An asymptotic structure of the bifurcation boundary of the perturbed Painlevé-2 equation
AAn asymptotic structure of the bifurcationboundary of the perturbed Painlev´e-2 equation
O. M. KiselevDecember 16, 2020
Abstract
Solutions of the perturbed Painlev´e-2 equation are typical for de-scribing a dynamic bifurcation of soft loss of stability. The bifurcationboundary separates solutions of different types before bifurcation andbefore loss of stability. This border has a spiral structure. The equa-tions of modulation of the bifurcation boundary depending on theperturbation are obtained. Both analytical and numerical results aregiven.
Here we construct an asymptotic perturbation theory for the Painlev´e-2 equa-tion in the form: u (cid:48)(cid:48) = − u + xu − εf ( u, u (cid:48) , x ) , < ε (cid:28) . (1)Usually, perturbed Painlev´e-2 equation appears in narrow layers whenstudying dynamic bifurcations of solutions of non-Autonomous differentialequations. It was shown, for example, in the work [1]. A careful study of therelationship between the pitchfork bifurcation and the Painlev´e-2 equationwas given in [2]. The asymptotic behaviour of the Painlev´e-2 equation dueto the hard loss of stability loss was constructed in [3] for the Painlev´e-2equation and in [4] for hard stability loss in the main resonance equation.From the point of view of a general approach for dynamic bifurcationsin second-order non-autonomous equations, similar questions are considered,for example, in [5]. A typical example is the doubling bifurcation in thetheory of parametric autoresonance [6]. It should be noted here that in thelisted works related to dynamic bifurcations [2], [5], [4], [6], [7], the Painlev´e-2 equation plays an important role in the narrow transition layer and the1 a r X i v : . [ n li n . S I] D ec u ( x ) x-4-3-2-1 0 1 2 3 4-50 -40 -30 -20 -10 0 10 20 Figure 1: In the left picture, the bifurcation boundary for the Painlev´e-2transcendent in the cross section of the phase space at x = −
50. The rightimage shows two numerical solutions of the unperturbed Painlev´e-2 equationwith initial conditions taken near the bifurcation boundary. For x <
0, thecurves almost coincide; for x >
0, they diverge as a result of soft loss ofstability.constructed asymptotics of solutions are used for matching the parametersof solutions before and after the bifurcations. In this case, in a narrow layer,it is sufficient to link the parameters of the asymptotics before and after thetransition with the parameters of the unperturbed Painlev´e transcendent.In general, the bifurcation structure of the hierarchy of Painlev´e equationswas discussed in [8]. An approach to scaling limits in the Painlev´e equationswas considered, for example, in [9].A typical picture for soft loss of stability of small oscillations for solutionsof the Painlev´e-2 equation has the form shown in figure 1. Here, on the leftside of the figure, the solutions of the unperturbed Painlev´e-2 equation aredefined by asymptotics of the form: u ∼ α √− x sin (cid:18)
