Notion Of Structural Stability Research Articles

Climate modelling and structural stability

Climate modelling plays a crucial role for understanding and addressing the climate challenge, in terms of both mitigation and adaptation. It is therefore of central importance to understand to what extent climate models are adequate for relevant purposes, such as providing certain kinds of climate change projections in view of decision-making. In this perspective, the issue of the stability of climate models under small relevant perturbations in their structure (or small relevant ‘structural model errors’ with respect to the target system) seems particularly important. Within this framework, a debate has emerged in the philosophy of science literature about the relevance for climate modelling of the mathematical notion of structural stability. This paper adresses several important foundational and epistemological questions that arise in this context, in particular about the the role of abstract mathematical considerations of a qualitative nature (in some precise, topological sense) for concrete modelling projects with mainly quantitative purposes.

Structural stability as a consistent predictor of phenological events.

The timing of the first and last seasonal appearance of a species in a community typically follows a pattern that is governed by temporal factors. While it has been shown that changes in the environment are linked to phenological changes, the direction of this link appears elusive and context-dependent. Thus, finding consistent predictors of phenological events is of central importance for a better assessment of expected changes in the temporal dynamics of ecological communities. Here we introduce a measure of structural stability derived from species interaction networks as an estimator of the expected range of environmental conditions compatible with the existence of a community. We test this measure as a predictor of changes in species richness recorded on a daily basis in a high-arctic plant-pollinator community during two spring seasons. We find that our measure of structural stability is the only consistent predictor of changes in species richness among different ecological and environmental variables. Our findings suggest that measures based on the notion of structural stability can synthesize the expected variation of environmental conditions tolerated by a community, and explain more consistently the phenological changes observed in ecological communities.

Codimension one structurally stable chain classes

The well-known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable, then the chain recurrent set is hyperbolic. It is natural to ask if this type of result is true for an individual chain class, that is, whether or not every structurally stable chain class is hyperbolic. Regarding the notion of structural stability, there is a subtle difference between the case of a whole system and the case of an individual chain class. The latter is more delicate and contains additional difficulties. In this paper we prove a result of this type for the latter, with an additional assumption of codimension 1. Precisely, let f f be a diffeomorphism of a closed manifold M M and let p p be a hyperbolic periodic point of f f of index 1 or dim ⁡ M − 1 \dim M-1 . We prove if the chain class of p p is structurally stable, then it is hyperbolic. Since the chain class of p p is not assumed in advance to be locally maximal, and since the counterpart of it for the perturbation g g is defined not canonically but indirectly through the continuation p g p_g of p p , the proof is quite delicate.

Generic scenarios of the accelerating Universe

Recent astronomical observations strongly indicate that the current Universe is undergoing an accelerated phase of its expansion. If the Universe evolution is described by the FRW model then the acceleration should be driven by some perfect substance violating the strong energy condition. Hence the negative pressure is required for the explanation of acceleration. While different candidates for the fluid termed dark energy have been suggested, the positive cosmological constant seems to be the simplest candidate for dark energy description. However the Lambda term treated as a quantum vacuum energy has no simple physical interpretation because of the fine tuning problem. In this paper we characterize a certain class of evolutional scenario of the accelerating universe from the point of view genericity expressed in notion of structural stability. It is shown that in this class all evolutional scenarios with transient acceleration effect are not generic. Only the LCDM model with the acceleration phase following the deceleration phase is structurally stable. Because the cosmological models with transient acceleration appear in string theory we conclude that they are not generic.

On models structurally stable in periodic motion

The results obtained for a class of models that are structurally stable in periodic motion were presented, and, in particular, the notion of structural stability was introduced. Feasibility of such structurally stable model was proved. For a model featuring space-time symmetry, the sufficient structural stability conditions were established, and examples were presented.

Paths of Friedmann–Robertson–Walker brane models

The dynamics of brane-world models of dark energy is reviewed. We demonstrate that simple dark energy brane models can be represented as two-dimensional dynamical systems of a Newtonian type. Hence a fictitious particle moving in a potential well characterizes the model. We investigate the dynamics of the brane models using methods of dynamical systems. The simple brane-world models can be successfully unified within a single scheme—an ensemble of brane dark energy models. We characterize generic models of this ensemble as well as exceptional ones using the notion of structural stability (instability). Then due to the Peixoto theorem we can characterize the class of generic brane models. We show that the global dynamics of the generic brane models of dark energy is topologically equivalent to the concordance ΛCDM model. We also demonstrate that the bouncing models or models in which acceleration of the universe is only transient phenomenon are nongeneric (or exceptional cases) in the ensemble. We argue that the adequate brane model of dark energy should be a generic case in the ensemble of Friedmann–Robertson–Walker dynamical systems on the plane.

Cosmological zoo—accelerating models with dark energy

Recent observations of type Ia supernovae indicate that the Universe is in an acceleratingphase of expansion. The fundamental quest in theoretical cosmology is to identify theorigin of this phenomenon. In principle there are two possibilities: (1) the presence of matterwhich violates the strong energy condition (a substantial form of dark energy) or(2) modified Friedmann equations (Cardassian models—a non-substantial form ofdark matter). We classify all these models in terms of two-dimensional dynamicalsystems of the Newtonian type. We search for generic properties of the models. It isachieved with the help of Peixoto’s theorem for dynamical systems on the Poincarésphere. We find that the notion of structural stability can be useful to distinguishthe generic cases of evolutional paths with acceleration. We find that, while theΛCDM models and phantom models are typical accelerating models, the cosmological models withbouncing phase are non-generic in the space of all planar dynamical systems. We derive theuniversal shape of the potential function which gives rise to presently accelerating models.Our results show explicitly the advantages of using the potential function (instead of theequation of state) to probe the origin of the present acceleration. We argue that simplicityand genericity are the best guide in understanding our Universe and its acceleration.

Novel Families of Toxin-like Peptides in Insects and Mammals: A Computational Approach

Most animal toxins are short proteins that appear in venom and vary in sequence, structure and function. A common characteristic of many such toxins is their apparent structural stability. Sporadic instances of endogenous toxin-like proteins that function in non-venom context have been reported. We have utilized machine learning methodology, based on sequence-derived features and guided by the notion of structural stability, in order to conduct a large-scale search for toxin and toxin-like proteins. Application of the method to insect and mammalian sequences revealed novel families of toxin-like proteins. One of these proteins shows significant similarity to ion channel inhibitors that are expressed in cone snail and assassin bug venom, and is surprisingly expressed in the bee brain. A toxicity assay in which the protein was injected to fish induced a strong yet reversible paralytic effect. We suggest that the protein may function as an endogenous modulator of voltage-gated Ca 2+ channels. Additionally, we have identified a novel mammalian cluster of toxin-like proteins that are expressed in the testis. We suggest that these proteins might be involved in regulation of nicotinic acetylcholine receptors that affect the acrosome reaction and sperm motility. Finally, we highlight a possible evolutionary link between venom toxins and antibacterial proteins. We expect our methodology to enhance the discovery of additional novel protein families.