Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 608252, 20 pages http://dx.doi.org/10.1155/2015/608252

Research Article Dynamics of Mixed Dark Energy Domination in Teleparallel Gravity and Phase-Space Analysis

Emre Dil1 and Erdinç Kolay2

1 Department of Physics, Sinop University, Korucuk, 57000 Sinop, Turkey 2Department of Statistics, Sinop University, Korucuk, 57000 Sinop, Turkey

Correspondence should be addressed to Emre Dil; [email protected]

Received 16 October 2015; Revised 26 November 2015; Accepted 29 November 2015

Academic Editor: Frank Filthaut

Copyright © 2015 E. Dil and E. Kolay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We consider a novel dark energy model to investigate whether it will provide an expanding universe phase. Here we propose a mixed dark energy domination which is constituted by tachyon, quintessence, and phantom scalar fields nonminimally coupled to gravity, in the absence of background dark matter and baryonic matter, in the framework of teleparallel gravity. We perform the phase-space analysis of the model by numerical methods and find the late-time accelerated attractor solutions implying the acceleration phase of universe.

1. Introduction quantity of dark energy; its equation of state parameter 𝜔DE = 𝑝DE/𝜌DE,where𝑝DE and 𝜌DE are the pressure and energy The universe is known to be experiencing an accelerating density of the dark energy, respectively. For cosmological expansion by astrophysical observations such as Supernova constant boundary 𝜔DE =−1,butforquintessencethe Ia [1, 2], large scale structure [3, 4], the baryon acoustic parameter 𝜔DE ≥−1,forphantom𝜔DE ≤−1,andfor oscillations [5], and cosmic microwave background radiation nonminimally coupled tachyon with gravity both 𝜔DE ≤−1 [6–9]. and 𝜔DE ≥−1[16–18]. In order to explain the late-time accelerated expansion The scenario of 𝜔DE crossing the cosmological constant of universe, an unknown form of energy, called dark energy, boundary is referred to as a “Quintom” scenario. The explicit is proposed. This unknown component of energy possesses construction of Quintom scenario has a difficulty, due to a some interesting properties; for instance, it is not clustered no-gotheorem.Theequationofstateparameter𝜔DE of a butspreadallovertheuniverseanditspressureisnegativefor scalar field cannot cross the cosmological constant boundary driving the current acceleration of the universe. Observations according to no-go theorem, if the dark energy described by show that the dark energy occupies 70% of our universe. the scalar field is minimally coupled to gravity in Friedmann- What is the constitution of the dark energy? One can- Robertson-Walker (FRW) geometry. The requirement for didate for the answer of this question is the cosmological crossing the cosmological constant boundary is that the dark constant Λ having a constant energy density filling the space energy should be nonminimally coupled to gravity; namely, it homogeneously [10–13]. But cosmological constant is not should interact with the gravity [19–24]. There are also mod- well accepted since the cosmological problem [14] and the els in which possible coupling between dark energy and dark age problem [15]. For this reason, many other dark energy matter can occur [25, 26]. In this paper, we consider a mixed models have been proposed instead of the cosmological dark energy model constituted by tachyon, quintessence, and constant. Other candidates for dark energy constitution are phantom scalar fields nonminimally coupled to gravity. quintessence,phantom,andtachyonfields.Wecanbriefly The mixed dark energy model in this study is considered classifythedarkenergymodelsintermsofthemostpowerful in the framework of teleparallel gravity instead of classical 2 Advances in High Energy Physics gravity. The teleparallel gravity is the equivalent form of with a nonminimal coupling to the gravity can be written as the classical gravity, but in place of torsion-less Levi-Civita [16, 38, 39] connection, curvature-less Weitzenbock one is used. The Lagrangian of teleparallel gravity contains torsion scalar 𝑇 𝑇 𝑆=∫ 𝑑4𝑥𝑒 [ +𝜉 𝑓(𝜓)𝑇 constructed by the contraction of torsion tensor, in contrast 2𝜅2 1 to the Einstein-Hilbert action of classical gravity in which [ contraction of the curvature tensor 𝑅 is used. In teleparallel 𝜕 𝜓𝜕 𝜓 gravity, the dynamical variable is a set of four tetrad fields 𝜇] 𝜇 ] −𝑉𝜓 (𝜓) √1+𝑔 +𝜉2𝑔(𝜙)𝑇 constructing the bases for the tangent space at each point of 𝑉𝜓 (𝜓) space-time [27–29]. The teleparallel gravity Lagrangian with (1) only torsion scalar 𝑇 corresponds to the matter-dominated 1 1 − 𝑔𝜇]𝜕 𝜙𝜕 𝜙−𝑉 (𝜙) + 𝜉 ℎ (𝜎) 𝑇+ 𝑔𝜇]𝜕 𝜎𝜕 𝜎 universe; namely, it does not accelerate. Therefore, to obtain 2 𝜇 ] 𝜙 3 2 𝜇 ] a universe with an accelerating expansion, we can either replace 𝑇 with a function 𝑓(𝑇), the so-called 𝑓(𝑇) gravity ] (teleparallel analogue of 𝑓(𝑅) gravity) [30–32], or add an −𝑉𝜎 (𝜎) , unknown form of energy, so-called dark energy, to the ] teleparallel gravity Lagrangian that allows also nonminimal 𝑖 𝑖 coupling between dark energy and gravity to overcome the where 𝑒=det(𝑒𝜇)=√−𝑔 and 𝑒𝜇 are the orthonormal tetrad no-go theorem. The interesting feature of 𝑓(𝑇) theories is the components, such that existence of second or higher order derivatives in equations. 𝑖 𝑗 Therefore, we prefer the second choice; adding extra scalar 𝑔𝜇] =𝜂𝑖𝑗 𝑒𝜇𝑒], (2) fields of the unknown energy forms as dark energy. where 𝑖, 𝑗 runover0,1,2,and3forthetangentspaceateach As different dark energy models, interacting teleparallel 𝜇 darkenergystudieshavebeenintroducedintheliterature;for point 𝑥 of the manifold and 𝜇, ] take the values 0, 1, 2, and instance, Geng et al. [33, 34] consider a quintessence scalar 3 and are the coordinate indices of the manifold. While 𝑇 is field with a nonminimal coupling between quintessence and the torsion scalar, it is defined as gravity in the context of teleparallel gravity. The dynamics of 1 1 𝑇= 𝑇𝜌𝜇]𝑇 + 𝑇𝜌𝜇]𝑇 −𝑇𝜌 𝑇]𝜇. this model has been studied in [35–37]. Tachyonic teleparallel 4 𝜌𝜇] 2 ]𝜇𝜌 𝜇] 𝜌 (3) dark energy is a generalization of the teleparallel quintessence 𝜌 dark energy by introducing a noncanonical tachyon scalar Here 𝑇𝜇] is the torsion tensor constructed by the Weitzenbock 𝜌 field in place of the canonical quintessence field [18, 38–40]. connection Γ𝜇], such that [41] In this study, the main motivation is that we consider 𝜌 𝜌 𝜌 𝜌 𝑖 𝑖 a more general dark energy model including three kinds 𝑇𝜇] =Γ]𝜇 −Γ𝜇] =𝑒𝑖 (𝜕𝜇𝑒] −𝜕]𝑒𝜇). (4) of dark energy models, instead of taking one dark energy model as in [16, 18, 38–40]. In order to explain the expansion All the information about the gravitational field is con- 𝜌 of universe by adding scalar fields as the dark energy tained in the torsion tensor 𝑇𝜇] in teleparallel gravity. The constituents, there has never been assumed a cosmological Lagrangian of the theory is set up according to the condi- model including three kinds of dark energy models. We tions of invariance under general coordinate transformations, assume tachyon, quintessence, and phantom fields as a global Lorentz transformations, and the parity operations mixed dark energy model which is nonminimally coupled to [42]. 2 gravity in the framework of teleparallel gravity. We make the Furthermore, 𝜅 =8𝜋𝐺in (1) and 𝑓(𝜓), 𝑔(𝜙),and dynamical analysis of the model in FRW space-time. Later ℎ(𝜎) are the functions responsible for nonminimal coupling on, we translate the evolution equations into an autonomous between gravity 𝑇 and tachyon, quintessence, and phantom dynamical system. After that the phase-space analysis of the fields, respectively. 𝜉1, 𝜉2,and𝜉3 are the dimensionless model and the cosmological implications of the critical (or coupling constants and 𝑉𝜓(𝜓), 𝑉𝜙(𝜙),and𝑉𝜎(𝜎) are the fixed)pointsofthemodelwillbestudiedfromthestability potentials for tachyon, quintessence, and phantom fields, behavior of the critical points. Finally, we will make a brief respectively. summary of the results. We consider a spatially flat FRW metric