23 ( − x ) / + 34 α log( − x ) + φ (cid:19) , x → −∞ . (2)The solution parameters are arbitrary constants α and φ .In the right part of the figure 1 solutions for x → ∞ oscillate in theneighbourhood of branches of the function ± (cid:112) x/ x = 0 for the unperturbed Painlev´e-2equation is determined by the monodromy data. The relationship betweenmonodromy data and the asymptotics of solutions of the Painlev´e-2 equationis considered in the monograph [10]. A detailed description is also availablein [11]. The boundary line defining the solution for x → −∞ , which for2 → + ∞ is determined by the asymptotic ± (cid:112) x/ x → −∞ ([12]):32 α log(2) − π − arg (cid:18) Γ (cid:18) iα (cid:19)(cid:19) = 0 . (3)When perturbation occurs, the bifurcation boundary is deformed, and thestructure of the set of initial values that pass through various stable branchesafter bifurcation becomes more complex.The effect of the disturbance is seen over a long period of time. Therefore,we can expect that it is sufficient to investigate the behaviour of the solutionfor large values of | x | . In this paper, we construct an asymptotic modelfor two parameters : the large parameter | x | (cid:29) < ε (cid:28) For a numerical study of the bifurcation boundary structure, it is convenientto study a family of trajectories released for a given value of the independentvariable − x (cid:28)
1. Among the studied family of trajectories, two sets areobtained. There are the trajectories passing into the neighbourhood of (cid:112) x/ − (cid:112) x/ x >
0. Because of the need toexplore large families, it is convenient to use parallel computing on the GPU.The figure (2) shows two boundaries of bifurcation curves. There are theunperturbed Painleve-2 and the perturbed one. The curve for the perturbedequation is similar to the deformed curve for the unperturbed Painleve-2equation.Depending on the type of perturbation, the boundary deformation maybe significant. For example, for a dissipative perturbation f = u (cid:48) at ε =0 . Figure 2: Here one can see the results of calculations of 2048x4096 trajectoriesby the Runge-Kutta method of the 4th order. The bifurcation boundary inthe section of the phase space ( u, u (cid:48) , x ) is given for x = −
50. The bold curvecorresponds to the Painlev´e-2 equation, the thin curve corresponds to theperturbed equation (1) with perturbation f = u ( u (cid:48) ) for ε = 0 . u, u (cid:48) , x ) at x = −
50 for the equation (1)with perturbation f = u (cid:48) at ε = 0 . x = 0,fall into the neighbourhood of (cid:112) x/
2. The bifurcation boundary of the setof trajectories is shown from the rectangle selected in the left image. In theright picture, the thin curves correspond to the Painlev´e-2 equation, while thethick one corresponds to the perturbed equation (1) with the perturbation f = u (cid:48) at ε = 0 .
1. 4
A formalism of perturbation theory
The asymptotics for the parameter ε will be constructed as: u ( x, ε ) ∼ ∞ (cid:88) k =0 ε k u k ( x, α, φ ) . (4)The main condition for representing corrections in the formula (4) is uniformboundedness in ε for x → −∞ .As the primary term of the asymptotics of perturbation theory in ε , theasymptotics of the Painlev´e-2 transcendent for x → −∞ , given in the formula(2), is not suitable. In solving the perturbed equation, the parameters α and φ are functions of the independent variable x and the parameter ε .We assume that α and φ can be decomposed into a series by the parameter ε and assume that the coefficients of this series depend on the slow variable ξ = εx : α ∼ ∞ (cid:88) k =0 ε k α k ( ξ ) , φ ∼ ∞ (cid:88) k =0 ε k φ k ( ξ ) . Asymptotics of the primary term of the perturbed Painlev´e transcendent: u ( x, α, φ ) ∼ α √− x sin (cid:32)
