Bifurcations analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions: the asymmetric case
aa r X i v : . [ m a t h . C A ] D ec Bifurcations analysis of the twist-Fr´eedericksztransition in a nematic liquid-crystal cell withpre-twist boundary conditions: the asymmetric case F.P. da Costa , M.I. M´endez † , J.T. Pinto Departamento de Ciˆencias e Tecnologia, Universidade Aberta, Lisboa, Portugal, and Centro de An´aliseMatem´atica, Geometria e Sistemas Dinˆamicos, Instituto Superior T´ecnico, Universidade de Lisboa, Lisboa,Portugal Departamento de Matem´aticas, IES Antonio L´opez, Getafe (Madrid), Spain Departamento de Matem´atica and Centro de An´alise Matem´atica Geometria e Sistemas Dinˆamicos,Instituto Superior T´ecnico, Universidade de Lisboa, Lisboa, Portugal
December 10, 2015
Abstract.
In the paper [Eur. J. of Appl. Math. , (2009) 269–287] by da Costa et al. the twist-Fr´eedericksz transition in a nematic liquid crystal one-dimensional cell of lenght L was studied imposingan antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper we extend thatstudy to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions.We use phase-plane analysis tools and appropriately defined time-maps to obtain the bifurcation diagramsof the model when L is the bifurcation parameter, and related these diagrams with the one in the symmetricsituation. The stability of the bifurcating solutions is investigated by applying the method of Kenjiro Maginu[J. Math. Anal. Appl. , (1978) 224–243]. Introduction
In the operation of liquid crystal devices the phenomena of Fr´eedericksz transitions in nematic liquidcrystal cells are of paramount technological importance [8, Chapter 5] and give rise to interesting andchallenging mathematical problems [7, Section 3.4].A nematic liquid crystal cell is basically a thin layer (a few microns) of a nematic liquid crystal containedbetween two glass plates whose inner surface is chemically treated in such a way as to force a certainallignment (anchoring) of the rod like nematic liquid crystal molecules lying close to the cell boundary. Thissurface alignment induces an allignment in the liquid crystal molecules filling the cell bulk so that the totalfree energy is minimized.When an exterior electric or magnetic field is applied to the cell a competition takes place between thereorienting effects of the field and the allignment imposed by the surface anchoring. Minimization of thetotal free energy (field, elastic bulk, and anchoring) then forces a reallignment of the molecules in the cellbulk (and, in the case of the so called weak anchoring conditions, also of those at the cell surface [2]) whenthe field intensity increases above a threshold value dependent on the physico-chemical characteristics of thedevice. This bifurcation phenomenon is called Fr´eedericksz transition, in honor of the soviet physicist whodiscovered it [4].If we model the rod-like nematic liquid crystal molecules by a “director vector field” n , with k n k ≡ , a system with strong anchoring of the molecules at the cell surface occupying a region Ω has a total freeenergy of the director field given by Z Ω w ( n , ∇ n ) , where the free-energy density w embodies the competition between the energy cost of distortions of thedirector field versus the energy reduction associated with aligning parallel (or perpendicular) to the magneticfield, and is givem by2 w = K (div n ) + K ( n · curl n ) + K k n × curl n k − µ ∆ χ ( H · n ) , where K , K , and K are phenomenological elastic constants, µ is the free-space magnetic permeability,∆ χ = χ k − χ ⊥ is the difference between the diamagnetic susceptibilities parallel to versus perpendicular tothe director, and H is the (constant) applied magnetic field. See, e.g., [7].We consider the geometry of the twist-Fr´eedericksz transition, with an asymmetric pre-twist at the bound-ary. Thus we consider a thin slab of nematic liquid crystal bounded by two parallel planes a distance d apart F.P. da C. and J.T.P. were partially supported by FCT/Portugal through UID/MAT/04459/2013. † Portions of this paper first appeared in the dissertation submited by the second author to the
Universidad Nacional deEducaci´on a Distancia,
Madrid, Spain, in June 2014, as part of the requirements for the degree of
M´aster en Matem´aticasAvanzadas. from each other, unbounded and extending to infinity in any direction parallel to these planes. Define a pos-itively oriented orthogonal coordinate system ( x, y, z ) such that z is perpendicular to the bounding planes.Let the director field be represented by(1.1) n = (cos φ ( z, τ ) , sin φ ( z, τ ) , , where φ denotes the (twist) angle of the director. We will assume that in the liquid crystal cell the director isfixed in opposing orientations − φ and φ at the two opposing planes bounding the device in the z direction.This induces a net twist of the director vector field n across the cell (see Figure 1).PSfrag replacements d Hnn n n − φ − φ − φ φ φ φ Figure 1.
