diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzghmj" "b/data_all_eng_slimpj/shuffled/split2/finalzzghmj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzghmj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nOne of interesting effects produced by flare energy release in the solar atmosphere is excitation of helioseismic waves, so-called sunquakes \\citep{Kosovichev1998}. Such waves usually propagate as expanding ripples from local impact sources occupying several pixels in photospheric Dopplergrams. The cause of these events is a subject of intensive debates \\citep[e.g.][]{Donea2011,Kosovichev2014}. Generally, the necessary condition for producing sunquakes is a sudden momentum enhancement in the lower solar atmosphere. One of the possible agents of such disturbance can be chromospheric heating due to injection of accelerated charged particles postulated by the standard model of solar flares \\citep[for a recent review see][]{Fletcher2011}. Models of the gas dynamics processes induced by nonthermal electron beams \\citep[e.g.][]{Kostiuk1975, Fisher1985, Kosovichev1986} predict formation of a shock wave or a chromospheric condensation moving towards the solar photosphere and, thus, transferring momentum to the dense plasma. \\cite{Kosovichev1995} discussed such beam-driven mechanism of sunquakes. However, the plasma momentum transfer is also possible by other mechanisms, such as sharp enhancement of pressure gradient due to flux-rope eruption \\citep[e.g.][]{Zharkov2013} or by an impulse of the Lorenz force which can be stimulated by electric currents in the lower solar atmosphere \\citep{Fisher2012}. Also, it is possible that different sunquake events are caused by different mechanisms. Usually, sunquakes are associated with M and X class flares. However, many X-class flares did not produce sunquakes \\cite[e.g.][]{Donea2011}, whereas these events had been noticed during relatively weak M-class flares \\citep{Martinez-Oliveros2008, Kosovichev2014}.\n\nIn this Letter, we discuss observations of the C7.0 flare of February 17, 2013, which produced a rather strong sunquake initiated during the HXR burst. We use data from four space instruments: EUV observations from SDO\/AIA \\citep{Lemen2012}, vector magnetic field measurements from SDO\/HMI \\citep{Scherrer2012}, integrated soft X-ray emission from GOES, and X-ray spectroscopic imaging data from RHESSI \\citep{Lin2002}. We investigate potential mechanisms of the sunquake initiation, and find that despite the precise temporal coincidence between the HXR impulse and the photospheric impact this event is not consistent with the standard flare model, because the HXR source and the sunquake impact were at different spatial locations, at two different footpoints of the flare loop. Our analysis leads to a suggestion that a significant role in the sunquake initiation may be played by electric currents in the low atmosphere.\n\n\\section{General description of the event and sunquake}\n\nThe flare event of Febrary 17, 2013, was observed in active region NOAA 11675. It consists of two subflares clearly separated in time and space: the first subflare has the C7.0 GOES X-ray class, and the second subflare reached M1.9 peak intensity. The duration of the whole double flare is about 8 min, starting at 15:46:00 UT and ending at 15:54:00 UT (Fig. 1). The highest energy of the hard X-ray (HXR) emission (maximum at 15:47:20 UT), detected by RHESSI during the first subflare is $\\sim 1$ MeV. The second subflare is characterized by weaker intensity and energy ($<$300 keV) of HXR emission which reached maximum at 15:50:30 UT.\n\nTop panels of Fig. 2 present the AIA images in the 94 $\\rm\\AA$ channel. The temporal and spatial resolutions are 12 seconds and $1.2^{\\prime\\prime}$ (with the angular pixel size of $0.6^{\\prime\\prime}$). The preflare state reveals a compact loop-like structure where the flare process occurs.\n\nThe sunquake is observed as an expanding circular wave in the HMI Dopplergrams filtered in the frequency high range, 5-6 mHz, to isolate the sunquake signal from the convective noise. The propagation of the sunquake wave is shown on the time-distance diagram \\citep{Kosovichev1998,Zharkova2007} presented in Fig.2 (middle right panel) comparing the observed signal with the theoretical ray-path theory prediction (dotted yellow line). The time-distance analysis shows that the sunquake is initiated by the first subflare with the initiation point at the flare impulse signal. The inclined wave pattern above the theoretical curve is associated with the frequency dispersion of the helioseismic wave packet.\n\nThe initiation of the sunquake is observed as a strong localized impulse in the HMI Dopplergrams and line-of-sight magnetogram at $\\approx$15:47:54 UT. The location of the impulse on the HMI magnetogram is illustrated in Fig. 2 (middle left panel). Because of the rapid variations during the flare impulse the Doppler velocity and magnetic field measurements in the impact pixels can be inaccurate. Therefore, we use the original level-1 HMI filtergram data from the two HMI cameras to locate the exact time and place of the flare impact. The bottom panels of Fig. 2 shows the time difference between the HMI filtergram images (from HMI Camera-1). We see that compared to the preflare time during the flare there is enhancement of emission in the pixels associated with the AIA brightnings and the place of the sunquake initiation. The timing of the photospheric impact is illustrated in Fig. 1 (bottom) which shows the signals from both HMI cameras as a function of time. The periodic variations of these curves are due to the line scanning. The plot shows that the photospheric impact coincides with the HXR impulse within 3 sec (the HMI camera resolution).