2 2 2 2 2 2 2. Dynamics of the Model 𝑑𝑠 =𝑑𝑡 −𝑎 (𝑡) [𝑑𝑟 +𝑟 𝑑Ω ], (5) 𝑖 Our model consists of three scalar fields as the three- and a tetrad field of the form 𝑒𝜇 = diag(1,𝑎,𝑎,𝑎).Thenthe component dark energy domination without background Friedmann equations for FRW metric read as dark matter and baryonic matter. These are the canonical 𝜙 𝜅2 scalar field, quintessence , and two noncanonical scalar 𝐻2 = (𝜌 +𝜌 +𝜌 ), fields, tachyon 𝜓 and phantom 𝜎, and all these scalar fields are 3 𝜓 𝜙 𝜎 (6) nonminimally coupled to gravity. Since we consider only the 2 dark energy dominated sector without the matter content of ̇ 𝜅 𝐻=− (𝜌𝜓 +𝑝𝜓 +𝜌𝜙 +𝑝𝜙 +𝜌𝜎 +𝑝𝜎), the universe, the action of the mixed teleparallel dark energy 2 Advances in High Energy Physics 3 where 𝐻=𝑎/𝑎̇ is the Hubble parameter, 𝑎 is the scale factor, values for the tachyon, quintessence, and phantom fields read and dot represents the derivative with respect to cosmic time as 𝑡. 𝜌 and 𝑝 are the energy density and the pressure of the 𝑉 (𝜓) corresponding scalar field constituents of the dark energy. 2 𝜓 𝜌𝜓 =−𝜉1𝑓(𝜓)6𝐻 + , Conservation of energy gives the evolution equations for 2 (12) √1−(𝜓̇ /𝑉𝜓 (𝜓)) thedarkenergyconstituents,suchas 𝜓̇ 2 𝜌̇ +3𝐻(𝜌 +𝑝 )=0, 𝑝 =𝜉𝑓(𝜓)6𝐻2 −𝑉 (𝜓) √1− 𝜓 𝜓 𝜓 𝜓 1 𝜓 𝑉 (𝜓) 𝜓 (13) 𝜌̇ +3𝐻(𝜌 +𝑝 )=0, 𝜙 𝜙 𝜙 (7) ̇ 󸀠 2 +4𝜉1𝐻𝑓 ( 𝜓 ) +4𝜉 1𝐻𝑓 (𝜓) 𝜓̇ , 𝜌̇ +3𝐻(𝜌 +𝑝 )=0. 𝜎 𝜎 𝜎 1 𝜌 =−𝜉𝑔(𝜙)6𝐻2 + 𝜙̇2 +𝑉 (𝜙) , 𝜙 2 2 𝜙 (14) The total energy density and the pressure of dark energy read 1 𝑝 =𝜉𝑔(𝜙)6𝐻2 + 𝜙̇2 −𝑉 (𝜙) + 4𝜉 𝐻𝑔̇ (𝜙) 𝜙 2 2 𝜙 2 (15) 𝜌tot =𝜌DE =𝜌𝜓 +𝜌𝜙 +𝜌𝜎, 󸀠 ̇ (8) +4𝜉2𝐻𝑔 (𝜙) 𝜙, 𝑝 =𝑝 =𝑝 +𝑝 +𝑝 , tot DE 𝜓 𝜙 𝜎 1 𝜌 =−𝜉ℎ (𝜎) 6𝐻2 − 𝜎̇2 +𝑉 (𝜎) , 𝜎 3 2 𝜎 (16) with the total equation of state parameter 1 𝑝 =𝜉ℎ (𝜎) 6𝐻2 − 𝜎̇2 −𝑉 (𝜎) +4𝜉 𝐻ℎ̇ (𝜎) 𝑝 𝜎 3 2 𝜎 3 𝜔 = tot =𝜔 Ω +𝜔 Ω +𝜔 Ω , (17) tot 𝜌 𝜓 𝜓 𝜙 𝜙 𝜎 𝜎 (9) 󸀠 tot +4𝜉3𝐻ℎ (𝜎) 𝜎,̇

2 where 𝜔𝜓 =𝑝𝜓/𝜌𝜓, 𝜔𝜙 =𝑝𝜙/𝜌𝜙,and𝜔𝜎 =𝑝𝜎/𝜌𝜎 are the respectively.Herewehaveusedtherelation𝑇=−6𝐻,and equation of state parameters and Ω𝜓 =𝜌𝜓/𝜌tot, Ω𝜙 =𝜌𝜙/𝜌tot, prime denotes the derivative of related coupling functions and Ω𝜎 =𝜌𝜎/𝜌tot are the density parameters for the tachyon, withrespecttotherelatedfieldvariables.Now,wecanfindthe quintessence, and phantom fields, respectively. Then the total equation of motions for three scalar fields from the variation density parameter is defined as of the field Lagrangians (11) with respect to the field variables 𝜓, 𝜙,and𝜎,suchthat 𝜅2𝜌 Ω =Ω +Ω +Ω = tot =1, 𝜓̇ 2 3 𝜓̇ 2 tot 𝜓 𝜙 𝜎 2 (10) 3𝐻 𝜓+3𝐻̈ 𝜓(1−̇ )+(1− )𝑉𝜓 (𝜓) 𝑉𝜓 (𝜓) 2 𝑉𝜓 (𝜓)

3/2 where we assume that three kinds of scalar fields constitute 𝜓̇ 2 the dark energy with an equal proportion of density parame- +6𝜉 (1 − ) 𝐻2𝑓󸀠 (𝜓) = 0, 1 𝑉 (𝜓) (18) ter such that Ω𝜓 =Ω𝜙 =Ω𝜎 =1/3. 𝜓 The Lagrangian of the scalar fields is reexpressed from the 𝜙+𝑉̈ 󸀠 (𝜙) + 6𝜉 𝐻2𝑔󸀠 (𝜙) + 3𝐻𝜙=0,̇ action in (1), as 𝜙 2 󸀠 2 󸀠 𝜎−𝑉̈ 𝜎 (𝜎) −6𝜉3𝐻 ℎ (𝜎) +3𝐻𝜎=0.̇ 𝜓̇ 2 𝐿 =𝜉𝑓(𝜓)𝑇−𝑉 (𝜓) √1− , 𝜓 1 𝜓 These equations of motions are for the tachyon, quintessence, 𝑉𝜓 (𝜓) and phantom constituents of dark energy, respectively. Here, 1 2 (11) the prime of potentials denotes the derivative of related field 𝐿 =𝜉𝑔(𝜙)𝑇+ 𝜙̇ −𝑉 (𝜙) , 𝜙 2 2 𝜙 potentialswithrespecttotherelatedfieldvariables.Allthese evolution equations in (18) can also be obtained by using the 1 𝐿 =𝜉ℎ (𝜎) 𝑇− 𝜎̇2 −𝑉 (𝜎) . relations (12)–(17) in the continuity equations (7). 𝜎 3 2 𝜎 We now perform the phase-space analysis of the model in order to investigate the late-time solutions of the universe Then the energy density and pressure values for three scalar considered here. fieldscanbefoundbythevariationofthetotalLagrangian 𝑒𝜇 in (1) with respect to the tetrad field 𝑖 .Afterthevariation 3. Phase-Space and Stability Analysis of Lagrangian, there come contributions from the geometric terms so the (0, 0)-component and (𝑖, 𝑖)-component of the We study the properties of the constructed dark energy stress-energy tensor give the energy density and the pressure, model by performing the phase-space analysis. Therefore respectively. Accordingly the energy density and pressure we transform the aforementioned dynamical system into its 4 Advances in High Energy Physics autonomous form [38, 39, 43–46]. To proceed we introduce form by using (12)–(17) in 𝜔=𝑝/𝜌for every scalar field, such the auxiliary variables as