Geometry of the liquid crystal cell with asymmetric pre-twist. The director n orientation inside the cell corresponds to the situation in branch C r (with k = 0) inSection 3.1 below.We will consider a magnetic vector field H applied along the constant direction (0 , ,
0) with intensity H = k H k and are interested in studying the effect it induces in the stationary director distribution, accordingto the Ericksen-Leslie theory [7].In terms of the angle representation 1.1, the simplest model for the dynamics of the director field inthe absence of flow is the gradient flow on the free energy of the system and, in dimensionless form, theinitial-boundary value problem governing the behaviour of the director field is then ∂φ∂s = ∂ φ∂ζ + λ sin φ cos φ, ( s, ζ ) ∈ R + × (0 , φ ( · ,
0) = − φ , φ ( · ,
1) = φ , (1.3) φ (0 , · ) = φ initial (1.4)where(1.5) s := K γ d τ , ζ := zd , λ := µ ∆ χH d K , with all the material parameters are positive for our system of interest. Observe that the dimensionlesscontrol parameter λ is proportional to the square of the magnetic field strength.The associated equilibrium problem is given by d φdζ + λ sin φ cos φ = 0 , < ζ < φ (0) = − φ , φ (1) = φ . (1.7)In the classical twist-Fr´eedericksz-transition problem, we have φ = φ = 0 the system possesses a simplesymmetry, φ ( ζ ) ↔ − φ ( ζ ) , and the ground-state solution ( φ = 0, which is invariant under this symmetry)loses stability to a pair of symmetric solutions at a pitchfork bifurcation at λ c = π . In [1] a system with antisymmetric pre-twist ( φ = φ = 0) was studied. We no longer have the simplesymmetry above. The problem still possesses Z symmetry, however it is now of the form φ ( ζ ) ↔ − φ (1 − ζ ).The ground-state solution (which is invariant under this symmetry) is no longer uniform. The problem stillhas a classical pitchfork bifurcation diagram, with the symmetric solution branch bifurcating at a value λ c , which is necessarily greater than π , as was showed in [1]. Observe that the antisymmetric nature of theboundary data is crucial to this scenario.In the present paper we consider the asymmetric case ( φ = φ , both nonzero). We conclude that nopitchfork bifurcation points remain: the pitchforks that had not been broken in the passage from the classicaltwist cenario to the antisymmetric one, are now broken when the φ becomes different from φ , and theresult is a bifurcation diagram with only saddle-node bifurcation points, branches emanating from them,and single nonbifurcating branch of solutions.The approach will be based on the time maps and phase-plane methods developed in [1] for the anti-symmetric case. The stability of these branches is also studied by applying the results of [6], also basedon the behaviour of time-maps, which allows the classification of the stationary solution branches as stable,asymptotically stable, or unstable. A more detailed study of the stability indices of the equilibria and thecharacterization of their connecting orbits will be the subject of a future paper.2. Preliminaries
We will be concerned with the stationary solutions to (1.2)-(1.4), i.e., solutions of (1.6)-(1.7). Considerthe change of variables t = t ( ζ ) := q λ (cid:0) ζ − (cid:1) , and let ζ ( t ) be its inverse function. Let(2.1) L := r λ . Then, φ ( ζ ) is a solution of (1.6)-(1.7) iff x ( t ) := φ ( ζ ( t )) is a solution of (cid:26) x ′ = yy ′ = − sin 2 x (2.2) x ( − L ) = − φ , x ( L ) = φ , (2.3)where φ , φ ∈ (0 , π ) , and ( t, x, y ) ∈ [ − L, L ] × [ − π/ , π/ × R . The bifurcation parameter is now
L > L ∝ H. We shall treat the independent variable t in (2.2)-(2.3) as the “time” of the dynamicalsystem associated with (1.2). Note that this “time” corresponds to the original spacial variable ζ and notto the original time s .The study of the bifurcation structure of solutions to (2.2)–(2.3) when φ = φ was done in [1]. We nowconsider the general case, where no relation between the values of φ and φ is imposed. As in [1], we shalluse the tools of time-maps and phase-plane analysis.The phase portrait of (2.2) is presented in Figure 2. −1.5 −1 −0.5 0.5 1 1.5−1.5−1−0.50.511.5 PSfrag replacements xy (cid:0) − π , (cid:1) (cid:0) π , (cid:1) x = φ x = − φ Figure 2.