\n\n\\section{Spatial structure of the flare region}\n\nHere we present a description of the spatial structure of the flare region according to the RHESSI and AIA\/SDO observations. RHESSI uses a Fourier technique to reconstruct X-ray emission sources \\citep{Hurford2002}. We apply the CLEAN algorithm to synthesize the X-ray images using detectors 1,3-6,8 (integration times are shown in Fig.3). In Fig. 3, the RHESSI HXR and SXR contour images are compared with the corresponding AIA 94 \\AA ~images for the time interval covering the HXR peaks of both flares. To compare positions of the EUV and X-ray sources with the structure of magnetic field we plot the polarity inversion line from the HMI magnetogram. The structure of the EUV emission sources is rather complicated. There are ribbon-like structures\nlocated on both sides of the magnetic field inversion line. During the HXR burst we observe a loop-like structure with one footpoint associated with very strong HXR emission (25-200 keV), and the other footpoint located in the place of the photospheric impact (sunquake initiation), which also coincides with a weak HXR emission source. The total emission intensity of the weaker X-ray source is approximately five times less than the emission intensity of the stronger HXR source. If the sunquake were initiated by an impact of high-energy electrons, then their impact would be in the place of the intensive energy loss of the accelerated particles, and coincide with the strongest HXR emission source. However, we observe the opposite situation when the sunquake impact correlates with a weaker HXR emission source. This indicates that the sunquake is unlikely be generated by the impact of high-energy electrons.\n\nThe second subflare has SXR source (6-12 keV) coinciding with the HXR source (25-50 keV) and saturated UV emission above the magnetic field inversion line. However, this subflare is located $\\sim$3 Mm away from the place of energy release in the first subflare and according to our analysis is not associated with the sunquake.\n\n\\section{Analysis of RHESSI spectra: accelerated particles and heating}\n\nTo determine properties of the accelerated particles, the plasma and their energetics we use the RHESSI data in the range 5-250 keV. We investigate two spectra taken during the HXR peaks of two subflares. The power-law approximation $f(E)=AE^{-\\gamma}$ ($A$ is normalization coefficient) is considered for the hard X-ray (HXR) nonthermal emission $\\gtrsim 20$ keV. To simulate the presence of the low-energy cutoff we use the broken power law \\citep{Holman2003} with fixed photon spectral index $\\gamma_0=1.5$ below the break energy ($E_{low}$). For the first subflare, an additional break energy ($E_{br}$) is considered, and, thus, for this case we have two spectral indices $\\gamma_1$($E_{low}E_{br}$). For the second subflare we consider only one spectral index $\\gamma_1$($E>E_{low}$) and also make a pileup correction as the count rate is sufficient to observe such effect.\n\nThe thermal soft X-ray (SXR) spectrum $\\lesssim 20$ keV is approximated by one-temperature thermal bremsstrahlung emission with two parameters: temperature ($T$) and emission measure ($EM$). The RHESSI spectra are fitted by means of the least squares technique implemented in the OSPEX package with 7 free parameters ($EM$, $T$, $A$, $E_{low}$, $\\gamma_1$, $E_{br}$ and $\\gamma_2$) for the first subflare and 5 parameters ($EM$, $T$, $A$, $E_{low}$ and $\\gamma_1$) for the second subflare. Fig. 4 displays results of the fitting.\n\nFrom the thermodynamics point of view the second subflare is hotter than the first one, but the emission measure is smaller. Volume $V$ of the UV loop estimated in the previous section is $10^{26}$ cm$^3$, so that plasma density $n_1 = \\sqrt{(EM_1\/V)}\\approx 6\\times 10^{10}$ cm$^{-3}$ for the first subflare. The plasma density for the second subflare, assuming the same flare region volume, is $n_2\\approx 2\\times 10^{11}$ cm$^{-3}$. Due to compactness of the flare region the plasma within the magnetic loops is rather dense.\n\nThe HXR photon spectrum is harder for the first HXR burst than for the second one. Normalization coefficient $A$ of the HXR spectrum is also one order of magnitude larger in the case of the first subflare. This means that the acceleration process is more efficient during the first subflare. The total flux $Fl$ [electrons s$^{-1}$] of accelerated electrons can be estimated following the work of \\cite{Syrovatskii1972}:\n\n$$\nFl(E_{low}=[dF_z\/dt]\/cL$, where $F_z$ is total magnetic flux inside a contour with length $L$, which covers the flare region. The evolution of $$ presented in Fig.1 (gray histograms in top panels) shows that both subflares correlate with the peaks of $$.\n\nTo calculate the vertical currents we use the disambiguated HMI vector magnetic field data \\citep{Centeno2014} with time cadence 720 seconds, and the same spatial resolution as of the line-of-sight magnetograms. The vertical electric current density is calculated from the HMI vector magnetograms using the Ampere's law \\citep[e.g.][]{Guo2013}:\n\n$$\nj_z=\\frac{c}{4\\pi}(\\nabla\\times\\vec{B})_z = \\frac{c}{4\\pi}\\left(\\frac{\\partial B_x}{\\partial y} - \\frac{\\partial B_y}{\\partial x}\\right)\n$$\n\nThe resulted $j_z$ map during the flare, effectively averaged over 12 min due to the HMI temporal resolution, is presented in Fig.~3. Figure 5 displays the evolution of $$ averaged through the flare region with area $\\approx 1.5\\times 10^{18}$ cm$^2$, and reveals a maximum corresponding to the flare. We estimated errors for $$ as the standard deviation of the $j_z$ distribution in the quite Sun regions.\n\nIn Fig.~3 we see that the place of sunquake generation correlates with the strong electric currents, and that there is no significant HXR emission in this place. The HXR is mostly emitted from the source located on the other side of the magnetic field polarity inversion line at the opposite footpoint of the flare loop. Such observation, and the time evolution of $$ and $$ can be an evidence of a non beam-driven origin of sunquake. The correlation with the location of the strongest electric currents suggests that the sunquake event could be initiated due to a local heating or impulsive Lorentz force in the flare region.\n\n\n\\section{Discussion}\n\nIn this section we will discuss the contributions of the electric currents and nonthermal electrons in flare energy release and the generation of the sunquake.