𝑝𝜓 𝜓̇ 𝜔𝜓 = 𝑥𝜓 = , 𝜌𝜓 √𝑉𝜓 (𝜓) −1 2 √ −𝜇 𝑦 +2𝜉1𝑢𝜓 [(2 3/3) 𝛼𝜓𝑥𝜓𝑦𝜓 +𝑢𝜓 (1−(2/3) 𝑠)] = 𝜓 , 2 2 𝜇𝑦𝜓 −2𝜉1𝑢𝜓 𝜅√𝑉𝜓 (𝜓) 𝑦 = , 𝑝 𝜓 √3𝐻 𝜙 𝜔𝜙 = 𝜌𝜙 (22) 𝑢 =𝜅√𝑓(𝜓), 2 2 √ 𝜓 𝑥 −𝑦 +2𝜉2𝑢𝜙 [(2 6/3) 𝛼𝜙𝑥𝜙 +𝑢𝜙 (1−(2/3) 𝑠)] = 𝜙 𝜙 , 2 2 2 𝑥𝜙 +𝑦𝜙 −2𝜉2𝑢𝜙 𝜅𝜙̇ 𝑥𝜙 = , 𝑝𝜎 √6𝐻 𝜔𝜎 = 𝜌𝜎 𝜅√𝑉 (𝜙) (19) −𝑥2 −𝑦2 +2𝜉 𝑢 [(2√6/3) 𝛼 𝑥 +𝑢 (1−(2/3) 𝑠)] 𝜙 = 𝜎 𝜎 3 𝜎 𝜎 𝜎 𝜎 , 𝑦𝜙 = , −𝑥2 +𝑦2 −2𝜉 𝑢2 √3𝐻 𝜎 𝜎 3 𝜎 󸀠 󸀠 where 𝛼𝜓 =𝑓(𝜓)/√𝑓(𝜓), 𝛼𝜙 =𝑔(𝜙)/√𝑔(𝜙), 𝛼𝜎 = 𝑢 =𝜅√𝑔(𝜙), 󸀠 2 𝜙 ℎ (𝜎)/√ℎ(𝜎) and 𝑠=−𝐻/𝐻̇ . From (9) and (20) and (22), we obtain 𝜔tot in the autonomous system, as 𝜅𝜎̇ 𝑥𝜎 = , −1 2 √6𝐻 𝜔tot =−𝜇 𝑦𝜓 2√3 2 𝜅√𝑉𝜎 (𝜎) 2 𝑦 = , +2𝜉1𝑢𝜓 [ 𝛼𝜓𝑥𝜓𝑦𝜓 +𝑢𝜓 (1 − 𝑠)] +𝜙 𝑥 𝜎 √3𝐻 3 3

2 2√6 2 𝑢 =𝜅√ℎ (𝜎), −𝑦 +2𝜉 𝑢 [ 𝛼 𝑥 +𝑢 (1 − 𝑠)] (23) 𝜎 𝜙 2 𝜙 3 𝜙 𝜙 𝜙 3 𝑁= 𝑎 𝐹 together with ln and for any quantity ,thetime −𝑥2 −𝑦2 derivative is 𝐹=𝐻(𝑑𝐹/𝑑𝑁)̇ . 𝜎 𝜎 We rewrite the density parameters for the fields 𝜓, 𝜙,and 2√6 2 𝜎 +2𝜉 𝑢 [ 𝛼 𝑥 +𝑢 (1 − 𝑠)] . in the autonomous system by using (10), (12), (14), and (16) 3 𝜎 3 𝜎 𝜎 𝜎 3 with (19) We can express 𝑠 in the autonomous system by using (6) and 𝜅2𝜌 Ω = 𝜓 =𝜇𝑦2 −2𝜉 𝑢2 , (23), such that 𝜓 3𝐻2 𝜓 1 𝜓 𝐻̇ 3 3 3 𝑠=− = (1 + 𝜔 )= − 𝜇−1𝑦2 𝜅2𝜌 𝐻2 2 tot 2 2 𝜓 Ω = 𝜙 =𝑥2 +𝑦2 −2𝜉 𝑢2 , (20) 𝜙 3𝐻2 𝜙 𝜙 2 𝜙 3 3 +𝜉 𝑢 [2√3𝛼 𝑥 𝑦 +𝑢 (3−2𝑠)]+ 𝑥2 − 𝑦2 1 𝜓 𝜓 𝜓 𝜓 𝜓 2 𝜙 2 𝜙 𝜅2𝜌 Ω = 𝜎 =−𝑥2 +𝑦2 −2𝜉 𝑢2 , 3 3 𝜎 3𝐻2 𝜎 𝜎 3 𝜎 +𝜉 𝑢 [2√6𝛼 𝑥 +𝑢 (3−2𝑠)]− 𝑥2 − 𝑦2 2 𝜙 𝜙 𝜙 𝜙 2 𝜎 2 𝜎 and the total density parameter is √ +𝜉3𝑢𝜎 [2 6𝛼𝜎𝑥𝜎 +𝑢𝜎 (3−2𝑠)], (24) 𝜅2𝜌 3 3 Ω = tot =Ω +Ω +Ω 𝑠={ − 𝜇−1𝑦2 +𝜉 𝑢 [2√3𝛼 𝑥 𝑦 +3𝑢 ] tot 3𝐻2 𝜓 𝜙 𝜎 2 2 𝜓 1 𝜓 𝜓 𝜓 𝜓 𝜓 2 2 2 2 2 2 2 (21) 3 2 3 2 3 2 =𝜇𝑦𝜓 −2𝜉1𝑢𝜓 +𝑥𝜙 +𝑦𝜙 −2𝜉2𝑢𝜙 −𝑥𝜎 +𝑦𝜎 + 𝑥 − 𝑦 +𝜉 𝑢 [2√6𝛼 𝑥 +3𝑢 ]− 𝑥 2 𝜙 2 𝜙 2 𝜙 𝜙 𝜙 𝜙 2 𝜎 2 −2𝜉3𝑢𝜎 =1, 3 − 𝑦2 +𝜉 𝑢 [2√6𝛼 𝑥 +3𝑢 ]} [1 + 2𝜉 𝑢2 2 𝜎 3 𝜎 𝜎 𝜎 𝜎 1 𝜓 ̇ 2 2 where 𝜇=1/√1−(𝜓 /𝑉𝜓(𝜓)) = 1/√1−𝑥𝜓.Thentheequa- −1 +2𝜉 𝑢2 +2𝜉 𝑢2 ] . tion of state parameters can be written in the autonomous 2 𝜙 3 𝜎 Advances in High Energy Physics 5