Orbits of (2.2) with the Dirichlet boundary condition (2.3) marked by the dashedvertical lines (with 0 < φ < φ < π/ . )For studying solutions to (2.2)-(2.3) we need to consider some time-maps measuring the time spent by theorbits. These maps are easily obtained from the fact that (2.2)-(2.3) is a conservative system with energy(2.4) V ( x, y ) = y − cos 2 x, which means that its orbits are subsets of the level sets of this function.Let α ∈ (cid:0) , π (cid:1) and denote by γ α the orbit that, at time t = 0, intersects the x -axis at ( α, . Clearly γ α is a periodic orbit (cf. Figure 2). Let P ( α ) be its period and define(2.5) T ( α ) := 14 P ( α ) = Z α dx √ cos 2 x − cos 2 α , where the second equality arises from the symmetry of the system with respect to reflexions in the x − and y − axis. Thus, T ( α ) is the time it takes for the point of intersection of γ α with the y − axis (which, byconservation of V along orbits, we easily conclude to be the point (0 , β ) with β = √ α ) to reach the x − axis (at the point ( α, , by construction).As in [1], two other time maps will be needed. The time-map(2.6) T ( α, φ ) := Z φ dx √ cos 2 x − cos 2 α , that measures the time spent by the point of intersection of the orbit γ α with the y − axis to reach the line x = φ α. Clearly T ( φ, φ ) = T ( φ ) . We will also consider a map T analogous to T but relevant for orbitscrossing the y − axis on or above the heteroclinic orbit γ h connecting ( − π/ ,
0) to ( π/ , , β ) with β > √
2, namely(2.7) T ( β, φ ) := Z φ dx p β + cos 2 x − . We can continuously extend this map to values β < √ T ( β, φ ) := T ( α ( β ) , φ ), where α ( β ) is definedto be the unique value of α for which the points ( α,
0) and (0 , β ) are on the same orbit. Since the orbitsare contained in the level sets of V , a brief inspection of Figure 2 allow us to conclude that β α ( β ) isa monotonically increasing function and thus, for each fixed φ , there is a smaller β for which T ( β, φ ) isdefined, which is the value β φ for which ( β φ ,
0) and ( φ,
0) are on the same orbit. For β below β φ no orbitsatisfies the boundary condition at t = L .Our analysis depend heavily on the following monotonicity properties of the time-maps defined above. Aproof of these results can be checked in [1]. Proposition 2.1.
Let α ∈ (cid:0) , π (cid:1) , φ ∈ (0 , α ) , and β > β φ . Then, the time-map T : (cid:0) , π (cid:1) → (0 , + ∞ ) defined by (2.5) is strictly increasing and converges to π √ as α → , and to + ∞ as α → π . for each fixed φ the time-maps T ( · , φ ) and T ( · , φ ) , defined by (2.6) and (2.7), respectively, arestrictly decreasing. The same holds true for T ( · , π ) . Bifurcation analysis
The study of (2.2)-(2.3) in the symmetric case φ = φ was done in [1] and will serve as a guide to ourpresent study. In the symmetric case a special role is played by the solutions of (2.2)-(2.3) that additionallysatisfy the homogeneous Neumann boundary condition y ( − L ) = y ( L ) = 0 (note that y = x ′ ). The valuesof L for which these solutions occur were termed “critical” (cf. [1, Figure 4 and Table 1]) and are pitchforkbifurcation points of the system [1, Figure 9]. The orbits corresponding to these values of L were denotedby γ ∗ . Due to the symmetry of the vector field of (2.2) and the asymmetry of the boundary condition (2.3) thereare no solutions to (2.2)-(2.3) satisfying homogeneous Neumann boundary conditions at both t = − L and t = L. However, there are solutions that satisfy such a condition at one, or the other, of the end points ofthe time interval. Although these do not correspond to bifurcation solutions, and the corresponding valuesof L are not bifurcation points of (2.2)-(2.3), they are important solutions that help us to organize theinformation and construct the bifurcation diagram in the asymmetric case, and relate it with the symmetriccase already studied.The two asymmetric cases φ < φ and φ > φ give rise to different bifurcation diagrams and will bestudied separately below. Since the approach for both cases is the same, we will present the first one in amore detailed way, and for the second will just briefly refer to the corresponding results.3.1. Case φ < φ . The “critical” cases.