\n\nFor the estimated fluxes of accelerated electrons for the first subflare, the total kinetic power $P_{nonth}\\approx 1.5\\times 10^{27}$ erg s$^{-1}$ in the HXR peak. To estimate the Joule heating in the sunquake generation region we need to estimate effective electric conductivity $\\sigma_{eff}$. In the regime of electric currents dissipation the magnetic Reynolds number $Re_m=4\\pi\\sigma_{eff}L^2\/(c^2\\tau)\\sim 1$, where $\\tau$ is a characteristic time of electric current dissipation ($\\sim 100$ s, the duration of the HXR burst), and $L$ is a characteristic length scale ($\\sim 1^{\\prime\\prime}$, the size of the impulsive region on the Dopplergrams). For these characteristic values we get $\\sigma_{eff}\\sim 10^6$ CGS units. This value is substantially higher than the theoretical Spitzer conductivity \\citep{Kopecky1966}. However, recent studies of the partially ionized plasma of the solar chromosphere show that the electric conductivity can be substantially reduced due to Pedersen resistivity \\citep[e.g.][]{Leake2012} or due to small scale MHD turbulence \\citep{Vishniac1999}. The volumetric energy release is $Q_j = j^2\/\\sigma_{eff}\\approx 8\\times 10^3$ erg s$^{-1}$ cm$^{-3}$ for $j\\approx 0.3$ A\/m$^2$. The total energy release due to dissipation of electric currents in the sunquake region is $Q_j^{tot}\\approx 3\\times10^{27}$ erg s$^{-1}$, estimating the volume for a box with length scale $L\\sim 1^{\\prime\\prime}$. So, we see that $P_{nonth}\\sim Q_j^{tot}$, and both types of energy release have the energy budget sufficient to explain heating in the flare according to the GOES data: the change of the plasma internal energy, $d(3nk_BTV)\/dt\\sim 10^{27}$ erg s$^{-1}$, and the radiation losses, $L_{rad}\\sim 5\\times 10^{26}$ erg s$^{-1}$.\n\nTo produce the sunquake we need a strong impulsive force in the lower solar atmosphere. The sunquake momentum can be estimated from the initial impact as $p_{sq}\\sim \\rho L^3v\\sim 10^{22}$ g$\\cdot$cm s$^{-1}$ for $\\rho\\sim 10^{-8}$ g$\\cdot$cm$^{-3}$ (photospheric value) and $v\\sim c_s\\sim 10$ km\/s, where $c_s$ is photospheric sound speed. In principle, the force generating sunquakes can be produced directly by energetic electron beams. The total momentum of injected nonthermal electrons is\n\n$$\np_e=\\tau\\sqrt{2m_e}\\int_{E_{low}}^\\infty f(E)\\sqrt{E}dE\n$$\n\nwhere $m_e$ is the electron mass, $f(E)$ is the distribution function of nonthermal electrons, and $\\tau$ is the characteristic time of the injection. For the first HXR burst, $p_e\\sim 10^{20}$ g$\\cdot$cm s$^{-1}$. As the emission intensity of the weaker HXR source associated with sunquake impact is five times less comparing with strong HXR source, than nonthermal electrons momentum in the footpoint associated with sunquake impact $\\sim 0.2\\times 10^{20}$ g$\\cdot$cm s$^{-1}$.\n\nMomentum of the accelerated protons can be much larger than in the case of electrons and lead to stronger disturbances in the solar atmosphere. Assuming that the protons roughly (not accounting collisions) have energy $E_p\\lesssim E_e$, the momentum contained in the proton beam $p_p\\lesssim p_e\\sqrt{(m_p\/m_e)}\\sim 45p_e\\sim 0.5\\times 10^{22}$ g$\\cdot$cm s$^{-1}$. We see that the momentum of accelerated protons represents a more probable agent of the sunquake initiation than the momentum of electrons.\n\nOur observations show that while the sunquake impact and the HXR impulse are simultaneous in time they are clearly separated in space, and located at the different footpoints of the flare magnetic loop. In addition, we find that the impact location correlates with the strongest electric currents. This suggests that, perhaps, energetic particles are accelerated by electric field in the place of sunquake initiation, and then the particles travel along the flare magnetic loop to the other footpoint and caused the HXR emission.\n\nThe impulsive plasma motion in the lower solar atmosphere may be caused by fast heating due to Joule dissipation or sharp increasing of the Lorentz force. In the first case we can estimate the plasma momentum as $p_J\\sim \\tau V\\nabla P\\sim P\\tau L^2$, where $\\nabla P$ is the pressure gradient on the length scale, $L$. The pressure can be estimated from the energy equation\n\n$$\n\\frac{dP}{dt}=\\frac{j^2}{\\sigma_{eff}}-L_{rad}\n$$\n\nwhere $L_{rad}$ is the radiation heat loss which is the main source of cooling in the lower solar atmosphere. From this equation $P\\lesssim j^2\\tau\/\\sigma_{eff}$ and, hence, $p_J \\lesssim (j\\tau L)^2\/\\sigma_{eff}\\sim 10^{23}$ g$\\cdot$cm s$^{-1}$.\n\nThe plasma momentum, associated with the Lorentz force, is $p_L\\sim jB\\tau L^3\/c\\sim 10^{22}$ g$\\cdot$cm s$^{-1}$, where $B\\sim 100$ G is the magnetic field in the sunquake source, and $c$ is the speed of light.\n\nFrom the estimated values of $Q_j$, $p_J$ and $p_L$ one can conclude that the appearance of strong electric currents in the lower solar atmosphere is sufficient to explain the flare energy release and generation of the sunquake. Moreover, these estimations show that the electric current driven disturbances are sufficiently strong, and also that the electric currents are concentrated in the place of the sunquake initiation while the strongest HXR impulse is $\\sim 3$ Mm away. Therefore, it is likely that not only high-energy particles play significant role in the flares, as assumed by the standard flare model, but also electric currents in the lower solar atmosphere can be also a significant part of flare energy release. In our recent paper, we discuss the relationship between electric currents and the fine structure of flare ribbons \\citep{Sharykin2014}.\n\n\\section{Summary and conclusion}\n\nThe main results of the work are as the following:\n\n\\begin{enumerate}\n\\item We observed a strong sunquake event in a weak C-class flare.\n\\item The sunquake is initiated exactly, within 3 sec observational accuracy, during the burst of the HXR emission.\n\\item The place of the photospheric impact associated with the sunquake generation corresponds to the weaker HXR emission source, while there is no significant photospheric impact in the stronger HXR emission source, which is located at the opposite footpoint of a flare loop observed in the EUV AIA images.\n\\item The place of the photospheric impact associated with sunquake initiation corresponds to the most intense electric currents.\n\\item The total (C7.0-M1.9) flare event temporarily correlates with the maxima of vertical and transversal electric currents estimated in the energy release site.