𝑠 √6 Here is only a jerk parameter used in other equations of 𝑢󸀠 = 𝛼 𝑥 , cosmological parameters. Then the deceleration parameter 𝑞 𝜙 2 𝜙 𝜙 is 󸀠 2 2 2 √ 𝑥𝜎 =−3𝑥𝜎 [1 + 𝑥𝜎 −𝑥𝜙 + 𝜉2 (𝑠𝑢𝜙 − 6𝛼𝜙𝑥𝜙)𝑢𝜙 𝐻̇ 1 3 3 𝑞=−1− = + 𝜔tot 2 𝜇 𝐻2 2 2 + 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 − 𝑥2 𝑦2 3 3 𝜎 𝜎 𝜎 𝜎 2 𝜓 𝜓 1 3 −1 2 √ = − 𝜇 𝑦 +𝜉1𝑢𝜓 [2 3𝛼𝜓𝑥𝜓𝑦𝜓 +𝑢𝜓 (3−2𝑠)] 2 √6 2 2 𝜓 + 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 ]+𝜆 𝑦2 (25) 3 1 𝜓 𝜓 𝜓 𝜓 𝜎 2 𝜎 3 3 + 𝑥2 − 𝑦2 +𝜉 𝑢 [2√6𝛼 𝑥 +𝑢 3−2𝑠] 𝜙 𝜙 2 𝜙 𝜙 𝜙 𝜙 ( ) √ 2 2 + 6𝜉3𝛼𝜎𝑢𝜎, 3 3 2 2 √ 󸀠 𝜇 2 2 2 2 − 𝑥 − 𝑦 +𝜉3𝑢𝜎 [2 6𝛼𝜎𝑥𝜎 +𝑢𝜎 (3−2𝑠)]. 𝑦 =3𝑦 [ 𝑥 𝑦 − 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 +𝑥 2 𝜎 2 𝜎 𝜎 𝜎 2 𝜓 𝜓 3 1 𝜓 𝜓 𝜓 𝜓 𝜙 Now we transform the equations of motions (6) and 2 − 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 −𝑥2 (18) into the autonomous system containing the auxiliary 3 2 𝜙 𝜙 𝜙 𝜙 𝜎 variables in (19) and their derivatives with respect to 𝑁=ln 𝑎. 𝑋󸀠 = 𝑓(𝑋) 𝑋 2 √6 Thus we obtain ,where is the column vector − 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 −𝜆 𝑥2 ], including the auxiliary variables and 𝑓(𝑋) is the column 3 3 𝜎 𝜎 𝜎 𝜎 𝜎 6 𝜎 󸀠 vector of the autonomous equations. After writing 𝑋 ,wefind 󸀠 the critical points 𝑋𝑐 of 𝑋,bysetting𝑋 =0. We expand 󸀠 √6 󸀠 𝑢 = 𝛼𝜎𝑥𝜎, 𝑋 = 𝑓(𝑋) around 𝑋=𝑋𝑐 +𝑈,where𝑈 isthecolumnvector 𝜎 2 𝛿𝑥 𝛿𝑦 of perturbations of the auxiliary variables, such as , ,and (26) 𝛿𝑢 for each scalar field. Thus, we expand the perturbation 󸀠 󸀠 󸀠 󸀠 equations up to the first order for each critical point as 𝑈 = where 𝜆𝜓 =−𝑉𝜓/𝜅𝑉𝜓, 𝜆𝜙 =−𝑉𝜙/𝜅𝑉𝜙,and𝜆𝜎 =−𝑉𝜎/𝜅𝑉𝜎. 𝑀𝑈,where𝑀 is the matrix of perturbation equations. For Henceforth, we assume the nonminimal coupling functions 𝑀 2 2 2 each critical point, the eigenvalues of perturbation matrix 𝑓(𝜓) ∝𝜓 , 𝑔(𝜙) ∝𝜙 ,andℎ(𝜎) ∝ 𝜎 ;thus𝛼𝜓, 𝛼𝜙,and determine the type and stability of the critical points [47–50]. 𝛼𝜎 are constant. Also the usual assumption in the literature −𝑘𝜓𝜆𝜓𝜓 −𝑘𝜙𝜆𝜙𝜙 Particularly, the autonomous form of the cosmological is to take the potentials 𝑉𝜓 =𝑉𝜓0𝑒 , 𝑉𝜙 =𝑉𝜙0𝑒 , system in (6) and (18) is [51–60] −𝑘𝜎𝜆𝜎𝜎 and 𝑉𝜎 =𝑉𝜎0𝑒 [43, 61–63]. Such potentials give also √3 constants 𝜆𝜓, 𝜆𝜙,and𝜆𝜎. 𝑥󸀠 = [𝜆 𝑥2 𝑦 +𝜆 (2 − 3𝑥2 )𝑦 𝜓 2 𝜓 𝜓 𝜓 𝜓 𝜓 𝜓 Now we perform the phase-space analysis of the model by finding the critical points of the autonomous system in (26). −3 −1 √ −2 −4𝛼𝜓𝜉1𝑢𝜓𝜇 𝑦𝜓 −2 3𝑥𝜓𝜇 ], To obtain these points, we set the left hand sides of (26) to zero. After some calculations, four critical points are found by √3 assuming 𝜔tot and 𝑞 as −1 for each critical point. The critical 𝑦󸀠 =[− 𝜆 𝑥 𝑦 +𝑠]𝑦 , 𝜓 2 𝜓 𝜓 𝜓 𝜓 points are listed in Table 1 with the existence conditions. Then we insert linear perturbations 𝑥→𝑥𝑐 +𝛿𝑥, 𝑦→ √3 𝑦𝑐 +𝛿𝑦,and𝑢→𝑢𝑐 +𝛿𝑢about the critical points for three 𝑢󸀠 = 𝛼 𝑥 𝑦 , 𝜓 2 𝜓 𝜓 𝜓 scalar fields 𝜓, 𝜙,and𝜎 in the autonomous system (26). Thus we obtain a 9×9perturbation matrix 𝑀 whose elements are 2 𝑥󸀠 =−3𝑥 [1 + 𝑥2 −𝑥2 + 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 given as 𝜙 𝜙 𝜎 𝜙 3 2 𝜙 𝜙 𝜙 𝜙 𝑀11 =−3, 2 𝜇 + 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 − 𝑥2 𝑦2 3 𝜎 𝜎 𝜎 𝜎 𝜓 𝜓 √ √ −2 3 2 𝑀12 = 3𝜆𝜓 +2 3𝛼𝜓𝜉1𝑦𝜓 𝑢𝜓,

2 √6 2 √ −1 + 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 ]+𝜆 𝑦 𝑀13 =−2 3𝛼𝜓𝜉1𝑦𝜓 , 3 1 𝜓 𝜓 𝜓 𝜓 𝜙 2 𝜙 2√3𝛼 𝜉 √ 𝜓 1 2 √3 2 − 6𝜉2𝛼𝜙𝑢𝜙, 𝑀 = 𝑢 𝑦 − 𝜆 𝑦 , 21 𝑃 𝜓 𝜓 2 𝜓 𝜓 𝜇 2 󸀠 2 2 √ 2 3 2 𝑦𝜙 =3𝑦𝜙 [ 𝑥𝜓𝑦𝜓 − 𝜉1 (𝑠𝑢𝜓 − 6𝛼𝜓𝑥𝜓)𝑢𝜓 +𝑥𝜙 𝑀 =− 𝑦 , 2 3 22 𝑃 𝜓 2 6𝜉 − 𝜉 (𝑠𝑢 − √6𝛼 𝑥 )𝑢 −𝑥2 𝑀 = 1 𝑢 𝑦 , 3 2 𝜙 𝜙 𝜙 𝜙 𝜎 23 𝑃 𝜓 𝜓 2 √6 √ √ 2 2 6𝛼𝜓𝜉2 − 𝜉3 (𝑠𝑢𝜎 − 6𝛼𝜎𝑥𝜎)𝑢𝜎 −𝜆𝜙 𝑥𝜙], 𝑀 = 𝑢 𝑦 , 3 6 24 𝑃 𝜙 𝜓 6 Advances in High Energy Physics

0.65 0.35 0.25 0.6 0.3 0.2 0.25 0.55 0.15 0.2 0.5 0.15 0.1 𝜙 𝜓 𝜓 x y 0.45 u 0.1 0.05 0.05 0.4 0 0 0.35 −0.05 −0.05 −0.1 −0.1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.2 −0.1 00.1 0.2 0.30.4 0.5

x𝜓 x𝜓 x𝜓 0.6 0.5 0.25 0.58 0.4 0.2 0.56 0.3 0.15 0.54 0.2 0.1 𝜙 𝜙 0.52 𝜎 y u 0.1 x 0.05 0.5 0.48 0 0 0.46 −0.1 −0.05 0.44 −0.2 −0.1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.2−0.1 0 0.1 0.2 0.30.4 0.5 −0.2−0.1 0 0.1 0.2 0.30.4 0.5 x x𝜓 x𝜓 𝜓 0.7 0 0.35 0.3 0.65 −0.05 −0.1 0.25 0.6 0.2 −0.15 0.55 0.15 𝜓 𝜎 𝜎 −0.2 u y 0.5 u 0.1 −0.25 0.05 0.45 −0.3 0 0.4 −0.35 −0.05 0.35 −0.4 −0.1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.35 0.4 0.45 0.5 0.55 0.6 0.65 y x𝜓 x𝜓 𝜓 0.25 0.6 0.5 0.2 0.58 0.4 0.15 0.56 0.3 0.54 0.1 0.2 𝜙 𝜙 𝜙 0.52 u x 0.05 y 0.1 0.5 0 0 0.48 −0.05 0.46 −0.1 −0.1 0.44 −0.2 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.60.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65

y𝜓 y𝜓 y𝜓 0.25 0.7 0 0.2 0.65 −0.05 −0.1 0.15 0.6 −0.15 0.1 0.55 𝜎 𝜎 𝜎 −0.2 u y x 0.05 0.5 −0.25 0 0.45 −0.3 −0.05 0.4 −0.35 −0.1 0.35 −0.4 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65 y𝜓 y𝜓 y𝜓

Figure 1: Continued. Advances in High Energy Physics 7

0.25 0.6 0.5 0.2 0.58 0.4 0.15 0.56 0.3 0.54 0.1 0.2 𝜙 𝜙 𝜙 0.52 x u 0.05 y 0.1 0.5 0 0.48 0 −0.05 0.46 −0.1 −0.1 0.44 −0.2 −0.05 0.05 0.15 0.25 0.35 −0.05 0.05 0.15 0.25 0.35 −0.05 0.05 0.15 0.25 0.35

u𝜓 u𝜓 u𝜓 0.25 0.7 0 0.2 0.65 −0.05 0.15 0.6 −0.1 −0.15 0.1 0.55 𝜎 𝜎 𝜎 −0.2 x u 0.05 y 0.5 −0.25 0 0.45 −0.3 −0.05 0.4 −0.35 −0.1 0.35 −0.4 −0.05 0.05 0.15 0.25 0.35 −0.05 0.05 0.15 0.25 0.35 −0.05 0.05 0.15 0.25 0.35

u𝜓 u𝜓 u𝜓 0.6 0.5 0.25 0.58 0.4 0.2 0.56 0.3 0.15 0.54 0.2 0.1 𝜙 𝜎 𝜙 0.52 u y 0.1 x 0.05 0.5 0.48 0 0 0.46 −0.1 −0.05 0.44 −0.2 −0.1 −0.1−0.05 0 0.05 0.1 0.15 0.2 0.25 −0.1−0.05 0 0.05 0.1 0.15 0.2 0.25 −0.1−0.05 0 0.05 0.1 0.15 0.2 0.25