Let γ ∗ be the orbit of (2.2)-(2.3) that satisfies the additional homogeneousNeumann condition y ( L ) = 0 . See Figure 3(a). It is clear from this figure and from the definition of thetime-maps in the previous section that the time spent by γ ∗ is T ∗ := T ( φ ) + T ( φ , φ ) . Since the total timespent by every orbit is 2 L , the corresponding half-length L is L ∗ = T ∗ . In a similar way, the orbit that satisfies the homogeneous Neumann boundary condition y ( − L ) = 0 willbe denoted by γ ∗ . See Figure 3(b). The time spent in by this orbit is T ∗ := 3 T ( φ ) − T ( φ , φ ), and thecorresponding half-lenght of the interval is L ∗ = T ∗ . PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 3.
Two “critical” orbits of (2.2)-(2.3) with φ < φ : (a) γ ∗ when 2 L = T ∗ := T ( φ ) + T ( φ , φ ); (b) γ ∗ when 2 L = T ∗ := 3 T ( φ ) − T ( φ , φ )From the definitions of the time-maps we easily observe that,(3.1) T ∗ = T ( φ ) + T ( φ , φ ) < T ( φ ) < T ( φ ) − T ( φ , φ ) = T ∗ , and these the inequalities turn to equalities if φ = φ , which, as already pointed out, was the case consideredin [1].By analogy to the terminology used in [1] for the symmetric case we shall call these solutions, orbits,etc., “critical”, although, as we shall see, they do not correspond to any critical feature in the bifurcationdiagrams. However, they will be very useful for the remaining constructions. In particular, as a matter ofterminology and when appropriate, we will keep denoting by subcritical [resp., supercritical] those situationswith values of L smaller [resp., larger] than L ∗ ou L ∗ .3.1.2. The subcritical case relative to γ ∗ . By Proposition 2.1 it is clear that the function (cid:0) φ , π (cid:1) ∋ α T A ( α ) := T ( α, φ ) + T ( α, φ ) is monotonically decreasing and T A ( α ) ↑ T ∗ as α ↓ φ . The correspondingorbit of (2.2)-(2.3) is a subset of the level set V ( α,
0) of V. Since the time it spents is 2 L = T A ( α ) < T ∗ , we call it subcritical relative to γ ∗ . Using the relation between the time-maps T and T we can extend thisapproach to orbits intersecting the y − axis above the heteroclinic orbit γ h . The time taken by these orbits isalso smaller than T ∗ and decreases as the ordinate of the intersection point increases.In Figure 4 we present two of these orbits subcritical relative to γ ∗ , together with the critical orbit γ ∗ . The monotonicity of the time-maps imply that, for each L ∈ (0 , T ∗ ) there is a single subcritical solution to(2.2)-(2.3). PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 4.
Two orbits of (2.2)-(2.3), with φ < φ , subcritical relative to the orbit γ ∗ .3.1.3. The supercritical case relative to γ ∗ . Consider again α ∈ (cid:0) φ , π (cid:1) and the level set V ( α,
0) of V. For α > φ but close to φ , we take an orbit of (2.2)-(2.3) close to γ ∗ which have its end point with y ( L ) < T C r ( α ) := 2 T ( α ) + T ( α, φ ) − T ( α, φ ) (the notation T C r was used in [1] for a branch of solutionswith a given symmetry relative to the origin. We use the same notation here because our C r solutions willcoincide with those of that paper when φ = φ ; the same will be done for other solution branches furtherdown the paper). Clearly T C r ( α ) → T ∗ as α → φ . PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 5.