\n\\end{enumerate}\n\nThe main conclusion of the presented observational results is that the helioseismic response (sunquake) and flare energy release in the lower solar atmosphere may have strong connection to photospheric electric currents. The sunquake impact may be initiated by a pressure gradient caused due a rapid current dissipation or impulsive Lorentz force. The discovery of the strong photospheric impact produced by a weak C7 flare, which initiated the helioseismic response, opens new perspectives for studying the flare energy release and transport because such flares usually have relatively simple magnetic topology and do not saturate detectors of space and ground-based telescopes. However, our results show that high spatial and temporal resolutions are needed for these studies.\n\nThe work was partially supported by RFBR grant 13-02-91165, President's grant MK-3931.2013.2, NASA grant NNX14AB70G, and NJIT grant.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nRecent cosmological observations point to an strong evidence for an spatially flat and accelerated expanding universe\n\\cite{Riess2011, Komatsu2011, Reid2010}. Despite the great agreement of observations with the concordance model\n\\cite{Li2011} \\footnote{The Cosmological Constant Model.}, it is a fact that quintom model, whose Equation of State\n(EoS) can cross the cosmological constant barrier $w=-1$, is not exclude by observations \\cite{Alam2004a,Feng2005,Huterer2005,Nesseris2007,Jassal2010,Novosyadlyj2012,Novosyadlyj2012a,Hinshaw2012}. A popular way\nto realize a viable quintom model and, at the same time, avoid the restrictions imposed by the \\textit{No-Go Theorem} \\cite{Vikman2005,Hu2005, Caldwell2005, Xia2008, Cai2010}\nis the introduction of extra degrees of freedom\\footnote{The only way to realize the crossing\nwithout any ghosts and gradient instabilities in standard gravity and with one single scalar degree of freedom was obtained in \\cite{Deffayet2010}. \n}. Following\nthis recipe, the simple quintom paradigm requires a canonical quintessence scalar\nfield $\\sigma$ and simultaneously a phantom scalar field $\\phi$ where the effective potential can be of arbitrary form,\nwhile the two components can be either coupled \\cite{Zhang2006} or decoupled \\cite{Feng2005, Guo2005}. \n\nThe properties of the quintom models have been studied from different points of view. Among them, the phase space studies,\nusing the dynamical systems tools, are very useful in order to analyze the asymptotic behavior of the model. In quintom models\n this program have been carried out in \\cite{Guo2005, Zhang2006, Lazkoz2006, Lazkoz2007, Setare2009, Setare2008a, Setare2009c, Cai2010}.\n In \\cite{Guo2005} the decoupled case between the canonical and phantom field with an exponential potential is studied shown that\n the phantom-dominated scaling solution is the unique late-time attractor. In \\cite{Zhang2006} the potential considers the interaction\n between the fields and shows that in the absence of interactions, the solution dominated by the phantom field should be the attractor\n of the system and the interaction does not affect its attractor behavior. This result is correct only in the case in which the\n existence of the phantom phase excludes the existence scaling attractors \\cite{Lazkoz2006}. Some of these\nresults were extended in \\cite{Lazkoz2007} for arbitrary potentials. In \\cite{Setare2009c} the authors showed that all quintom models\n with nearly flat potentials converge to a single expression for EoS of dark energy, in addition, the necessary conditions for the\n determination of the direction of the $w=-1$ crossing was found.\n\n\nThe aim of this paper is to extend the study of Refs. \\cite{Guo2005,\nLazkoz2006, Lazkoz2007, Setare2009, Cai2010} -investigation of the\ndynamics of quintom cosmology- to include a wide variety of\npotential beyond the exponential potential without interaction\nbetween the fields, all of them can be constructed using the Bohm\nformalism \\cite{Guzman2007, Socorro2010, Bohm1952} of the quantum\nmechanics under the integral systems premise, which is known as\nquantum potential approach. This approach makes it possible to\nidentify trajectories associated with the wave function of the\nuniverse \\cite{Guzman2007} when we choose the superpotential\nfunction as the momenta associated to the coordinate field $q$. This\ninvestigation was undertaken within the framework of the\nminisuperspace approximation to quantum theory when we investigate\nthe dynamics of only a finite number of models. Here we make use of\nthe dynamical systems tools to obtain useful information about the\nasymptotic properties of the model. In order to be able to analyze\nself-interaction potentials beyond the exponential one, we rely on\nthe method introduced in Ref. \\cite{Fang2009} in the context of\nquintessence models and that have been generalized to several cosmological contexts\nlike: Randall-Sundrum II and DGP branes \\cite{Leyva2009, Escobar2012a,\nEscobar2012e}, Scalar Field Dark Matter models \\cite{Matos2009},\ntachyon and phantom fields \\cite{Quiros2010, Fang2010a,\nFarajollahi2011} and loop quantum gravity \\cite{Xiao2011j}.\n\nThe plan of the paper is as follow: in section \\ref{ss1} we introduce the quintom model for arbitrary potentials and in section \\ref{ss2}\n we build the corresponding autonomous system. The results of the study of the corresponding critical points,\n their stability properties and the physical discussion are shown in section \\ref{s2}. The section \\ref{ss4} is devoted to conclusions. Finally, we include in the two appendices \\ref{apen1} and \\ref{apen2} the center manifold calculation of the solutions dominated by either the phantom or quintessence potential. \n\n\\section{The model}\\label{ss1}\nThe starting action of our model, containing the canonical field $\\sigma$ and the phantom field $\\phi$, is \\cite{Feng2005, Guo2005, Zhang2006}:\n\n\\begin{eqnarray}\\label{action}\nS=\\int d^{4}x\\sqrt{-g}\\left(\\frac{1}{2}R-\\frac{1}{2}g^{\\mu\\nu}\\partial_{\\mu}\\sigma\\partial_{\\nu}\\sigma+V_{\\sigma}(\\sigma) + \\right. \\nonumber\\\\\n \\left. + \\frac{1}{2}g^{\\mu\\nu}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi+V_{\\phi}(\\phi)\\right),\n\\end{eqnarray}\n\nwhere we used natural units ($8 \\pi G=1$) and $V_{\\sigma}(\\sigma)$ and $V_{\\phi}(\\phi)$ are respectively the self interactions\npotential of the quintessence