x𝜙 x𝜙 x𝜙 0.7 0 0.5 0.65 −0.05 0.4 0.6 −0.1 0.3 −0.15 0.55 0.2 𝜎 𝜙 𝜎 −0.2 y u 0.5 u 0.1 −0.25 0.45 −0.3 0 0.4 −0.35 −0.1 0.35 −0.4 −0.2 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

x𝜙 x𝜙 y𝜙 0.25 0.7 0 0.2 0.65 −0.05 0.15 0.6 −0.1 −0.15 0.1 0.55 𝜎 𝜎 𝜎 −0.2 u x 0.05 y 0.5 −0.25 0 0.45 −0.3 −0.05 0.4 −0.35 −0.1 0.35 −0.4 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

y𝜙 y𝜙 y𝜙

Figure 1: Continued. 8 Advances in High Energy Physics

0.25 0.7 0 0.2 0.65 −0.05 0.15 0.6 −0.1 −0.15 0.1 0.55 𝜎 𝜎 𝜎 −0.2 x u 0.05 y 0.5 −0.25 0 0.45 −0.3 −0.05 0.4 −0.35 −0.1 0.35 −0.4 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.2−0.10 0.1 0.2 0.3 0.4 0.5 −0.2−0.1 0 0.1 0.2 0.3 0.4 0.5

u𝜙 u𝜙 u𝜙 0.7 0 0 0.65 −0.05 −0.05 0.6 −0.1 −0.1 −0.15 −0.15 0.55 𝜎 𝜎 −0.2 𝜎 −0.2 y u 0.5 u −0.25 −0.25 0.45 −0.3 −0.3 0.4 −0.35 −0.35 0.35 −0.4 −0.4 −0.1 −0.05 0 0.05 0.10.15 0.2 0.25 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

x𝜎 x𝜎 y𝜎

Figure 1: Two-dimensional projections of the phase-space trajectories for 𝜉1 =𝜉2 =−𝜉3 = 0.5, 𝛼𝜓 =𝛼𝜙 =−𝛼𝜎 = 1.5.Allplotsbeginfrom the critical point 𝐴 being a stable attractor.

Table 1: Critical points and existence conditions.

Label 𝑥𝜓 𝑦𝜓 𝑢𝜓 𝑥𝜙 𝑦𝜙 𝑢𝜙 𝑥𝜎 𝑦𝜎 𝑢𝜎 𝜔tot 𝑞 Existence 1 1 1 𝐴0 00 00 0−1−1𝜆 =𝜆 =𝜆 =0 √3 √3 √3 𝜓 𝜙 𝜎 −1 −1 −1 𝐵0 00 00 0−1−1𝜆 =𝜆 =𝜆 =0 √3 √3 √3 𝜓 𝜙 𝜎

𝛼𝜓 =𝛼𝜙 =𝛼𝜎 =0 1 1 1 𝐶00 00 00 −1 −1 and 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ 󵄨 󵄨 √ 󵄨 󵄨 √ 󵄨 󵄨 𝜉1,𝜉2,𝜉3 <0, 6 󵄨𝜉1󵄨 6 󵄨𝜉2󵄨 6 󵄨𝜉3󵄨 𝜆𝜓,𝜆𝜙,𝜆𝜎 <0

𝛼𝜓 =𝛼𝜙 =𝛼𝜎 =0 −1 −1 −1 𝐷00 00 00 −1 −1 and 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 √ 󵄨 󵄨 √ 󵄨 󵄨 √ 󵄨 󵄨 𝜉1,𝜉2,𝜉3 <0, 6 󵄨𝜉1󵄨 6 󵄨𝜉2󵄨 6 󵄨𝜉3󵄨 𝜆𝜓,𝜆𝜙,𝜆𝜎 >0

3 𝑀 =−3, 𝑀 =− 𝑦 𝑦 , 44 25 𝑃 𝜙 𝜓 6𝜉 𝑀 = √6𝜆 𝑦 , 𝑀 = 2 𝑢 𝑦 , 45 𝜙 𝜙 26 𝑃 𝜙 𝜓 √ 2√6𝛼 𝜉 𝑀46 =− 6𝛼𝜙𝜉2, 𝑀 = 𝜎 3 𝑢 𝑦 , 27 𝑃 𝜎 𝜓 4√3𝛼 𝜉 3 √ 𝜓 1 𝑃−1 𝑀 =− 𝑦 𝑦 , 𝑀51 =2 6𝛼𝜓𝜉1𝑢𝜓𝑦𝜙 − ( )𝑢𝜓𝑦𝜓𝑦𝜙, 28 𝑃 𝜎 𝜓 𝑃 2 6𝜉 6 𝑃−1 𝑀 = 3 𝑢 𝑦 , 𝑀 = ( )𝑦 𝑦 , 29 𝑃 𝜎 𝜓 52 𝑃 2 𝜓 𝜙 12𝜉 𝑃−1 √3 𝑀 =− 1 ( )𝑢 𝑦 , 𝑀 = 𝛼 𝑦 , 53 𝑃 2 𝜓 𝜙 31 2 𝜓 𝜓 Advances in High Energy Physics 9

0 6 0.25 −0.1 5 0.2 −0.2 4 −0.3 0.15 3 −0.4 0.1 𝜙 𝜓 𝜓 −0.5 2 x u y 0.05 −0.6 1 −0.7 0 0 −0.8 −0.05 −0.9 −1 −1 −2 −0.1 −1−0.5 0 0.5 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x𝜓 x𝜓 x𝜓 0.1 0.25 1 0.8 0 0.2 0.6 0.15 −0.1 0.4 0.1 −0.2 0.2 𝜙 𝜎 𝜙 0.05 0 u y −0.3 x 0 −0.2 −0.4 −0.4 −0.05 −0.6 −0.5 −0.1 −0.8 −0.6 −0.15 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1−0.6 −0.2 0.2 0.6

x𝜓 x𝜓 x𝜓 1 1 3 0.8 2.5 0.6 0.5 2 0.4 0 0.2 1.5 𝜎 𝜎 0 −0.5 𝜓 1 u y u −0.2 0.5 −0.4 −1 0 −0.6 −1.5 −0.8 −0.5 −1 −2 −1 −1 −0.5 0 0.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0

x𝜓 x𝜓 y𝜓 0.3 0 0.3 0.25 −0.1 0.25 0.2 0.2 −0.2 0.15 0.15 −0.3 0.1 𝜙 𝜙 0.1 𝜙 −0.4 0.05 u y x 0.05 −0.5 0 −0.05 0 −0.6 −0.1 −0.05 −0.7 −0.15 −0.1 −0.8 −0.2 −2−1.5 −1 −0.5 0 −2−1.5 −1 −0.5 0 −2−1.5 −1 −0.5 0

y𝜓 y𝜓 y𝜓 1.5 0 0 −0.5 −0.05 1 −1 −0.1 −1.5 −0.15 −2 −0.2 𝜎 𝜎 0.5 𝜎 u y x −2.5 −0.25 −3 −0.3 0 −3.5 −0.35 −4 −0.4 −0.5 −4.5 −0.45 −2 −1.5−1 −0.5 0 −2 −1.5−1 −0.5 0 −2 −1.5−1 −0.5 0

y𝜓 y𝜓 y𝜓

Figure 2: Continued. 10 Advances in High Energy Physics

0.25 0 0.25 0.2 0.2 −0.1 0.15 0.15 −0.2 0.1 0.1 𝜙 𝜙 𝜙 −0.3 0.05 u y x 0.05 0 −0.4 0 −0.05 −0.05 −0.5 −0.1 −0.1 −0.15 −2 −1 031 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1 01234 56

u𝜓 u𝜓 u𝜓 2 0 0 −0.2 1.5 −0.5 −0.4 −1 −0.6 1 −0.8 𝜎 𝜎 𝜎 −1.5 −1 x u 0.5 y −1.2 −2 −1.4 0 −2.5 −1.6 −1.8 −0.5 −3 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1 1.5 2 −2 −1 031 2

u𝜓 u𝜓 u𝜓 0.1 0.25 20 18 0 0.2 16 0.15 −0.1 14 0.1 −0.2 12 𝜎 𝜙 𝜙 0.05 10 x u y −0.3 0 8 −0.4 6 −0.05 4 −0.5 −0.1 2 −0.6 −0.15 0 −0.1 −0.05 00.05 0.10.15 0.2 −0.1 −0.05 00.05 0.10.15 0.2 0.25 −0.1 −0.05 0 0.05 0.1 0.15 0.2