An orbit of (2.2)-(2.3), with φ < φ , supercritical relative to the orbit γ ∗ .We shall prove that T C r ( α ) > T ∗ for α > φ , thus providing a justification for calling these orbitssupercritical relative to γ ∗ . From the expression of T C r and Proposition 2.1 we conclude that(3.2) dT C r dα ( α ) = T ′ ( α ) + ∂T ∂α ( α, φ ) − ∂T ∂α ( α, φ ) > ∂T ∂α ( α, φ ) − ∂T ∂α ( α, φ ) . To conclude the sign of the right-hand side observe that ∂∂φ ∂T ∂α = ∂∂α ∂T ∂φ = ∂∂α √ cos 2 φ − cos 2 α = − sin 2 α (cos 2 φ − cos 2 α ) / < . From this inequality and the assumption that φ < φ we infer that ∂T ∂α ( α, φ ) > ∂T ∂α ( α, φ ) , and plugging this into (3.2) gives that T C r is strictly increasing with α , concluding the proof.3.1.4. The supercritical case relative to γ ∗ . Consider an orbit in V ( α, α ∈ (cid:0) φ , π (cid:1) , as representedin Figure 6(a). From this figure and the definition of the time-maps we immediately conclude that the timespent to travel this orbit is T D ( α ) := 4 T ( α ) − T ( α, φ ) − T ( α, φ ) (see subsection 3.1.3 for a justificationof this notation).PSfrag replacements ( a )( b ) xy Ω − Ω + x = − φ x = φ − π π γ ∗ γ ∗ PSfrag replacements ( a ) ( b ) xy Ω − Ω + x = − φ x = φ − π π γ ∗ γ ∗ Figure 6. (a) An orbit of (2.2)-(2.3), with φ < φ , supercritical relative to the orbit γ ∗ .(b) The Ω − and Ω + regions.Clearly T D ( α ) → T ∗ as α → φ . We shall prove that T D ( α ) > T ∗ . In order to prove this, consider thestrips Ω − := ( − π/ , − φ ) × R , and Ω + := [ − φ , π/ × R . Let γ ∗± := γ ∗ ∩ Ω ± . Denoting by D an orbit ofthe type represented in Figure 6, let also D ± = D ∩ Ω ± . Since γ ∗ + = γ ∗ , the time spent in γ ∗ + is equal to T ∗ . Thus, in Ω + we just need to compare T ∗ with the time spent by D + . But D + is really an orbit of type C r with α > φ and thus, by the previous subsection, T D + ( α ) > T ∗ . In Ω − we need to compare the time taken by the orbit D − with that taken by γ ∗− , which a brief inspectionto Figure 6(b) shows it is equal to 2 T ( φ ) − T ( φ , φ ) . Since T D − ( α ) = 2 T ( α ) − T ( α, φ ) , α ∈ ( φ , π/ , we have ∂T D − ∂α ( α ) = 2 T ′ ( α ) − ∂T ∂α ( α, φ ) , and the monotonicity results in Proposition 2.1 imply that thisderivative is positive, and thus T D − ( α ) > T ( φ ) − T ( φ , φ ) . Finally, from the above we have T D ( α ) = T D − ( α ) + T D + ( α ) > T ( φ ) − T ( φ , φ ) + T ∗ = 3 T ( φ ) − T ( φ , φ )= T ∗ , as we wanted to prove.3.1.5. The subcritical case relative to γ ∗ . To complete the analysis, let us consider orbits in V ( α, α ∈ (cid:0) φ , π (cid:1) , as represented in Figure 7.PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 7.
An orbit of (2.2)-(2.3), with φ < φ , subcritical relative to the orbit γ ∗ .It is clear from this plot that the time spent by this orbit is T C ℓ ( α ) := 2 T ( α ) − T ( α, φ ) + T ( α, φ ) (seesubsection 3.1.3 for a justification of this notation T C ℓ ).It is also clear that T C ℓ ( α ) → T ∗ as α → φ . We shall prove that, for α > φ sufficiently close to φ , wehave T C ℓ ( α ) < T ∗ . This is not as easy to prove as in the previous cases. We start by considering in (2.5) and(2.6) a new variable ˜ α := sin α, and changing in (2.5) the integration variable x θ where sin x = √ ˜ α sin θ. This allow us to write(3.3) T C ℓ ( α ) = ˜ T C ℓ (˜ α ) := √ Z π/ dθ p − ˜ α sin θ + 1 √ Z φ φ dx p ˜ α − sin x , Differentiating we obtain d ˜ T C ℓ d ˜ α = 1 √ Z π/ sin θ (cid:0) − ˜ α sin θ (cid:1) / dθ − √ Z φ φ dx (cid:0) ˜ α − sin x (cid:1) / , and computing the second derivative we obtain d ˜ T C ℓ d ˜ α = 32 √ Z π/ sin θ (cid:0) − ˜ α sin θ (cid:1) / dθ + 34 √ Z φ φ dx (cid:0) ˜ α − sin x (cid:1) / > . Hence, ˜ T C ℓ is a convex function of ˜ α := sin α ∈ (sin φ , . From the definition of T C ℓ and ˜ T C ℓ , theabove expressions, and Proposition 2.1, we also conclude that ˜ T C ℓ → + ∞ as ˜ α → , and d ˜ T Cℓ d ˜ α → −∞ as˜ α → sin φ ; however, note that ˜ T C ℓ → T ∗ as ˜ α → sin φ (see the start of this paragraph).This behaviour obviously implies the existence of a single local extrema (a minimum) of ˜ T C ℓ , and hence of T C ℓ , in the interior of their respetive intervals of definition, and thus T C ℓ ( α ) < T ∗ when α > φ sufficientlyclose to φ . This justifies us calling this situation a (local) subcritical case relative to γ ∗ . We emphasize thatthe situation is local : if α is larger than the minimizer of T C ℓ ( α ) , the value of this function increases withoutbound as α → π/ , and thus,at some point, it will certainly be larger than T ∗ .Collecting the results obtained in the subsections 3.1.1–3.1.5 we obtain the bifurcation diagram in Figure 8.Remark that, due to the symmetry of the system, the value of y ( − L ) of the orbits γ ∗ and γ ∗ have the sameabsolute value (and different signs).3.1.6. Other solution branches.