and phantom fields. \n\nFrom this action the Friedmann equations for a flat geometry reads \\cite{Guo2005, Zhang2006}:\n\\begin{equation}\\label{F1}\n H^{2}=\\frac{1}{3}\\left( \\frac{\\dot{\\sigma}^{2}}{2}+V_{\\sigma}(\\sigma)-\\frac{\\dot{\\phi}^{2}}{2}+V_{\\phi}(\\phi)\\right)\n\\end{equation}\n\\begin{equation}\\label{F2}\n \\dot{H}=-\\frac{1}{2}\\left(\\dot{\\sigma}^{2}-\\dot{\\phi}^{2} \\right)\n\\end{equation} where \nwhere $H=\\frac{\\dot{a}}{a}$ is the Hubble parameter and the dot denotes derivative with respect the time. \n\nThe evolution of the quintessence and phantom field are:\n\\begin{equation}\\label{KG1}\n \\ddot{\\sigma}+3H\\dot{\\sigma}+V_{\\sigma}'(\\sigma)=0\n\\end{equation}\n\\begin{equation}\\label{KG2}\n \\ddot{\\phi}+3H\\dot{\\phi}- V_{\\phi}'(\\phi)=0,\n\\end{equation} where the coma denotes the derivative of a function with respect to their argument. \n\nAdditionally we can introduce the total energy density and pressure as:\n\\begin{equation}\n \\rho_{DE}=\\rho_{\\sigma}+\\rho_{\\phi},\\;\\;\\;p_{DE}=p_{\\sigma}+p_{\\phi}\n\\end{equation}\nwhere\n\\begin{equation}\n \\rho_{\\sigma}=\\frac{\\dot{\\sigma}^2}{2}+V_{\\sigma}(\\sigma),\\;\\;\\;\\rho_{\\phi}=-\\frac{\\dot{\\phi}^2}{2}+V_{\\phi}(\\phi)\n\\end{equation}\n\\begin{equation}\n p_{\\sigma}=\\frac{\\dot{\\sigma}^2}{2}-V_{\\sigma}(\\sigma),\\;\\;\\;p_{\\phi}=-\\frac{\\dot{\\phi}^2}{2}-V_{\\phi}(\\phi)\n\\end{equation}\nand its equation of state parameter is given by\n\\begin{equation}\\label{weff}\n w_{eff}=\\frac{p_{\\sigma}+p{\\phi}}{\\rho_{\\sigma}+\\rho_{\\phi}}=\\frac{\\dot{\\sigma}^{2}-\\dot{\\phi}^{2}-2V_{\\sigma}(\\sigma)-2V_{\\phi}(\\phi)}{\\dot{\\sigma}^{2}-\\dot{\\phi}^{2}+2V_{\\sigma}(\\sigma)+2V_{\\phi}(\\phi)}\n\\end{equation}\nand \n\\begin{equation}\\label{omegaaa}\n \\Omega_{\\sigma}=\\frac{\\rho_{\\sigma}}{\\rho_{DE}},\\;\\;\\Omega_{\\phi}=\\frac{\\rho_{\\phi}}{\\rho_{DE}}\n\\end{equation}\n\\begin{equation}\n \\Omega_{\\sigma}+ \\Omega_{\\phi}=1\n\\end{equation}\nare the the individual and total dimensionless densities parameters. \n\n\\section{The autonomous system}\\label{ss2}\nIn order to study the dynamical properties of the system\n(\\ref{F1}-\\ref{KG2}) we introduce the following dimensionless phase\nspace variables to build an autonomous system \\cite{Copeland1998, Chen2009}:\n\\begin{equation}\\label{nv1}\nx_{\\sigma}=\\frac{\\dot{\\sigma}}{\\sqrt{6}H}, \\\\\nx_{\\phi}=\\frac{\\dot{\\phi}}{\\sqrt{6}H}, \\\\\ny_{\\sigma}=\\frac{\\sqrt{V_{\\sigma}(\\sigma)}}{\\sqrt{3}H}, \\\\\n\\end{equation}\n\\begin{equation}\n\\lambda_{\\sigma}=-\\frac{V_{\\sigma}'(\\sigma)}{V_{\\sigma}(\\sigma)}, \\\\\n\\lambda_{\\phi}=-\\frac{V_{\\phi}'(\\phi)}{V_{\\phi}(\\phi)}, \\\\\n\\end{equation}\n\nNotice that the phase space variables $\\lambda_{\\sigma}$ and\n$\\lambda_{\\phi}$ are sensitive of the kind of self interactions\npotential chosen for quintessence and phantom component,\nrespectively and are introduced in order to be able to study\narbitrary potentials. Applying the above dimensionless variables to\nthe system (\\ref{F1}-\\ref{KG2}) we obtain the following autonomous\nsystem:\n\n\\begin{eqnarray}\n \\frac{d x_{\\sigma}}{dN}&=& -3 x_{\\sigma} \\left(1+x_\\phi^2-x_\\sigma^2 \\right)+\\sqrt{\\frac{3}{2}} y_{\\sigma}^2 \\lambda_\\sigma \\label{auto1}\\\\\n \\frac{d x_{\\phi}}{dN}&=& -3 x_{\\phi} \\left(1+x_\\phi^2-x_\\sigma^2 \\right)+\\nonumber\\\\&&-\\sqrt{\\frac{3}{2}}\\left(1+x_\\phi^2-x_\\sigma^2-y_\\sigma^2\\right) \\lambda_\\phi \\label{auto2}\\\\\n\\frac{d y_{\\sigma}}{dN} &=&\\frac{1}{2}y_\\sigma \\left(6x_\\sigma^2-\\sqrt{6} x_\\sigma\n \\lambda_\\sigma -6 x_\\phi^2\\right) \\label{auto3}\\\\\n \\frac{d \\lambda_{\\sigma}}{dN}&=& -\\sqrt{6} x_{\\sigma} f(\\lambda_\\sigma) \\label{auto4}\\\\\n\\frac{d \\lambda_{\\phi}}{dN} &=& -\\sqrt{6}x_\\phi g(\\lambda_\\phi)\\label{auto5}\n\\end{eqnarray}\nwhere $N=\\ln a$ is the number of e-foldings and $f(\\lambda_\\sigma)= \\lambda_\\sigma\n^2(\\Gamma_{\\sigma}-1)$ and $ g(\\lambda_\\phi)= \\lambda_\\phi\n^2(\\Gamma_{\\phi}-1) $ where:\n\\begin{equation}\n\\Gamma_{\\sigma}=\\frac{V_{\\sigma}(\\sigma)V_{\\sigma}''(\\sigma)}{(V_{\\sigma}'(\\sigma))^{2}},\n\\;\\;\\; \\Gamma_{\\phi}=\\frac{V_{\\phi}(\\phi)V_{\\phi}''(\\phi)}{(V_{\\phi}'(\\phi))^{2}}\n\\end{equation}\nIn order to get from the autonomous equation (\\ref{auto1}-\\ref{auto5}) a closed system of ordinary differential equation\nwe have assumed that the funtions $\\Gamma_{\\sigma}$ and $\\Gamma_{\\phi}$ can be written as a function of the\nvariables $\\lambda_{\\sigma}\\in\\mathbb{R}$ and $\\lambda_{\\phi}\\in\\mathbb{R}$ respectively \\cite{Fang2009}.\n\nThe phase space for the autonomous dynamical system driven by de evolutions of Eqs. (\\ref{auto1}-\\ref{auto5}) can be defined as follows:\n\\begin{eqnarray}\n \\Psi=\\{(x_{\\sigma},x_{\\phi},y_{\\sigma}):y_{\\sigma}\\geq 0, x_\\sigma^2-x_\\phi^2+y_\\sigma^2\\leq 1 \\}\\times\\nonumber\\\\\n \\times\\{(\\lambda_{\\sigma},\\lambda_{\\phi})\\in\\mathbb{R}^{2} \\}\n\\end{eqnarray}\n\nWith the aim of explain the physical significance of the critical points of the autonomous system (\\ref{auto1}-\\ref{auto5}) we need to obtain the relevant cosmological parameters in terms of the dimensionless phase space variables (\\ref{nv1}). Following this, the cosmological parameter (\\ref{weff}) and (\\ref{omegaaa}) can be expressed as\n\\begin{equation}\n w_{eff}=-1+2x_{\\sigma}^2-2x_{\\phi}^2\n\\end{equation}\n\\begin{equation}\n \\Omega_{\\sigma}=x_{\\sigma}^2+y_{\\sigma}^2,\\;\\;\\ \\Omega_{\\phi}=1-x_{\\sigma}^2-y_{\\sigma}^2,\n\\end{equation}\nwhile the deceleration parameter becomes\n\\begin{equation}\n q=-\\left[1+\\frac{\\dot{H}}{H^2}\\right]=-1+3x_{\\sigma}^2-3x_{\\phi}^2.\n\\end{equation}\n\n\n\n\\section{Critical points and stability}\\label{s2}\nThe critical points of the system (\\ref{auto1}-\\ref{auto2}) are\nsummarized in Table \\ref{tab1}. The eigenvalues of the corresponding\nJacobian matrices are show in Table \\ref{tab2}. In both cases\n$\\lambda_{\\sigma}^{\\ast}$ and $\\lambda_{\\phi}^{\\ast}$ are the values\nwhich makes the functions\n$f(\\lambda_{\\sigma})=\\lambda_\\sigma^2\\left(\\Gamma_{\\sigma}-1\\right)$\nand $g(\\lambda_{\\phi})=\\lambda_\\phi^2\\left(\\Gamma_{\\phi}-1\\right)$\nvanish respectively.