x𝜙 x𝜙 x𝜙 0 0 0.25 −0.2 0.2 −5 −0.4 0.15 −10 −0.6 −0.8 0.1 𝜎 𝜎 −15 −1 𝜙 0.05 u y u −1.2 0 −20 −1.4 −0.05 −25 −1.6 −1.8 −0.1 −30 −2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

x𝜙 x𝜙 y𝜙 30 0 0 −0.2 25 −10 −20 −0.4 20 −30 −0.6 −40 −0.8 𝜎 𝜎

15 𝜎

x −50 u y −1 10 −60 −70 −1.2 5 −80 −1.4 −90 −1.6 0 −100 −1.8 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

y𝜙 y𝜙 y𝜙

Figure 2: Continued. Advances in High Energy Physics 11

3 0 0 −0.2 −0.05 2.5 −0.4 −0.1 2 −0.6 −0.15 −0.8 −0.2 𝜎 𝜎 1.5 𝜎 −1 −0.25 x u y −1.2 −0.3 1 −1.4 −0.35 0.5 −1.6 −0.4 −1.8 −0.45 0 −2 −0.5 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 00.1 0.2 0.3 −0.2 −0.1 00.1 0.2 0.3

u𝜙 u𝜙 u𝜙 0 0 0.1 −0.1 −0.1 −0.2 0 −0.2 −0.3 −0.1 −0.4 −0.3 𝜎 𝜎 −0.5 −0.4 𝜎 −0.2 y u u −0.6 −0.5 −0.7 −0.3 −0.6 −0.8 −0.4 −0.9 −0.7 −1 −0.8 −0.5 −0.1−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 −3 −2.5 −2 −1.5 −1 −0.5 0

x𝜎 x𝜎 y𝜎

Figure 2: Two-dimensional projections of the phase-space trajectories for 𝜉1 =𝜉2 =−𝜉3 = 0.5, 𝛼𝜓 =𝛼𝜙 =−𝛼𝜎 = 1.5.Allplotsbeginfrom the critical point 𝐵 being a stable attractor.

√ 12𝜉 𝑃−1 4 6𝛼𝜙𝜉2 𝑃−1 1 𝑀 =2√6𝛼 𝜉 𝑢 𝑦 − ( )𝑢 𝑦 𝑀83 =− ( )𝑢𝜓𝑦𝜎, 54 𝜙 2 𝜙 𝜙 𝑃 2 𝜙 𝜙 𝑃 2 4√6𝛼 𝜉 √ 𝜙 2 𝑃−1 6𝜆𝜙 𝑀 =2√6𝛼 𝜉 𝑢 𝑦 − ( )𝑢 𝑦 , − 𝑦 , 84 𝜙 2 𝜙 𝜎 𝑃 2 𝜙 𝜎 2 𝜙 6 𝑃−1 6 𝑃−1 𝑀85 = ( )𝑦𝜙𝑦𝜎, 𝑀 = ( )𝑦2, 𝑃 2 55 𝑃 2 𝜙 12𝜉2 𝑃−1 𝑀86 =− ( )𝑢𝜙𝑦𝜎, 12𝜉2 𝑃−1 𝑃 2 𝑀56 =− ( )𝑢𝜙𝑦𝜙, 𝑃 2 4√6𝛼 𝜉 𝑃−1 𝑀 =2√6𝛼 𝜉 𝑢 𝑦 − 𝜎 3 ( )𝑢 𝑦 4√6𝛼 𝜉 𝑃−1 87 𝜎 3 𝜎 𝜎 𝑃 2 𝜎 𝜎 𝑀 =2√6𝛼 𝜉 𝑢 𝑦 − 𝜎 3 ( )𝑢 𝑦 , 57 𝜎 3 𝜎 𝜙 𝜎 𝜙 √ 𝑃 2 6𝜆𝜎 − 𝑦𝜎, 6 𝑃−1 2 𝑀58 = ( )𝑦𝜎𝑦𝜙, 6 𝑃−1 𝑃 2 𝑀 = ( )𝑦2, 88 𝑃 2 𝜎 12𝜉3 𝑃−1 𝑀59 =− ( )𝑢𝜎𝑦𝜙, 12𝜉 𝑃−1 𝑃 2 𝑀 =− 3 ( )𝑢 𝑦 , 89 𝑃 2 𝜎 𝜎 √6 𝑀 = 𝛼 , √6 64 2 𝜙 𝑀 = 𝛼 , 97 2 𝜎 𝑀 =−3, 77 (27) 𝑀 =−√6𝜆 𝑦 , 78 𝜎 𝜎 and all the other matrix elements except those given above 𝑃=1+2𝜉𝑢2 +2𝜉𝑢2 + √ in (27) are zero. Here also 1 𝜓 2 𝜙 𝑀79 = 6𝛼𝜎𝜉3, 2 2𝜉3𝑢𝜎. Then we calculate the eigenvalues of perturbation 4√3𝛼 𝜉 𝑃−1 matrix 𝑀 for four critical points given in Table 1 with the 𝑀 =2√6𝛼 𝜉 𝑢 𝑦 − 𝜓 1 ( )𝑢 𝑦 𝑦 , 81 𝜓 1 𝜓 𝜎 𝑃 2 𝜓 𝜓 𝜎 corresponding existing conditions. We obtain four sets of eigenvalues for four perturbation matrices for each of the four 6 𝑃−1 𝑀 = ( )𝑦 𝑦 , critical points. To determine the type and stability of critical 82 𝑃 2 𝜓 𝜎 points, we examine the sign of the real parts of eigenvalues. 12 Advances in High Energy Physics

0.6 0.25 0.5 0.2 0.6 0.4 0.15 0.5 0.3 0.1 0.4 0.2 0.05 𝜙 𝜓 𝜓 0.1 0 x y 0.3 u 0 −0.05 0.2 −0.1 −0.1 −0.2 −0.15 0.1 −0.3 −0.2 0 −0.4 −0.25 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1.5−1−0.5 0 0.5 1 1.5

x𝜓 x𝜓 x𝜓

1 0.25 0.8 0.2 0.5 0.6 0.15 0.4 0.4 0.2 0.1 𝜎 𝜙 𝜙 0.3 0 x u y −0.2 0.05 0.2 −0.4 0 −0.6 0.1 −0.05 −0.8 0 −1 −0.1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

x𝜓 x𝜓 x𝜓

0.8 0 0.25 0.7 −0.05 0.2 0.6 −0.1 0.15 0.1 0.5 −0.15 0.05 𝜓 𝜎 𝜎 0.4 −0.2 0 u y u −0.05 0.3 −0.25 −0.1 0.2 −0.3 −0.15 0.1 −0.35 −0.2 −0.25 0 −0.4 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x𝜓 x𝜓 y𝜓

0.25 0.5 0.2 0.5 0.4 0.15 0.3 0.1 0.4 0.2 0.05 0.1 𝜙 0 𝜙 0.3 x y 𝜙 0

−0.05 u −0.1 0.2 −0.1 −0.15 −0.2 −0.2 0.1 −0.3 −0.25 −0.4 0 −0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y𝜓 y𝜓 y𝜓 0.2 0.8 0 0.7 −0.01 0.15 −0.02 0.6 0.1 −0.03 0.5 −0.04 𝜎 𝜎 0.05 𝜎 0.4 −0.05 y u x 0.3 −0.06 0 −0.07 0.2 −0.05 −0.08 0.1 −0.09 −0.1 0 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.250.3 0.35 0.4 0.45 0.5 0.55 0.6

y𝜓 y𝜓 y𝜓

Figure 3: Continued. Advances in High Energy Physics 13

0.25 0.5 0.2 0.5 0.4 0.15 0.3 0.1 0.4 0.2 0.05 0.1 𝜙 𝜙 𝜙

0 0.3 u 0 x −0.05 y −0.1 −0.1 0.2 −0.2 −0.15 −0.3 −0.2 0.1 −0.4 −0.25 −0.5 0 −0.2 −0.1 0 0.1 0.2 0.3 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.5 0 0.5

u𝜓 u𝜓 u𝜓 0.3 0.8 0 0.25 −0.01 0.7 −0.02 0.2 0.6 −0.03 0.15 −0.04 𝜎 𝜎 𝜎 0.1 0.5 −0.05 y u x 0.05 −0.06 0.4 −0.07 0 0.3 −0.08 −0.05 −0.09 −0.1 0.2 −0.1 −0.4−0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2

u𝜓 u𝜓 u𝜓

0.6 0.5 0.55 0.6 0.4 0.5 0.4 0.3 0.45 0.2 0.2 0.4 0.1 𝜙 𝜙 0.35 0 𝜎 0 y u 0.3 x −0.1 −0.2 0.25 −0.2 0.2 −0.4 −0.3 0.15 −0.6 −0.4 0.1 −0.5 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2