In addition to the solution branches studied above and represented in Fig. 8,(2.2)–(2.3) has an infinite number of solution families, each corresponding to orbits circling the origin acomplete k number of times ( k = 1 , , . . . ) . As in the cases studied above, it is convenient to start byconsidering orbits corresponding to solutions that satisfy the additional boundary condition y ( L ) = 0 , and,as before, we denote those orbits by a star, in this case by γ ∗ k and γ ∗ k . Although they do not correspond tobifurcating points, they are very useful in organizing or knowledge about the solution branches. In Table 1we present the orbit γ ∗ k and those which form a connected branch with it when L changes from the valuecorresponding to γ ∗ k . In Table 2 we present the analogous picture concerning the orbit γ ∗ k . PSfrag replacements AC ℓ C r D Ly ( − L ) γ ∗ γ ∗ Figure 8.
Solid lines: portion of the bifurcation diagram when φ < φ constructed fromthe analysis of the time-maps about γ ∗ and γ ∗ presented in subsections 3.1.1–3.1.5. Dashedlines: the corresponding diagram when φ = φ (from [1]). The designation of the orbits byletters A , C ℓ , C r and D correspond to those used in [1]: see Table 1 and Figure 8 of thatarticle. Table 1.
Branch of solutions to (2.2)-(2.3), with φ < φ , winding k full times around and containing the solution γ ∗ k (For k = 0 the portion of the orbits with a thin trace shouldbe disregarded.)Orbit γ α,k Time taken by the orbit γ α,k (winds k times around )PSfrag replacements A T A ( α ) := 4 kT ( α ) + T ( α, φ ) + T ( α, φ )PSfrag replacements γ ∗ k T ∗ k ( φ ) := (4 k + 1) T ( φ ) + T ( φ , φ )PSfrag replacements C r T C r ( α ) := (4 k + 2) T ( α ) + T ( α, φ ) − T ( α, φ )Observe that these orbits are analogous to those studied in the previous subsections, which can be con-sidered the case k = 0 in this description (i.e., the orbits do not complete a full turn around the origin).The amounts of time spent by each of these orbits are exactly those of the corresponding ones in subsec-tions 3.1.1–3.1.5 with the addition of 4 kT ( α ) , which is the time of k full turns about the origin.The following conclusions are easily drawn: a: From the definitions of the time-maps it follows that T ∗ k ( φ ) < T ( k +1) ∗ ( φ ) . b: From (3.1) we immediately get T k ∗ ( φ ) < T ∗ k ( φ ) . c: From the results in subsections 3.1.3 and 3.1.4 and the fact that the time spent by the orbits with k > k = 0 plus 4 kT ( α ) , we easily conclude that T C r ( α ) > T ∗ k ( φ )and T D ( α ) > T ∗ k ( φ ) . Table 2.