\n\n\n\\begin{table*}\\caption[crit]{Properties of the critical points for the\nautonomous system (\\ref{auto1}-\\ref{auto5})}\n\\begin{tabular}\n{l c c c c c c c c c c}\n\\hline\\hline\\\\[-0.3cm]\n$Label$&$x_{\\sigma}$&$y_{\\sigma}$&$x_{\\phi}$&$\\lambda_{\\sigma}\n$&$\\lambda_{\\phi}$&Existence&$\\Omega_{\\sigma}$&$\\Omega_{\\phi}\n$&$q$&$w_{eff}$\\\\\n\\hline\\\\[-0.2cm]\n\n$P_1^\\pm$ &$0$ &$0$ &$\\pm i$ &$\\lambda_{\\sigma}$ &\n$\\lambda_{\\phi}^{\\ast}$ & Non real &$0$ &$1$& $2$ & $1$\\\\[0.2cm]\n\n$P_2^\\pm$ &$\\pm1$ & $0$ &$0$ &$\\lambda_{\\sigma}^{\\ast}$\n&$\\lambda_{\\phi}$& Always &$1$ &$0$ &$2$ &$1$ \\\\[0.2cm]\n\n$P_3^\\pm$ &$\\pm\\sqrt{1+x_\\phi^2}$ &$0$ & $x_\\phi$ &$\\lambda_{\\sigma}^{\\ast}$ &\n$\\lambda_{\\phi}^{\\ast}$& \\textquotedblright &$1+x_{\\phi}^{2}$ &$-x_{\\phi}^{2}$\n&$2$ &$1$ \\\\[0.2cm]\n\n$P_{4}$ &$\\frac{\\lambda_{\\sigma}^{\\ast}}{\\sqrt{6}}$\n&$\\sqrt{1-\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{6}}$&$0$\n&$\\lambda_{\\sigma}^{\\ast}$ &$\\lambda_{\\phi}$& $-\\sqrt{6}\\leq \\lambda_{\\sigma}^{\\ast} \\leq \\sqrt{6}$ &$1$ &$0$\n&$-1+\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{2}$ &\n$-1+\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{3}$ \\\\[0.2cm]\n\n$P_5$ & $0$ & $0$ & $0$ & $\\lambda_{\\sigma}$ &\n$0$& Always & $0$ & $1$ &$-1$ & $-1$ \\\\[0.2cm]\n\n$P_6$ &$0$ &$1$ &$0$ &$0$ &$\\lambda_{\\phi}$ & \\textquotedblright & $1$ &\n$0$ & $-1$ &$-1$ \\\\[0.2cm]\n\n$P_{7}$ &$0$ &$y_{\\sigma}$ &$0$ & $0$\n&$0$ & $0< y_{\\sigma}< 1$ &$y_{\\sigma}^{2}$ &$1-y_{\\sigma}^{2}$ &\n$-1$ &$-1$\\\\ [0.2cm]\n\n$P_{8}$&$0$ &$0$ &$-\\frac{\\lambda_{\\phi}^{\\ast}}{\\sqrt{6}}$\n&$\\lambda_{\\sigma}$ & $\\lambda_{\\phi}^{\\ast}$ & $\\lambda_{\\sigma}\n\\in \\mathbb{R}$ &$0$ &$1$\n&$-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{2}$\n&$-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{3}$ \\\\ [0.2cm] \\hline \\hline\n\\end{tabular}\\label{tab1}\n\\\\ [0.2cm]\n\\end{table*}\n\n\\begin{table*}\\caption[crit2]{Eigenvalues of the linear perturbation matrix associated to each of the critical points displayed in Table \\ref{tab1}}\n\\begin{tabular}\n{l c c c c c c c c c c}\n\\hline\\hline\\\\[-0.3cm]\n$Label$&$m_1$&$m_2$&$m_3$&$m_4$&$m_5$\\\\\n\\hline\\\\[-0.2cm]\n\n$P_1^\\pm$ & $3$ & $0$ & $0$ & $\\mp {i}\\sqrt{6}g'(\\lambda_{\\phi}^{\\ast})$ & $6\\mp{ i}\\sqrt{6}\\lambda_{\\phi}^{\\ast}$\\\\[0.2cm]\n\n$P_2^\\pm$ & $6$ & $0$ & $0$ & $\\mp\\sqrt{6}f'(\\lambda_{\\sigma}^{\\ast})$ & $3\\mp\\sqrt{\\frac{3}{2}}\\lambda_\\sigma^{\\ast}$\\\\[0.2cm]\n\n$P_3^\\pm$ & $0$ & $-\\sqrt{6} g'(\\lambda_{\\phi}^{\\ast}) {x_\\phi}$&$\\mp\\sqrt{6} {f'(\\lambda_{\\sigma}^{\\ast})}\n \\sqrt{x_\\phi^2+1}$&$3\\mp\\sqrt{\\frac{3}{2}} \\sqrt{x_\\phi^2+1}\n \\lambda_\\sigma ^{\\ast}$& $6-\\sqrt{6} x_\\phi \\lambda_\\phi ^{\\ast}$ \\\\[0.2cm]\n\n$P_4$ & $0$ &$-{f'(\\lambda_{\\sigma}^{\\ast})} \\lambda \\sigma ^{\\ast}$ & $\\left(\\lambda \\sigma\n ^{\\ast}\\right)^2$ &$\\frac{1}{2} \\left(\\left(\\lambda \\sigma\n ^{\\ast}\\right)^2-6\\right)$ &$\\frac{1}{2} \\left(\\left(\\lambda \\sigma\n ^{\\ast}\\right)^2-6\\right)$\\\\ [0.2cm]\n\n$P_5$ & $-3$ & $0$ & $0$ & $-\\frac{3}{2}\\left(1+\\sqrt{1+\\frac{4}{3}g(0)}\\right)$ & $-\\frac{3}{2}\\left(1-\\sqrt{1+\\frac{4}{3}g(0)}\\right)$\\\\[0.2cm]\n\n$P_6$ & $-3$ & $0$ & $0$ & $-\\frac{3}{2}\\left(1+\\sqrt{1-\\frac{4}{3}f(0)}\\right)$ & $-\\frac{3}{2}\\left(1-\\sqrt{1-\\frac{4}{3}f(0)}\\right)$\\\\[0.2cm]\n\n$P_7$ & $0$ & $\\frac{1}{2} \\left(-\\sqrt{9-12 f(0) y_\\sigma^2}-3\\right)$ & $\\frac{1}{2} \\left(\\sqrt{9-12 f(0) y_\\sigma^2}-3\\right)$ & $\\frac{1}{2} \\left(-\\sqrt{9-12 g(0) \\left(y_\\sigma^2-1\\right)}-3\\right)$ & $\\frac{1}{2} \\left(\\sqrt{9-12 g(0)\n \\left(y_\\sigma^2-1\\right)}-3\\right)$\\\\[0.2cm]\n\n$P_8$ & $0$ &${g'(\\lambda_{\\phi}^{\\ast})} \\lambda_\\phi ^{\\ast}$& $-\\frac{1}{2} \\left(\\lambda_\\phi\n ^{\\ast}\\right)^2$&$-\\frac{1}{2} \\left(\\left(\\lambda_\\phi\n ^{\\ast}\\right)^2+6\\right)$&$-\\frac{1}{2} \\left(\\left(\\lambda_\\phi\n ^{\\ast}\\right)^2+6\\right)$\\\\ [0.2cm]\n\n\n\n\\hline \\hline\\\\[-0.3cm]\n\\end{tabular}\\label{tab2}\n\\end{table*}\nAs we see from Table \\ref{tab1}, the points $P_1^\\pm$ do not exist in the strict sense ($x_\\phi$ is purely imaginary at the fixed points).\nPoint $P_5$ is associated with a combination of a phantom potential whose first $\\phi$-derivative vanishes at some\/several point\/points, i.e.,\n$\\lambda_\\phi=0$ (this case include the exponential potential whose $\\phi$-derivative at any order vanish everywhere) and an arbitrary self\ninteraction potential for the quintessence component (arbitrary value of $\\lambda_{\\sigma}$). Point $P_6$ is associated with a combination\n of a quintessence potential whose first $\\sigma$-derivative vanishes at some\/several point\/points, i.e., $\\lambda_\\sigma=0$\n (this case include the exponential potential whose $\\sigma$-derivative at any order vanish everywhere) and an arbitrary self\n interaction potential for the phantom component (arbitrary value of $\\lambda_{\\phi}$). Point $P_7$ is associated with a\n combination of a phantom potential whose first $\\phi$-derivative vanishes at some\/several point\/points, i.e., $\\lambda_\\phi=0$\n (this case include the exponential potential whose $\\phi$-derivative at any order vanish everywhere) and a self interaction\n potential for the quintessence component whose first $\\sigma$-derivative vanishes also at some\/several point\/points, i.e., $\\lambda_\\phi=0.$ \nIt is worth noticing that the existence of points $P_2^\\pm$, $P_3^\\pm$, $P_4$, $P_8$ and $P_9$ depends of the concrete form\nof the potential. From the table of the eigenvalues, notice, besides, that all the points belongs to nonhyperbolic sets of\ncritical point with a least one null eigenvalue.\n\n\\subsection{Stability of the critical points}\n\nAlthough all these critical points are shown in the Tables \\ref{tab1} here we have summarized their basic properties:\n\\begin{itemize}\n \\item $P_{1}^\\pm$, $P_{2}^\\pm$ and $P_{3}^\\pm$ correspond to a solution dominated by the kinetic energy of the scalar fields\n (stiff fluid solution: $q=2$ and $\\omega=1$). The exact dynamical behavior differs for each points. $P_1^\\pm$ corresponds to\n a phantom kinetic energy dominated ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). However, these points have a purely imaginary\n value of $x_\\phi,$ thus, they do not exists in the strict sense. They have a three$-$dimensional center subspace and a two-dimensional\n unstable manifold ($m_1=3>0,\\; \\Re(m_5)=6>0$). Thus they cannot be late-time attractors. $P_3^\\pm$ represents an scaling regimen\n between the kinetics energies of the quintessence and phantom fields ($\\Omega_{\\sigma}=1+x_{\\phi}^{2}$ and $\\Omega_{\\phi}=-x_{\\phi}^{2}$).