x𝜙 x𝜙 x𝜙 0.8 0.01 0 0.6 0.7 −0.01 0.4 0.6 −0.02 −0.03 0.2 0.5 −0.04 𝜎 𝜎 𝜙 0 y u 0.4 −0.05 u −0.06 −0.2 0.3 −0.07 −0.4 0.2 −0.08 −0.09 −0.6 0.1 −0.1 −0.2 −0.1 0 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

x𝜙 x𝜙 y𝜙 0.2 1 0 0.9 −0.01 0.15 0.8 −0.02 0.1 0.7 −0.03 0.6 −0.04 𝜎 𝜎 0.05 𝜎 0.5 −0.05 y u x 0.4 −0.06 0 0.3 −0.07 −0.05 0.2 −0.08 0.1 −0.09 −0.1 0 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

y𝜙 y𝜙 y𝜙

Figure 3: Continued. 14 Advances in High Energy Physics

0.2 0.8 0 0.7 −0.01 0.15 −0.02 0.6 0.1 −0.03 0.5 −0.04 𝜎 𝜎 0.05 𝜎 0.4 −0.05 y u x 0.3 −0.06 0 −0.07 0.2 −0.05 −0.08 0.1 −0.09 −0.1 0 −0.1 −0.4 −0.2 0 0.2 0.4 0.6 −0.4 −0.2 00.2 0.4 0.6 −0.2 −0.1 0 0.1 0.2

u𝜙 u𝜙 u𝜙 1 0 0 −0.01 0.9 −0.02 −0.02 0.8 −0.04 −0.03 0.7 −0.04 𝜎 −0.06 𝜎 𝜎 u −0.05 y 0.6 u −0.08 −0.06 0.5 −0.07 −0.1 −0.08 0.4 −0.12 −0.09 0.3 −0.1 −0.1−0.05 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.5 0.55 0.6 0.65 0.7 x 𝜎 x𝜎 y𝜎

Figure 3: Two-dimensional projections of the phase-space trajectories for 𝜉1 =𝜉2 =𝜉3 = −0.5, 𝛼𝜓 =𝛼𝜙 =𝛼𝜎 = −1.5.Allplotsbeginfrom the critical point 𝐶 being a stable attractor.

Table 2: Eigenvalues of the perturbation matrix for each critical Table 3: Stability of the critical points. point. Critical points Stability 3 Critical points Eigenvalues 0< <𝜉 1 3 Stable point, if 2 1, 2 4𝛼𝜓 √9−12𝜉1𝛼 − , 2 𝜓 2 3 1 3 0< <𝜉 − √9−12𝜉 𝛼2 − 𝐴 𝐵 4𝛼2 2 , 2 1 𝜓 2 , and 𝜙 1 3 −3 2 𝜉 < <0 √9−12𝜉2𝛼𝜙 − , and 3 2 2 2 4𝛼𝜎 𝐴 𝐵 1 3 𝜉 ,𝜉 <0 and − √9−12𝜉 𝛼2 − Saddle point, if 1 2 2 2 𝜙 2 , 1 3 𝐶 and 𝐷 Stable point for all 𝜉, 𝛼,and𝜆 √12𝜉 𝛼2 +9− 2 3 𝜎 2 , 1 3 − √12𝜉 𝛼2 +9− 2 3 𝜎 2 , −1 In order to analyze the cosmological behavior of each critical point, the attractor solutions in scalar field cosmology 𝐶 𝐷 and −3 should be noted [64]. In modern theoretical cosmology it is common that the energy density of one or more scalar fields exerts a crucial influence on the evolution of the universe and some certain conditions or behaviors naturally affect this The critical point is stable for negative real part of eigenvalues. evolution. The evolution and the affecting factors on this evo- The physical meaning of negative eigenvalue is always a late- lution meet in the term of cosmological attractors: the scalar time stable attractor; namely, if the universe reaches this field evolution approaches a certain kind of behavior by the solution, it keeps its state forever and thus it can attract the dynamical conditions without finely tuned initial conditions universe at a late-time. Accelerated expansion occurs here, [65–77], either in inflationary cosmology or in late-time dark because 𝜔tot =−1<−1/3. If a convenient condition is energy models. Attractor behavior is a situation in which a provided, an accelerated contraction can even exist for 𝜔tot = collection of phase-space points evolve into a certain region −1 < −1/3 value. Eigenvalues of the matrix 𝑀 are represented and never leave. in Table 2 for each of the critical points 𝐴, 𝐵, 𝐶,and𝐷. As seen in Table 2, the first two critical points 𝐴 and 𝐵 Critical Point A. This point exists for all values of 𝜉1, 𝜉2, 𝜉3, havethesameeigenvaluesas𝐶 and 𝐷 have same eigenvalues. 𝛼𝜓, 𝛼𝜙,and𝛼𝜎 while 𝜆𝜓 =𝜆𝜙 =𝜆𝜎 =0which means the The stability conditions of each critical point are listed in potentials 𝑉𝜓, 𝑉𝜙,and𝑉𝜎 are constant. Acceleration occurs at Table 3, according to the sign of the eigenvalues. this point since 𝜔tot =−1<−1/3and this is an expansion Advances in High Energy Physics 15

−0.5 0.4 0.4 −0.55 0.3 0.35 −0.6 0.2 0.3 −0.65 0.1 0.25 −0.7 0 0.2 𝜙 𝜓 𝜓 −0.75 −0.1 0.15 x y −0.8 u −0.2 0.1 −0.85 −0.3 0.05 −0.9 −0.4 0 −0.95 −0.5 −0.05 −1 −0.1 −0.4 −0.2 0 0.2 0.4 0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0

x𝜓 x𝜓 x𝜓 −0.5 0.4 0.25 0.35 −0.55 0.2 0.15 0.3 0.1 0.25 −0.6 0.05 𝜙 𝜎 𝜙 0 0.2 u y x −0.65 −0.05 0.15 −0.1 −0.15 0.1 −0.7 −0.2 0.05 −0.25 −0.75 0 −0.4 −0.2 0 0.2 0.4 0.6 −1−0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6

x𝜓 x𝜓 x𝜓

−0.5 0 0.4 −0.55 −0.1 0.3 −0.6 0.2 −0.65 −0.2 0.1 −0.7 −0.3 𝜎 𝜎 −0.75 𝜓 0 y u −0.4 u −0.8 −0.1 −0.85 −0.5 −0.2 −0.9 −0.6 −0.95 −0.3 −1 −0.4 −0.8−0.6−0.4 −0.2 0 0.2 0.4 0.6 0.8 −1−0.5 0 0.5 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5

x𝜓 x𝜓 y𝜓

0.01 −0.5 0.008 −0.51 0.25 0.2 0.006 −0.52 0.15 0.004 −0.53 0.1 0.002 −0.54 0.05 𝜙 𝜙 0 𝜙 −0.55 0 u y x −0.002 −0.56 −0.05 −0.004 −0.57 −0.1 −0.15 −0.006 −0.58 −0.2 −0.008 −0.59 −0.25 −0.01 −0.6 −0.6 −0.59 −0.58 −0.57 −0.56 −1 −0.9−0.8−0.7 −0.6 −0.5 −1 −0.9 −0.8 −0.7 −0.6 −0.5

y𝜓 y𝜓 y𝜓

0.4 −0.5 0 0.35 −0.55 −0.05 −0.6 0.3 −0.1 −0.65 0.25 −0.7 −0.15 𝜎 𝜎 0.2 𝜎 −0.75 −0.2 y u x 0.15 −0.8 −0.25 −0.85 0.1 −0.3 −0.9 0.05 −0.95 −0.35 0 −1 −0.4 −1 −0.9−0.8 −0.7 −0.6 −0.5 −1 −0.9 −0.8−0.7 −0.6 −0.5 −1 −0.9 −0.8 −0.7 −0.6 −0.5

y𝜓 y𝜓 y𝜓

Figure 4: Continued. 16 Advances in High Energy Physics

−0.5 0.5 0.25 0.4 −0.55 0.2 0.3 0.15 0.2 −0.6 0.1 0.1 0.05 𝜙 0 −0.65 𝜙 0 y u 𝜙 −0.1 −0.05 x −0.2 −0.7 −0.1 −0.3 −0.15 −0.4 −0.75 −0.2 −0.5 −0.25 −0.8 −0.5 0 0.5 −0.5 0 0.5 −0.4 −0.2 0 0.2 0.4 0.6

u𝜓 u𝜓 u𝜓 1 −0.5 0 0.9 −0.55 0.8 −0.6 −0.1 0.7 −0.65 −0.2 0.6 −0.7 𝜎 𝜎 0.5 𝜎 −0.75 −0.3 y u x 0.4 −0.8 0.3 −0.85 −0.4 0.2 −0.9 −0.5 0.1 −0.95 0 −1 −0.5 0 0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

