUATION] By setting [MATH] and [MATH] we get the probability [MATH] to produce one promoter within [MATH] sampling steps: |
[EQUATION] Only one of these possibilities is the correct one (production of a promoter with the last sampling step), i.e. for the derivation of [MATH] we have to divide Eq. ( ) by the binomial coefficient, leading to |
[EQUATION] which is a decaying exponential distribution with [MATH] III.3 Network connectivity In this section, we relate the previously derived properties of the artificial genome to characteristic parameters of the resulting networks. |
III.3.1 Average connectivity For a given TF, the probability to match to a random “DNA” sequence of length [MATH] is given by [MATH] There are [MATH] possibilities to bind to a binding region of average length [MATH] thus the probability that the TF provides at least one input to the gene defined by the promoter sequen... |
[EQUATION] Since we have [MATH] binding regions, the average connectivity (averaged over the whole ensemble of possible random genomes) scales linear with the number of genes, |
[EQUATION] and the slope depends on [MATH] [MATH] and [MATH] Notice, however, that the average connectivity [MATH] obtained from a particular genome realization can substantially deviate from this typical value, since the possible values of [MATH] are Gaussian distributed around |
[MATH] III.3.2 In- and outdegree distribution From the above considerations, it is straight-forward to derive the statistical distributions for the number of ingoing and outgoing links in randomly constructed genomes. Since a TF has equal a priori probability to bind at any region of the base string, generation of out-... |
[EQUATION] The number of inputs a gene receives, however, is proportional to the length [MATH] of its associated binding region, hence, it follows from Eq. |
[EQUATION] i.e. the indegree distribution is exponential. Both results are confirmed by numerical simulations (Fig. 2a and 2b). III.4 Relevance to biology |
Clearly, random genome realizations ar far from being a realistic model of real biological genetic systems. However, it can be shown that even this extreme oversimplification has some relevance for biology. In Figure , the predicted number of genes in a genome, [MATH] , is plotted as a function of genome size for [MATH... |
[MATH] [MATH] is reasonable (not shown), but little observed data exists. On the other hand, statistical distributions of regulatory inputs and outputs do not match biological data particularly well; here, more realistic statistics can be obtained by constructing artificial genomes from duplication and divergence event... |
IV Stabilizing selection for a phenotype - an evolutionary scenario Though evolved by the random processes of genetic drift and selection pressure from changing environments, real genetic systems are far from being random. Common wisdom is that this is often due to the highly non-linear nature of the genotype-phenotype... |
Next, different limit cycles of the associated RTNs are identified by running network dynamics, as defined in section 2.1.1, from 10000 different random initial state configurations. This process is stopped when a RTN is identified which has a fixed point |
[MATH] (a limit cycle of length one), and at least 5 additional attractors; the relative weight of the basin of attraction leading to [MATH] should be small (less than [MATH] of the tested configurations). The last two criteria are chosen to rule out a to quick convergence of evolutionary dynamics (i.e., to make the pr... |
We now apply stabilizing selection as follows: 1. Create a mutant [MATH] by random single base mutations, occurring with a probability [MATH] per base. |
2. Run RTN dynamics from a random initial state, until an attractor is reached, otherwise stop after 200 iterations. 3. If dynamics has converged to [MATH] , keep [MATH] , otherwise keep [MATH] |
4. For the next generation, iterate from (1). We note that we disregard mutations of promoter sites, as well as mutation leading to new ”genes”, to avoid complications resulting from a varying genome size. Notice that, in step (2), we test only one initial configurations, corresponding to the fact that biological organ... |
[EQUATION] defines the robustness against fluctuations, where [MATH] is the fraction of initial states that lead to [MATH] at generation [MATH] A second measure of robustness is associated to the capacitance to buffer the system against disadvantageous mutations (mutational robustness [MATH] Ciliberti2007 ). At each ge... |
[EQUATION] If [MATH] , and hence [MATH] increases with [MATH] , this indicates restructuring of the genome such that the probability of neutral or advantageous mutations with respect to [MATH] has increased. Fig. 5 shows both quantities in a typical evolutionary run. Both [MATH] and [MATH] increase rapidly, however, ex... |
Bornholdt1998 , indicating metastability of the evolutionary dynamics. In fact, in all evolutionary runs we studied [MATH] and [MATH] could be stabilized only over a finite number of generations, as indicated in Fig. 5 by the sharp decrease of both quantities around [MATH] [MATH] and [MATH] are positively correlated, s... |
Figure 6 shows the evolution of the distributions of regulatory input and output numbers per gene, in the same evolutionary run as shown in Fig. 5 with regards to adaptation dynamics. Evidently, considerable reorganization of network structure is taking place: while the indegree-distribution tends to become narrower, t... |
# Source: arxiv 0808.0077 # Title: Recent Structural Evolution of Early-Type Galaxies: Size Growth from z=1 to z=0 # Sections: all # Downloaded: 2026-03-02T07:58:50.176347+00:00 |
Recent Structural Evolution of Early-Type Galaxies: Size Growth from [MATH] to [MATH] Abstract Strong size and internal density evolution of early-type galaxies between [MATH] and the present has been reported by several authors. Here we analyze samples of nearby and distant ( [MATH] galaxies with dynamically measured ... |
[MATH] field and cluster early-type galaxies with typical masses [MATH] . Sizes ( [MATH] ) are determined with Advanced Camera for Surveys imaging. We compare the distant sample with a large sample of nearby ( [MATH] early-type galaxies extracted from the Sloan Digital Sky Survey for which we determine sizes, masses, a... |
distributions of the nearby and distant samples, regardless of sample selection effects. The implied evolution in [MATH] at fixed mass between [MATH] and the present is a factor of [MATH] This is in qualitative agreement with semianalytic models; however, the observed evolution is much faster than the predicted evoluti... |
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