23 ( − x ) / + 34 (cid:90) ξ/ε α ( ζ ) dζζ + φ (cid:33) , x → −∞ . Note that the integral in the argument of the function sin for the unperturbedPainlev´e-2 equation gives the standard formula (2).To construct an asymptotic for the parameter ε , consider the equationfor the first correction term for ε : u (cid:48)(cid:48) = − u u + xu − f ( u , u (cid:48) , x ) − α (cid:48) ∂ α u (cid:48) − φ (cid:48) ∂ φ u (cid:48) . Here the function u is the solution of the inhomogeneous linearizedPainlev´e-2 equation. The linearized Painlev´e-2 equation has the form: v (cid:48)(cid:48) = − u v + xv. Two linearly independent solutions to this equation can be obtained fromthe derivatives of the Painlev´e-2 transcendent in the parameters α and φ .Asymptotic behaviour of solutions of the linearized equation: v ∼ √− x sin (cid:18)
23 ( − x ) / + 34 α log( − x ) + φ (cid:19) , x → −∞ , ∼ √− x cos (cid:18)
23 ( − x ) / + 34 α log( − x ) + φ (cid:19) , x → −∞ . The Wronskian of these solutions can be calculated as follows: w = v v (cid:48) − v (cid:48) v ∼ − α √− x (cid:18) (cid:18)
43 ( − x ) / + 32 α log( − x ) + φ (cid:19)(cid:19) . Since Wronskian of linearly independent solutions is constant, then passingto the limit at x → −∞ in the right part of the asymptotic formula forWronskian we obtain: w = 1 . For a linearized Painlev´e-2 equation with a modulated Painvel´e transcendent,the x asymptotic of two linearly independent solutions can be obtained bydifferentiating the asymptotics of the primary term in the parameters α and φ . We denote these linearly independent solutions u α and u φ , respectively: u α ∼ √− x sin (cid:32)
23 ( − x ) / + 34 (cid:90) ξ/ε α ( ζ ) dζζ + φ (cid:33) , x → −∞ .u φ ∼ √− x cos (cid:32)
23 ( − x ) / + 34 (cid:90) ξ/ε α ( ζ ) dζζ + φ (cid:33) , x → −∞ . The solution of the equation for the first correction term can be repre-sented using the formula: u = u α (cid:90) x ( f ( u , u (cid:48) , y ) − α (cid:48) ∂ α u (cid:48) − φ (cid:48) ∂ φ u (cid:48) ) u φ ( y ) dy − u φ (cid:90) x ( f ( u , u (cid:48) , y ) − α (cid:48) ∂ α u (cid:48) − φ (cid:48) ∂ φ u (cid:48) ) u α ( y ) dy. (5)We calculate the integrals of the derivatives of the perturbed transcendentof Painlev´e-2: (cid:90) x ∂ α u (cid:48) u φ ( y ) dy ∼ (cid:90) cos (cid:32)
23 ( − y ) / + 34 (cid:90) ξ/ε α ( ζ ) dζζ + φ (cid:33) dy ∼ x , (cid:90) x ∂ φ u (cid:48) u φ ( y ) dy = 12 α ( ∂ φ u ( x, α, φ )) , (cid:90) x ∂ α u (cid:48) u α ( y ) dy = 12 α ( ∂ α u ( x, α, φ )) ;6 Figure 4: Bifurcation boundary for the parameters of the Painlev´e-2 tran-scendent in the polar coordinate system, here r is the distance from thecoordinate axis, ((3 / r log(2) − π/ − arg(Γ( ir /
2) is the angle relative tothe abscissa axis. (cid:90) x ∂ φ u (cid:48) u α ( y ) dy = u α ( x ) ∂ φ u ( x, α, φ ) − (cid:90) x u α ( y ) ∂ φ u (cid:48) ( y, α, φ ) dy ∼ − x . Consequently, in the first correction term, secular terms may arise duringintegration.A condition for discard the linear growth in the first correction term canbe obtained by averaging: α (cid:48) = lim x →−∞ x (cid:90) x f ( u , u (cid:48) , y ) u φ ( y ) dy,φ (cid:48) = − lim x →−∞ x (cid:90) x f ( u , u (cid:48) , y ) u α ( y ) dy. The equations for higher corrections have the form: u (cid:48)(cid:48) k = 6 u u k + xu k − f k ( u , . . . , u k − , u (cid:48) , . . . , u (cid:48) k − , x ) − α (cid:48) k − ∂ α u (cid:48) − φ (cid:48) k − ∂ φ u (cid:48) . The averaging equations for the higher corrections are obtained similarly: α (cid:48) k − = lim x →−∞ x (cid:90) x ( u , . . . , u k − , u (cid:48) , . . . , u (cid:48) k − , x ) u φ ( y ) dy,φ (cid:48) k − = − lim x →−∞ x (cid:90) x ( u , . . . , u k − , u (cid:48) , . . . , u (cid:48) k − , x ) u α ( y ) dy. In the theory of Painlev´e transcendent, it is known that for x → ∞ solutions,two families can be divided according to the asymptotic behaviour. These7amilies and their relation to monodromy data were established in the alreadymentioned works [12] and [13]. The sign of this expression κ = sin (cid:0) ((3 / α log(2) − π/ − arg(Γ( iα / − φ ) (cid:1) . (6)defines a bifurcation transition for x → ∞ : u ∼ − sgn( κ ) (cid:114) x . (7)The boundary in terms of α, φ is shown in figure 4.The value of the parameters α (0) and φ (0) determines the type of bifur-cation transition for the perturbed equation. Namely, the well-known α (0)and φ (0) can be used to determine the sign of the expression (6).Let’s consider the inverse problem. Determine the deformation of thebifurcation boundary for some ξ <
0. To do this, let’s parametrize the initialcurve in the polar coordinate system: α ( r,
0) = r, φ ( r,
0) = ((3 / r log(2) − π/ − arg(Γ( ir / . Then, for − x (cid:29)
1, the bifurcation boundary in the x section of the phasespace for small ε will be defined by the formulas: u ( x, α ( r, ξ ) , φ ( r, ξ )) ∼ α ( r,ξ ) √− x sin (cid:16) ( − x ) / + (cid:82) ξ/ε α ( r, ζ ) dζζ + φ ( r, ξ ) (cid:17) ,u (cid:48) ( x, α ( r, ξ ) , φ ( r, ξ )) ∼ α ( r, ξ ) √− x cos (cid:16) ( − x ) / + (cid:82) ξ/ε α ( r, ζ ) dζζ + φ ( r, ξ ) (cid:17) . Here we consider an example of using the theory of non-integrable perturba-tions of the second transcendent of Painlev´e developed above. Consider thePainlev´e-2 equation with a small dissipative term: u (cid:48)(cid:48) = − u + xu − εu (cid:48) . (8)The linearization in the neighbour of zero for this equation leads to theAiry equation with a dissipative term: y (cid:48)(cid:48) = xy − εy (cid:48) . (9)8 Figure 5: The numerical solution of the equation (8) on the figure practicallycoincides with the constructed asymptotic solution. Therefore, the figureshows the numerical solution, which is a thin line, and the difference betweenthe numerical solution and the constructed asymptotic solution, which is abold line near the abscissa axis.The solution of the equation (9) can be represented as an integral: y = 1 π (cid:90) ∞ cos (cid:18) kx + k (cid:19) e − εk / dk. Asymptotic solution of this function for x → − inf ty : y ∼ e − εx/ √ π cos (cid:0) ( − x ) / (cid:1) ( − x ) / . Therefore, the amplitude of the solution oscillations decreases when the in-dependent variable on the negative half-axis changes towards larger values,i.e. towards the point x = 0. Similar results can be expected for solutions ofthe small-amplitude Painlev´e-2 equation.For solutions of the unperturbed Painlev´e-2 equation, formulas are knownabout the relationship between the parameters of the asymptotic solutions ofthe Painlev´e-2 equation and the monodromy data for it, see [12], [13]. Here,for simplicity of calculations, solutions from [12], [13] are considered only inthose regions of the independent variable x where they are bounded or small.According to the calculations in the Section 3 the equation for modulatingthe parameters of the Painlev´e trencendent asymptotic is: α (cid:48) = − lim x →−∞ x (cid:90) x u (cid:48) ( y, α, φ ) u φ ( y ) dy Figure 6: Here one can see the cross-section of the u, u (cid:48) bifurcation boundaryat x = −
50 for solutions of the perturbed Painlev´e-2 equation with smalldissipation (8) at ε = 0 .