u𝜓 u𝜓 u𝜓 −0.5 1 −0.51 0.25 0.9 −0.52 0.2 0.8 0.15 −0.53 0.1 0.7 −0.54 0.05 0.6 𝜎 𝜙 −0.55 𝜙 0 0.5 y x −0.56 u −0.05 0.4 −0.57 −0.1 0.3 −0.15 −0.58 −0.2 0.2 −0.59 −0.25 0.1 −0.6 0 −0.2 −0.1 0 0.1 0.2 −0.5 0 0.5 −0.5 0 0.5

x𝜙 x𝜙 x𝜙 −0.5 0 −0.55 0.25 −0.6 −0.1 0.2 0.15 −0.65 −0.2 0.1 −0.7 0.05 𝜎 𝜎 −0.75 −0.3 𝜙 0 y u −0.8 u −0.05 −0.85 −0.4 −0.1 −0.15 −0.9 −0.5 −0.2 −0.95 −0.25 −1 −0.2 −0.10 0.1 0.2 0.3 −0.4 −0.2 0 0.2 0.4 0.6 −0.6 −0.58 −0.56 −0.54 −0.52 −0.5

x𝜙 x𝜙 y𝜙 1 −0.5 0 0.9 −0.55 −0.05 0.8 −0.6 −0.1 0.7 −0.65 −0.15 0.6 −0.7 −0.2 𝜎 𝜎 0.5 𝜎 −0.75 −0.25 y u x 0.4 −0.8 −0.3 0.3 −0.85 −0.35 0.2 −0.9 −0.4 0.1 −0.95 −0.45 0 −1 −0.5 −0.8 −0.7 −0.6−0.5 −0.4 −0.65 −0.6 −0.55 −0.5 −0.8 −0.75−0.7 −0.65−0.6 −0.55 −0.5

y𝜙 y𝜙 y𝜙

Figure 4: Continued. Advances in High Energy Physics 17

1 −0.5 0 0.9 −0.55 −0.1 0.8 −0.6 −0.2 0.7 −0.65 −0.3 0.6 −0.7 −0.4 𝜎 𝜎 0.5 𝜎 −0.75 −0.5 y u x 0.4 −0.8 −0.6 0.3 −0.85 −0.7 0.2 −0.9 −0.8 0.1 −0.95 −0.9 0 −1 −1 −0.2 −0.10 0.1 0.2 0.3 −0.2 −0.1 0 0.1 0.2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

u𝜙 u𝜙 u𝜙

−0.5 0 0 −0.55 −0.05 −0.6 −0.1 −0.05 −0.65 −0.15 −0.1 −0.7 −0.2 𝜎 𝜎 𝜎 −0.75 −0.25 −0.15 y u −0.8 −0.3 u −0.85 −0.35 −0.2 −0.9 −0.4 −0.25 −0.95 −0.45 −1 −0.5 0 0.05 0.10.15 0.20.25 0.3 0 0.2 0.4 0.6 0.8 1 −1 −0.9 −0.8−0.7 −0.6 −0.5

x𝜎 x𝜎 y𝜎

Figure 4: Two-dimensional projections of the phase-space trajectories for 𝜉1 =𝜉2 =𝜉3 = −0.5, 𝛼𝜓 =𝛼𝜙 =𝛼𝜎 = −1.5.Allplotsbeginfrom the critical point 𝐷 being a stable attractor.

phase because 𝑦𝜓, 𝑦𝜙,and𝑦𝜎 have positive values; therefore 𝛼𝜎 = −1.5,andthreeauxiliary𝜆 values for each field 𝜓, 𝜙,and 𝐻 is positive, too. Point 𝐴 is stable (meaning that universe 𝜎. A stable attractor starting from the critical point 𝐶 is seen 2 2 keeps its evolution), if 0 < 3/(4𝛼𝜓)<𝜉1, 0 < 3/(4𝛼𝜙)<𝜉2 and from the plots in Figure 3. 2 𝜉3 < −3/(4𝛼𝜎)<0, but it is a saddle point (meaning that the Critical Point D. This point exists for 𝜉1,𝜉2,𝜉3 <0and universe evolves between different states) for values 𝜉1,𝜉2 <0. 𝜆𝜓,𝜆𝜙,𝜆𝜎 >0while 𝛼𝜓 =𝛼𝜙 =𝛼𝜎 =0implying constant In Figure 1, we represent the 2-dimensional projections of 9- 𝑓(𝜓), 𝑔(𝜙), ℎ(𝜎) coupling functions. Acceleration phase is dimensional phase-space trajectories for the values of 𝜉1 = valid due to 𝜔tot <−1/3.Point𝐷 is also stable for all 𝜉, 𝛼, 𝜉2 =−𝜉3 = 0.5, 𝛼𝜓 =𝛼𝜙 =−𝛼𝜎 = 1.5 and three auxiliary 𝜆 and 𝜆 values of fields 𝜓, 𝜙,and𝜎.Two-dimensionalplotsof values for each field 𝜓, 𝜙,and𝜎. This state corresponds to a phase-space trajectories are shown in Figure 4 for 𝜉1 =𝜉2 = stable attractor starting from the critical point 𝐴,asseenfrom 𝜉3 = −0.5, 𝛼𝜓 =𝛼𝜙 =𝛼𝜎 = −1.5,andthreeauxiliary𝜆 values the plots in Figure 1. for each field 𝜓, 𝜙,and𝜎.Thisstateagaincorrespondstoa stable attractor starting from the point 𝐶,asinFigure4. 𝐵 𝜉 𝜉 𝜉 Critical Point B.Point also exists for any values of 1, 2, 3, AlltheplotsinFigures1–4havethestructureofstable 𝛼 𝛼 𝛼 𝜆 =𝜆 =𝜆 =0 𝜓, 𝜙,and 𝜎 while 𝜓 𝜙 𝜎 which means the attractor, since each of them evolves to a single point which 𝑉 𝑉 𝑉 potentials 𝜓, 𝜙,and 𝜎 are constant. Acceleration phase is is in fact one of the critical points in Table 1. These evolutions 𝜔 =−1<−1/3 again valid here since tot , but this point refers to the critical points are the attractor solutions in mixed dark 𝑦 𝑦 𝑦 to contraction phase because 𝜓, 𝜙,and 𝜎 have negative energy domination cosmology of our model which imply an sign. Stability of the point 𝐵 is same with the point 𝐴 for expanding universe. same conditions. Therefore the stable attractor behavior for contraction is represented in Figure 2. We plot 2-dimensional projections of phase-space trajectories for same values as 4. Conclusion point 𝐴, 𝜉1 =𝜉2 =−𝜉3 = 0.5, 𝛼𝜓 =𝛼𝜙 =−𝛼𝜎 = 1.5,and auxiliary 𝜆 values. Mixed dark energy is a generalized combination of tachyon, quintessence, and phantom fields nonminimally coupled to Critical Point C. Critical point 𝐶 occurs for 𝜉1,𝜉2,𝜉3 <0and gravity[33,34,38,39].Thesethreescalarfieldsarethe 𝜆𝜓,𝜆𝜙,𝜆𝜎 <0while 𝛼𝜓 =𝛼𝜙 =𝛼𝜎 =0meaning constant constituents of mixed dark energy. Firstly, the action integral coupling functions 𝑓(𝜓), 𝑔(𝜙),andℎ(𝜎).Thecosmological of nonminimally coupled mixed dark energy model is set behavior is again an acceleration phase since 𝜔tot <−1/3. up to study its dynamics. Here we consider that our dark Point 𝐶 is stable for any values of 𝜉, 𝛼,and𝜆 of fields 𝜓, energy constituents interact only with the gravity. There 𝜙,and𝜎. Two-dimensional projections of phase-space are could also be chosen some interactions between the dark represented in Figure 3 for 𝜉1 =𝜉2 =𝜉3 = −0.5, 𝛼𝜓 =𝛼𝜙 = constituents, but in order to start from a pedagogical order we 18 Advances in High Energy Physics prefer to exclude the dark interactions. We obtain the Hubble a cosmological constant,” The Astronomical Journal,vol.116,no. parameter and Friedmann equations of model in spatially 3, pp. 1009–1038, 1998. flat FRW geometry. Energy density and pressure values [3] U. Seljak, A. Makarov, P. McDonald et al., “Cosmological with the evolution equations for tachyon, quintessence, and parameter analysis including SDSS Ly𝛼 forest and galaxy phantom fields are obtained from the variation of action bias: constraints on the primordial spectrum of fluctuations, and Lagrangian of model. 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