Branch of solutions to (2.2)-(2.3), with φ < φ , winding k full times around and containing the solution γ ∗ k (For k = 0 the portion of the orbits with a thin trace shouldbe disregarded.)Orbit γ α,k Time taken by the orbit γ α,k (winds k times around )PSfrag replacements C ℓ T C ℓ ( α ) := (4 k + 2) T ( α ) − T ( α, φ ) + T ( α, φ )PSfrag replacements γ ∗ k T ∗ k ( φ ) := (4 k + 3) T ( φ ) − T ( φ , φ )PSfrag replacements D T D ( α ) := 4( k + 1) T ( α ) − T ( α, φ ) − T ( α, φ ) d: The study of the relation between T C ℓ ( α ) and T ∗ k ( φ ), for α > φ sufficiently close to φ proceeds exactlyas in subsection 3.1.5, paying attention to the fact that we need to add 4 kT ( α ) to those computations. Since T ′ ( α ) → T ( φ ) ∈ (0 , + ∞ ) as α ↓ φ , and ˜ T ′′ (˜ α ) > , the addition of 4 kT ( α ) to the right-hand side of (3.3)does nor change the conclusion. Hence we have T C ℓ ( α ) < T ∗ k ( φ ) , for α − φ > α ˜ T C ℓ (˜ α ) remain valid. e: Finally, it remains to study the relation between T A ( α ) and T ∗ k ( φ ) . The analysis also follows thatpresented in subsection 3.1.5. Changings variables as in subsection 3.1.5 we can write an expression for T A ( α ) similar to (3.3), namely T A ( α ) = ˜ T A (˜ α ) :=:= 2 √ k Z π/ dθ p − ˜ α sin θ + 1 √ Z φ dx p ˜ α − sin x + 1 √ Z φ dx p ˜ α − sin x . Now, the convexity argument employed in subsection 3.1.5 and also used in case d above, can again beapplied to conclude that, for α − φ > A orbits satisfy T A ( α ) < T ∗ k ( φ ) and thecorresponding branch in the diagram L vs. y ( − L ) is convex. Note that, in contrast to the case studied insubsection 3.1.5, but as was the case in [1], the branches of type A solutions have a (unique, by convexity)saddle-node, since we know that, from Proposition 2.1, T A ( α ) → + ∞ as α → π . Thus, we conclude from these results that, for each k , the relation of the various types of orbits amongthemselves is the same as existed in the case k = 0 illustrated in Figure 8. We collect the results obtainedthus far in the bifurcation diagram of Figure 9. Observe that, due to the symmetry of the system, the valueof y ( − L ) of the orbits γ ∗ k are the same for all k , and the same happens for γ ∗ k ; as was the case when k = 0,for all k these values in γ ∗ k and in γ ∗ k have the same absolute value (and different signs).3.2. Case φ > φ . The analysis of the case φ > φ proceeds in a way entirely similar to the case φ < φ and so we will not present the details of the arguments in what follows. We will, in the next figures, exhibitthe plots of the several types of orbits and the bifurcation diagram obtained. We start, in Figure 10, bythe orbits that, at t = − L , satisfy the additional boundary condition y ( − L ) = 0, which we designate by“critical” orbits, as done in the similar situation in subsection 3.1.1.Due to the symmetry of the problem relative to the transformations x
7→ − x and φ ↔ φ , we concludethat, from each “critical” orbit emerges two branches, a subcritical and a supercritical, with exactly the sameproperties as obtained for the corresponding branches in subsections 3.1.2–3.1.5. These orbits are illustratedin figures 11 and 12.In an entirely analogous way to what was presented in section 3.1.6, we also have the solution branchescorresponding to orbits circling the origin a complete number k > PSfrag replacements AC ℓ C r D Ly ( − L ) γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ Figure 9.
Solid lines: portion of the bifurcation diagram when φ < φ constructed fromthe analysis presented in subsections 3.1.1–3.1.6. Dashed lines: the corresponding diagramwhen φ = φ (from [1]). The designation of the orbits by letters A , C ℓ , C r and D correspondto those used in [1]: see Table 1 and Figure 8 of that article.PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ PSfrag replacements ( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 10.
Two “critical” orbits of (2.2)-(2.3) with φ > φ : (a) γ ∗ when 2 L = T ∗ := T ( φ ) + T ( φ , φ ); (b) γ ∗ when 2 L = T ∗ := 3 T ( φ ) − T ( φ , φ )PSfrag replacements ( b )( a )( b ) xy x = − φ x = φ − π π γ ∗ PSfrag replacements ( b )( a )( b ) xy x = − φ x = φ − π π γ ∗ Figure 11.
Orbits of (2.2)-(2.3), with φ > φ which are: (a) subcritical relative to theorbit γ ∗ ; (b) supercritical relative to the orbit γ ∗ .Collecting these results we can plot the bifurcation diagram corresponding to the case φ > φ . Thisis done in Figure 13. To understand the apparently drastic difference relative to the diagram for the case φ < φ presented in Figure 9 we need to bear in mind the fact that in both cases what is being plot in thevertical axis is the value of y ( − L ) of the corresponding orbit. If, in the case φ > φ , we choose to plotthe value of y ( L ) instead, by the symmetry considerations alluded to above, the corresponding bifurcationdiagram will be equal to that of Figure 9.4. Stability analysis of the equilibria
In this section we present a brief study of the stability of the equilibria using the approach of Maginu[6]. We believe it is possible, by a modification of methods originally developed for homogeneous Neumannboundary conditions (see, e.g., [3] and [5, Section 4.3]) to provide more detailed information about the PSfrag replacements ( b )( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ PSfrag replacements ( b )( a )( b ) xy x = − φ x = φ − π π γ ∗ γ ∗ Figure 12.