\n These points depend of the form of the potentials and under certain conditions they have a four dimensional unstable subspace which could\n correspond to the past attractor. However, this point is unphysical since $\\Omega_{\\phi}<0$ . $P_{2}^\\pm$ is dominated by the\n quintessence kinetic term ($\\Omega_{\\sigma}=1$ and $\\Omega_{\\phi}=0$). Since they are non-hyperbolic due to the existence of\n two null eigenvalues, we are not able to extract information about their stability by using the standard tools of the linear\n dynamical analysis. However, since these points seems to be particular cases of $P_3^\\pm,$ they should share the same dynamical\n behavior. Because all of these points are nonhyperbolic, as we notice before, we cannot rely on the standard linear\n dynamical systems analysis for deducing their stability. Thus, we need to rely our analysis on numerical inspection of\n the phase portrait for specific potentials or use more sophisticated techniques like Center Manifold theory.\n \\item $P_4$ is an scaling solution between the kinetic and the potential energy of the quintessence component of dark energy. This solution in sensitive to the explicit form of the potential. This is always a saddle equilibrium point in the phase space since $m_2=(\\lambda_{\\sigma}^{\\ast})^{2}$ and $m_4=\\frac{1}{2}((\\lambda_{\\sigma}^{\\ast})^{2}-6)$ are of opposite sign in the existence region of this point. It represents an accelerated solution for a potential $V_\\sigma(\\sigma)$ whose function $f(\\lambda_{\\sigma})$ vanish for $\\lambda_{\\sigma}=\\lambda_{\\sigma}^{\\ast}$ in the interval $-\\sqrt{2}<\\lambda_{\\sigma}^{\\ast}<\\sqrt{2}$, leading to a $-1\\leq w_{eff}<-1\/3$. When $\\lambda_{\\sigma}^{\\ast}=0$ the critical point $P_4$ becomes in $P_6$. In the regions $-\\sqrt{6}\\leq\\lambda_{\\sigma}^{\\ast}\\leq-\\sqrt{2}$ or $\\sqrt{2}\\leq\\lambda_{\\sigma}^{\\ast}\\leq\\sqrt{6}$, the critical point $P_4$ represents a non-accelerated phase. A very interesting issue of this critical point appears when, for an specific form of\nthe quintessence potential, $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3}$, driving to $w_{eff}=0$. This means that the quintessence field is able to mimic the dark matter behavior.\n \\item $P_5$, $P_6$ and $P_7$ represents solutions dominated by the potential energies of the potentials (all of them represent de Sitter solutions: $q=-1$ and $w_{eff}=-1$). Once again the exact dynamical nature differs from one point to the other: $P_5$ is dominated by the potential energy of the phantom component ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). Because of the existence of two null eigenvalues is not possible to conclude about its dynamics. However it has a three-dimensional stable manifold for $g(0)<0$ (in the interval $g(0)<-\\frac{3}{4}$ it has to complex conjugated eigenvalues with negative real parts). In this cases it is worthy to analyze its stability using the center manifold theory. $P_6$ is a critical point dominated by the quintessence potential energy term ($\\Omega_{\\sigma}=1$ and $\\Omega_{\\phi}=0$), despite its nonhyperbolicity, it has three-dimensional stable manifold for $f(0)>0$ (in the case $f(0)>\\frac{3}{4}$ it has to complex conjugated eigenvalues with\nnegative real parts), thus, it is worthy to analyze its stability using the center manifold theory. $P_7$ denotes a segment (curve) of non-isolated fixed points, representing a scaling regimen between the quintessence and phantom potential ($\\Omega_{\\sigma}=y_{\\sigma}^{2}$ and $\\Omega_{\\phi}=1- y_{\\sigma}^{2}$). The existence of one non-zero eigenvalue is due to the fact that it is a curve of fixed points. As an invariant set of non-isolated singular points it is normally-hyperbolic, since the eigenvector associated to the zero eigenvalue, $(0,0,1,0,0)^T,$ is tangent to the curve. Thus its stability is determined by the sign of the remaining non-null eigenvalues. Hence, it is stable for $00, \\; g(0)<0$ or a saddle otherwise.\n\\item $P_8$ is a line of fixed points parameterized by $\\lambda_\\sigma\\in\\mathbb{R}$. The existence of one non-zero eigenvalue is due to the fact that it is a curve of fixed points. As an invariant set of non-isolated singular points it is normally-hyperbolic, since the eigenvector associated to the zero eigenvalue, $(0,0,0,1,0)^T,$ is tangent to the curve. Thus its stability is determined by the sign of the remaining non-null eigenvalues. From table \\ref{tab2} follows that $P_8$ admits a four dimensional stable subspace provided $g'(\\lambda_{\\phi}^{\\ast}) \\lambda_{\\phi}^{\\ast}<0$, thus, the invariant curve is stable. It represents accelerated solutions dominated by the phantom potential providing a crossing through the phantom divide ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). For every value of $\\lambda_{\\phi}^{\\ast}$ this point provide the typical superaccelerated expansion of quintom paradigm ($w=-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{3}$) the only exception occurs when $\\lambda_{\\phi}^{\\ast}=0$\nrecovering the behavior of the de Sitter solution $P_5$ ($\\omega=-1$). This line of critical point corresponds to the stable point $P$ in \\cite{Guo2005} and B in \\cite{Cai2010} (phantom dominated solution). Summarizing, the line $P_8$ is the late time stable attractor provided $g'(\\lambda_{\\phi}^{\\ast}) \\lambda_{\\phi}^{\\ast}<0$, otherwise, it is a saddle point.\n\\end{itemize}\n\n \\subsection{Cosmological consequences}\n\nAs was shown in the previous subsection the autonomous systems only admits\nseven classes of critical points (some of them are actually curves) \\footnote{$P_{1}^\\pm$ and $P_{3}^\\pm$ are ruled out. The first one because of they lead to imaginary values of dimensionless variable $x_\\phi$. And the last one because of is outside of the physical phase space, representing a critical point with a negative energy density $\\Omega_{\\phi}<0$.