1. the bold line is the boundary obtained numericallyfrom 2048x4096 trajectories, with the beginning at x = −
50. A thin line isa boundary calculated from perturbation theory.Substituting the asymptotics to the right-hand side of this formula gives: α (cid:48) = − α lim x →−∞ x (cid:90) x cos (cid:32)
23 ( − y ) / + 34 (cid:90) ξ/ε α ( ζ ) dζζ + φ (cid:33) dy. Integrating we get: α (cid:48) ∼ − α . The equation for the modulation of φ is: φ = lim x →−∞ x (cid:90) x u (cid:48) ( y, α, φ ) u α ( y ) dy after substituting the main terms, the asymptotic gives: φ (cid:48) ∼ lim x →−∞ x (cid:90) x sin (cid:32)
43 ( − y ) / + 32 (cid:90) ξ/ε α ( ζ ) dζζ + 2 φ (cid:33) dy. Going to the limit leads to the equation φ (cid:48) = 0 . Then the asymptotic behaviour of the primary term of the Painlev´e-2 tran-scendent with small dissipation has the form: u ∼ a e − εx/ √− x sin (cid:18)
23 ( − x ) / + 3 a (cid:90) x e − εz z dz + p (cid:19) , x → −∞ . a and p are solution parameters.A comparison of the constructed asymptotics and the numerical solutionfor α = 1 , p = 0 is shown in figure 5.The constructed asymptotics allows us to obtain a bifurcation boundaryfor any value of − x (cid:29)
1. In particular, figure reffigDP2NumAndAsympBordershows a comparison between the bifurcation boundary obtained by numericalanalysis of 2048x4096 trajectories starting at x = −
50 and the bifurcationboundary calculated using the asymptotic formula. Figure 5 shows that thesecurves are close. -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0
Figure 7: The numerical solution of the equation (10) on the figure prac-tically coincides with the constructed asymptotic solution. Therefore, thefigure shows the numerical solution, which is a thin line, and the differencebetween the numerical solution and the constructed asymptotic solution,which is a bold line near the abscissa axis.Here we consider another example of the perturbed Painlev´e-2 equation: u ” = − u + xu − ε ( u (cid:48) ) u. (10)For this equation, modulating the parameter α is follow: α (cid:48) ∼ lim x →−∞ x (cid:82) x α cos (cid:16) ( − y ) / + (cid:82) ξ/ε α ( ζ ) dζζ + φ (cid:17) ×× sin (cid:16) ( − y ) / + (cid:82) ξ/ε α ( ζ ) dζζ + φ (cid:17) dy Going to the limit gives: α (cid:48) = 0 . Figure 8: Here one can see the cross-section of the u, u (cid:48) bifurcation boundaryat x = −
50 for solutions of the non-linearly perturbed Painlev´e-2 equation(10) at ε = 0 .
1. The boundary obtained numerically for2048x4096 trajec-tories, with the beginning at x = −
50 , and the boundary calculated fromperturbation theory almost coincide. Differences can be observed away fromthe center. The rectangle highlighted in the left drawing is enlarged in theright drawing. In the right drawing, the boundary obtained numerically cor-responds to short vertical dashes, the boundary obtained by perturbationtheory is indicated by continuous lines.Equation for modulation φ is: φ (cid:48) ∼ lim x →−∞ x (cid:82) x α cos (cid:16) ( − y ) / + (cid:82) ξ/ε α ( ζ ) dζζ + φ (cid:17) ×× sin (cid:16) ( − y ) / + (cid:82) ξ/ε α ( ζ ) dζζ + φ (cid:17) dy Going to the limit gives: φ (cid:48) ∼ − α . That is, the perturbation leads to a shift: u ∼ α √− x sin (cid:18)
23 ( − x ) / + 3 a − x ) − α εx + p (cid:19) , x → −∞ . (11)Here α and p are solution parameters.The figure 7 shows the numerical solution of the equation (10) and thedifference between the numerical solution and the constructed asymptotics.The figure 8 shows a cross-section for x = −
50 of the bifurcation boundaryfor solutions of the perturbed Painlev´e-2 equation (10). The figure showsthat the boundary constructed from numerical results and the boundaryconstructed from asymptotic formulas are close.12
Conclusion
The equations for the parameters of the asymptotic behaviour of the Painlev´e-2 transcendent at x → −∞ derived in 3 allow us to obtain a formula for thebifurcation boundary of solutions for a perturbed equation with a soft loss ofstability in the neighbourhood of x = 0. This makes it possible to divide thesolutions of the perturbed equation into solutions close to (cid:112) x/ − (cid:112) x/ x → ∞ . The results are illustrated by computing perturbationsof various classes in the sections 5 and 6. References [1] R. Haberman. Nonlinear transition layers - second painelev´e trancen-dent.
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