Orbits of (2.2)-(2.3), with φ > φ which are: (a) subcritical relative to theorbit γ ∗ (for α − φ > γ ∗ .PSfrag replacements AC ℓ C r D Ly ( − L ) γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ Figure 13.
Solid lines: portion of the bifurcation diagram when φ > φ constructedfrom what was presented and discussed in subsection 3.2. Dashed lines: the correspondingdiagram when φ = φ (from [1]). The designation of the orbits by letters A , C ℓ , C r and D correspond to those used in [1]: see Table 1 and Figure 8 of that article. unstable solutions, in particular clarifying, for each unstable equilibrium, which directions are unstable, andto characterize their heteroclinic connections. This will be postponed to a latter work.We pretend to classify as stable, asymptotically stable, or unstable the branches of equilibria determinedin the last section. The results of [6] relevant to our case are the theorems 3.1, 3.2 and 3.3. What the firsttwo of these theorems state is that solutions ( x ( t ) , y ( t )) of (2.2)–(2.3) are asymptotically stable (as stationarysolutions of the corresponding partial differential equation (1.2)–(1.4)) if y ( t ) has no zeros in [ − L, L ) or in( − L, L ]; and it is unstable if y ( t ) has two or more zeros.Clearly, these results take care of the stability characterization of all the branches of solutions with k > k = 0 of the branch denoted by A (which is asymptotically stable),and by D (which is unstable).Theorem 3.3 of [6] is one of a series or results characterizing the case when y ( t ) has a single zero in[ − L, L ] , located in ( − L, L ). Maginus’ result states that such an equilibrium E is asymptotically stable if thecorresponding time map T E ( α ) is strictly increasing, and is unstable if it is strictly decreasing.Applied to our case, this result will allow us to determine the stability of the remaining cases, namely:the branches C r and C ℓ when k = 0.Consider φ < φ . Let us start with the C r branch. Clearly such solutions are of the type considered in[6, Theorem 3.3] (the existence of a single time instant for which y ( t ) = 0). In Subsection 3.1.3 we concludedthat dT Cr dα > . Hence, Maginu’s result imply the branch is asymptotically stable.Let us consider now the case of the C ℓ branch. The relevant computations are the ones in subsection 3.1.5,where we concluded that T C ℓ ( α ) is convex, with a single local minimum. This means that dT Cℓ dα < C ℓ branch to the left of the γ ∗ and to the right of the leftmost point of the branch, i.e., thesaddle-node bifurcation point (which corresponds to the orbit for which T C ℓ ( α ) attains its unique minimum.)So, by [6, Theorem 3.3], these equilibria are unstable. For the remaining part of the C ℓ branch, i.e, for points of the orbit below the saddle-node bifurcation point, we have dT Cℓ dα >
0, and thus, again by [6, Theorem 3.3],the corresponding equilibria are asymptotically stable.These stability conclusions for the k = 0 branches are collected in Figure 14.PSfrag replacements A C ℓ C r D LSN ss su uγ ∗ γ ∗ Figure 14.
Enlargement of the bifurcation diagram of Figure 8. The saddle-node bifurca-tion point refered to in the text is denoted by SN , and the letters s and u denote branchesof stable and unstable solutions, respectively. The remaining notation is as in Figure 8.Exactly the same results can be applied to the case when φ > φ with analogous results: by theorems3.1 and 3.2 of [6] all the k > D branch, whereas the A branch isasymptotically stable. An analysis corresponding to that in subsections 3.1.3 and 3.1.5 and the applicationof theorem 3.3 of [6] results in the conclusion that C ℓ is an asymptotically stable branch, and the portionof the C r branch between y ∗ and the leftmost point (a saddle-node) of the branch corresponds to unstableequilibria, whereas the points above this last point are asymptotically stable equilibria.These conclusions about the stability of the k = 0 branches are collected in Figure 15.PSfrag replacements AC ℓ C r D LSNsss uuγ ∗ γ ∗ Figure 15.
Enlargement of the bifurcation diagram of Figure 13. The saddle-node bifurca-tion point refered to in the text is denoted by SN , and the letters s and u denote branchesof stable and unstable solutions, respectively. The remaining notation is as in Figure 13. References
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