}. The curves $P_{2}^\\pm$ correspond to decelerated solutions, with $q=2$, where the Friedmann constraint (\\ref{F1}) is dominated by the kinetic energy of the quintessence field with an equation of state of stiff type, $w_{eff}=1$. These solutions are only relevant a early times and should be unstable \\cite{Copeland1998}. Unfortunately these critical points are nonhyperbolic (it has two zero eigenvalues) meaning that is not possible to obtain conclusions about its stability with the previous linear analysis. However the numerical analysis performed in the next subsection with a particular potentials confirm the previous results in \nliterature.\n\nAn important result come from the stability of critical point $P_4$. This points exists if $-\\sqrt{6}\\leq\\lambda_{\\sigma}^{\\ast}\\leq\\sqrt{6}$ and always behave as a saddle fixed point. The latter means that under certain initial conditions the orbits in the phase space will approach to this point spending some time in its vicinity before being repelled toward the attractor solution of the system. In the case of this point, as we mentioned before, if the quintessence potential fulfill the condition:\n\\begin{equation}\\label{condiDEDM}\n \\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3}\n\\end{equation}\nthen the effective equation of state of this dark energy component would mimic pressureless fluid ($w_{eff}=0$), in other words: it will dynamically behave exactly as cold dark matter. The possibility of this dynamical characteristic impose a fine tunning over the shape of quintessence potentials and a priori there is no guarantee that all possible quintessence potentials may satisfy the above condition (\\ref{condiDEDM}). Let's note that in order to obtain the lower possible dimensionality of the phase space and to studying in a relatively simple way the effects of include arbitrary quintom potentials, we have neglected the contribution of the usual matter fields: radiation and baryonic matter in our model \\footnote{See Eqs. (\\ref{action}-\\ref{F1}).}. As a result, a full study of important aspects, derived from realization of condition (\\ref{condiDEDM}), such as: transition redshift between the decelerated and accelerated expansion phase and the clustering properties of this \\textit{effective dark matter} \nare beyond the present study and will be left for a future paper.\n\nAnother important characteristics of the model is the presence of three accelerated solutions, described by critical points $P_5$, $P_6$ and $P_7$. All of them are de Sitter solutions ($w_{eff}=-1$) dominated by the potentials of the scalar fields. As in the case of $P_4$, they behave as saddle points and, depending on the initial conditions, the orbits can evolve from the unstable fixed point ($P_{2}^\\pm$ in our case) towards one or the other of the saddle points. A favorable scenario would be one in which the initial condition lead to an evolution from $P_{2}^\\pm$ to the saddle point $P_4$ \\footnote{we are assuming that if (\\ref{condiDEDM}) is fulfilled, then quintessence field behave as the Dark Matter.} and then, the orbits tend to one of the de Sitter solutions $P_5$, $P_6$ or $P_7$ or to the late time phantom attractor ($P_8$). In terms of the cosmological evolution of the Universe, the above favorable scenario implies that the Universe started at early times from an stage dominated by the kinetic \nterm of \nthe quintessence, then evolve into an epoch dominated by the \\textit{effective dark matter} and finally enter in the final phase of accelerated expansion. This accelerated phase can be the de Sitter solutions or a phantom dominated solution ($w_{eff}<-1$) \\footnote{In fact, these models admits the possibility of having two stable solutions: a de Sitter solution ($P_7$) and a phantom solution ($P_8$), each one within their basin of attraction as was shown in previous subsection.}. This final stage of evolutions towards critical point $P_8$ is consistent with the recent joint results from \\textit{WMAP}+\\textit{eCMB}+\\textit{BAO}+$H_0$+\\textit{SNe} \\cite{Hinshaw2012} which suggest a mild preference for a dark energy equation-of-state parameter in the phantom region ($w_{eff}<-1$). \n\nFinally, in order to examine the stability of the nonhyperbolic points that cannot consistently be studied via the present linear analysis, we present a concrete example. We provide a numerical elaboration of the phase space orbits of the corresponding quintom model.\n\n\\subsection{$V(\\sigma,\\phi)=V_{0}\\sinh^{2}(\\alpha\\sigma)+V_{1}\\cosh^{2}(\\beta\\phi)$}\\label{cosh}\nThis potential is derived, in a Friedmann-Robertson-Walker\ncosmological model, from canonical quantum cosmology under determined\nconditions in the evolution of our universe\\footnote{This is part of a forthcoming paper.}, using the bohmian\nformalism \\cite{Guzman2007}. For this potential:\n\\begin{equation}\\label{f1}\n f(\\lambda_{\\sigma})=-\\frac{\\lambda_{\\sigma}^{2}}{2}+ 2 \\alpha^{2}, \\;\\;\\lambda_{\\sigma}^{\\ast}=\\pm 2 \\alpha, \\;\\;f'(\\lambda_{\\sigma}^{\\ast})=-{\\lambda_{\\sigma}^{\\ast}}\n\\end{equation}\nand\n\\begin{equation}\\label{f2}\n g(\\lambda_{\\phi})=-\\frac{\\lambda_{\\phi}^{2}}{2}+ 2 \\beta^{2}, \\;\\;\\lambda_{\\phi}^{\\ast}=\\pm 2 \\beta, \\;\\;g'(\\lambda_{\\phi}^{\\ast})=-{\\lambda_{\\phi}^{\\ast}}.\n\\end{equation}\nFrom the Table \\ref{tab2} and the equation (\\ref{f2}) we see that the condition\nto ensure that Point $P_8$ has a four dimensional stable subspace is\nalways satisfied due to the opposite signs between\n$\\lambda_{\\phi}^{\\ast}$ and $g'(\\lambda_{\\phi}^{\\ast})$. In order to\nhaving achieved success scalar field dark matter domination era we\nneed that $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$ since this is the only way to have a standard transient matter dominated solution ($P_4$). Recall that for the choice $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$ the standard quintessence dominated solution mimics dark matter ($w_{eff}=0$).\nImposing the condition $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$ we have as a degree of freedom the potential\nparameter $\\alpha$ that can be adjusted using (\\ref{f1}). Furthermore, we impose\none of the following conditions:\n\\begin{equation}\n \\lambda_{\\sigma}^{\\ast}=\\sqrt{3},\\;\\lambda_\\phi ^{\\ast}\\leq -\\sqrt{6}, \\; 1