We consider the behaviour of attractors near invariant subspaces on varying a parameter that changes dynamics in subspace dynamical system. refer to such as `non-normal'. In presence chaos is fragile, we find blowout bifurcations are blurred over range values. demonstrate this can occur set positive measure space.

Increasing evidence is becoming available about not only the surface differential rotation of rapidly rotating cool stars but, in a small number cases, also temporal variations, which possibly are analogous to solar torsional oscillations. Given present difficulties resolving precise nature such due both short length and poor resolution data, theoretical input vital help assess modes behaviour that might be expected, will facilitate interpretation observations. Here we take first step this...

We examine global dynamics and bifurcations occurring in a truncated model of stellar mean-field dynamo. This has symmetry-forced invariant subspaces for the we find examples transient type I intermittency blowout to on-off intermittency, involving laminar phases submanifold. In particular, our provides that occur on varying non-normal parameter; is, parameter varies within subspace at same time as normal it. As consequence this Lyapunov exponents do not vary smoothly bifurcation occurs over...

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out in both partial differential equation (PDE) and ordinary (ODE) models mean field dynamos. This type intermittency [introduced P. Ashwin, E. Covas, R. Tavakol, Nonlinearity 9, 563 (1999)] occurs systems with invariant submanifolds and, opposed on-off which can also occur skew product systems, it requires an absence structure. By this we that the dynamics on attractor intermittent manifold cannot...

Analyses of recent helioseismic data have produced ample evidence for substantial dynamical variation the differential rotation within solar convection zone. Given inevitable difficulties in resolving precise nature variations at deeper layers, much effort has recently gone into determining theoretically expected modes behaviour, using nonlinear dynamo models. Two important limitations these models are that they so far included only one form nonlinearity, and as yet not taken account density...

We attempt to forecast the Sun's sunspot butterfly diagram in both space (i.e. latitude) and time, instead of usual one-dimensional time series forecasts prevalent scientific literature. use a prediction method based on non-linear embedding data high dimensions. this latitude (space) using full spatial-temporal from 1874 2015. The analysis results shows that it is indeed possible reconstruct overall shape amplitude pattern sunspots, but its current form does not have real predictive power....

In this paper, we show empirical evidence on how to construct the optimal feature selection or input representation used by layer of a feedforward neural network for propose forecasting spatial-temporal signals. The approach is based results from dynamical systems theory, namely non-linear embedding theorems. We demonstrate it variety signals, with one spatial and temporal dimensions, that consists grid, spatial/temporal lags determined minimum mutual information signals number points taken...

We investigate the detailed dynamics of a truncated $αω$ dynamo model with dynamic $α$ effect. find presence multiple attractors, including two chaotic attractors fractal basin boundary which merge to form single attractor as control parameter is increased. By considering phase portraits and scaling averaged times transitions between we demonstrate that this merging accompanied by crisis-induced intermittency. also range values over system has dependence for fixed initial conditions. This...

Recent analyses of helioseismological observations seem to suggest the presence two new phenomena connected with dynamics solar convective zone. Firstly, there are present torsional oscillations periods about 11 years, which penetrate significantly into convection zone and secondly, oscillatory regimes exist near base markedly different from those observed top, having either reduced or being non-periodic. Recently spatiotemporal fragmentation/bifurcation has been proposed as a possible...

We study the effects of noise on a recently discovered form intermittency, referred to as in-out intermittency. This type which reduces on-off in systems with skew product structure, has been found dynamics maps, (ODE) and (PDE) simulations that have symmetries. It shows itself trajectories spend long time near symmetric state interspersed short bursts away from symmetry. In contrast there are clearly distinct mechanisms approach towards state, this needs be taken into account order properly...

We find concrete evidence for the presence of crisis-induced and Pomeau-Manneville type-I intermittencies in an axisymmetric partial differential equation (PDE) mean-field dynamo model. These findings are potential importance two different reasons. First, as far we aware, this is first time detailed has been produced occurrence these types intermittency such deterministic PDE models. And second, despite rather idealized nature models, more than one type models makes it principle possible...

Recent analysis of the helioseismic observations indicate that previously observed surface torsional oscillations extend significantly downwards into solar convection zone. In an attempt to understand these oscillations, we study nonlinear coupling between magnetic field and differential rotation in context a mean dynamo model, which nonlinearity is due action azimuthal component Lorentz force generated on angular velocity. The underlying zero order velocity chosen be consistent with most...

Recent analyses of the helioseismic data have produced evidence for a variety interesting dynamical behaviour associated with torsional oscillations. What is not so far clear whether these oscillations extend all way to bottom convection zone and, if so, oscillatory at top and different. Attempts been made understand such modes within framework nonlinear dynamo models which include action Lorentz force generated magnetic field on solar angular velocity. One aspect that remains uncertain...

Analyses of recent helioseismic data indicate that the dynamical regimes at base convection zone can be different from those observed top, having either significantly shorter periods or non-periodic behaviour. Recently spatiotemporal fragmentation/bifurcation has been proposed as a mechanism to account for multi-mode behaviour is possibly in solar zone, without requiring separate physical mechanisms with time scales depths. Here we study robustness this respect changes zero order rotation...

In a recent paper [Ossendrijver, 2000] numerical simulations of 2D mean-field model where shown to produce grand minima, typical the long-term behavior solar magnetic activity. The consisted dynamo that features an α effect based on buoyancy instability flux tubes, which gives rise switching back and forth from minima "regular" behavior. this Letter, we report evidence time-series analysis for presence crisis-induced intermittency due attractor-widening. We support finding by showing average...

Employing some recent results in dynamics of systems with invariant subspaces we find evidence both truncated and full axisymmetric mean-field dynamo models a recently discovered type intermittency, referred to as in-out intermittency. This is generalised form on-off intermittency that can occur are not skew products. As far aware this the first time detailed has been produced for occurrence particular such deterministic PDE their truncations. The specific signatures make it possible...

For both public and private firms, comparable companies' analysis is widely used as a method for company valuation. In particular, the of great value valuation equity companies. The several approaches to usually rely on qualitative approach identifying similar peer companies, which tend use established industry classification schemes and/or analyst intuition knowledge. However, more quantitative methods have started being in literature industry, machine learning clustering, natural language...

In this paper (and two others referenced) we stated that fitted the dynamo generated poloidal magnetic fields to a vacuum exterior solution.We now realize instead inadvertently applied condition azimuthal component of vector potential A = 0 at r R, with consequence field lines do not penetrate surface R.