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4Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 1, pp. 119-135. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220108
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 34E10
Asymptotics of Dynamical Saddle-node Bifurcations
L. A. Kalyakin
Dynamical bifurcations occur in one-parameter families of dynamical systems, when the parameter is slow time. In this paper we consider a system of two nonlinear differential equations with slowly varying right-hand sides. We study the dynamical saddle-node bifurcations that occur at a critical instant. In a neighborhood of this instant the solution has a narrow transition layer, which looks like a smooth jump from one equilibrium to another. The main result is asymptotics for a solution with respect to the small parameter in the transition layer. The asymptotics is constructed by the matching method with three time scales. The matching of the asymptotics allows us to find the delay of the loss of stability near the critical instant.
Keywords: nonlinear equation, small parameter, asymptotics, equilibrium, dynamical bifurcation
1. Introduction 1.1. Problem
The main object of the paper is a system of two ordinary differential equations with a small parameter 0 < e ^ 1 at derivatives. In a vector form, the equations read as
dx
e— =f(x;a); x g R2, r g M1. (1.1)
dr
In what follows, in some formulas we shall make use of the components of the vector columns x =
= (x, y)T, f = (f, g)T.
The right-hand side f (x; a) is a smooth vector function, namely, it is infinitely differentiable with respect to all variables. It depends on a finite-dimensional parameter a £ Rra. In an autonomous case when a = const the system has fixed points (equilibrium): x = p(a). Such solutions are determined by the functional equation
f(x; a) = 0. (1.2)
Received December 14, 2020 Accepted February 22, 2022
Leonid A. Kalyakin klenru@mail.ru
Institute of mathematics with computing center UFRC RAS ul. Chernyshevskogo 112, Ufa, 450008 Russia
We assume that there exists a bifurcation surface S c Mra through which the number of equilibria changes.
The aim of this paper is to study a nonautonomous system in which the parameter varies along a smooth line L = {a £ Rra: a = A(t)} with the property |A'(r)| = 0. We assume that the deformation line L crosses the bifurcation surface S at a critical instant t0, so that A(t0) = = a0 £ S. Due to the invariance of the type of the equations with respect to T-shift we can restrict the study to the case t0 = 0. A nonautonomous system may have no equilibrium. In similar problems the original equations are sometimes written in terms of a fast time t = e-1T, and the dependence a = A(et) is interpreted as a slow deformation of the parameter. We construct asymptotics with respect to a small parameter for the solution, which coincides with one of the roots of the functional equation: x(t; e) = p(A(t)) + O(e) as e ^ 0, t < 0.
It is possible to formulate another statement of the problem by using the triangular normal form [2]. But this leads to a more general problem with a small parameter which is not considered here. The equations as given in (1.1) are more suitable for applications.
1.2. Results
Numerical experiments with equations similar to (1.1) at small e detect solutions with narrow transient layers that arise near a critical instant. Examples of such solutions for the Landau -Lifshitz equations taken from [3, 4]
are given in Fig. 1. Equations (1.3) correspond to (1.1) as x = (z, , a = (A, B, ().
Fig. 1. Examples of a transient layer for the system (1.3). Left: the component z(t) in the transient layer has a smooth jump between different roots (dashed curve) of the functional equation for z as $ = 0. Right: the time shift of the jump depends on the small parameter: e = 0.01, 0.03
In dynamical problems, these solutions are linked with the loss of stability under slowly deformation of parameters. The main aim of this paper is to construct asymptotics with respect
(1.3)
-1.0
to a small parameter e ^ 0 on a time interval which includes a critical instant. The structure of asymptotics in the transition layer depends on the type of bifurcation. Here we consider a saddle-stable node bifurcation. While we construct an asymptotic solution,1 two fast time scales are detected. The asymptotic construction at the initial stage of the transition layer is described in terms of the time scale d = e-2/3 t and there appears the Riccati equation in a problem for the first correction of the asymptotics. The main restructuring of the solution is described in terms of the fast time n = e-1T — s(e). There appears the autonomous system (1.1) with a = a0 £ S for the leading term of the asymptotics. The coefficients of the asymptotics are solutions of differential equations. They are defined uniquely under additional matching conditions [5]. In this way the time shift s(e) is defined. The value s(e) ~ e-1/3 is not equal to zero. This phenomenon of the time shift of the fast transition layer is interpreted as a delay of the loss of stability, see the right panel of Fig. 1. In the slow time scale the shift es(e) & e2/3 is small. It is a specific feature of the saddle-node bifurcation. Another situation takes place for the Andronov-Hopf bifurcations which was investigated by A.Neishtadt [6-8]. In that case the slow time shift has the order of unity as e ^ 0.
1.3. Overview of related issues
Equations with a small parameter at derivatives are commonly called singularly perturbed. Some methods to study them are given, for example, in [5, 9]. The construction of the leadingorder term of asymptotics as e ^ 0 looks trivial, since it reduces to solving a functional equation obtained as e = 0. Such asymptotics, which we call algebraic asymptotics, correspond to isolated solutions. In a stable case these exceptional solutions serve as attractors for other solutions. In mathematical constructions, a fast transition to the algebraic asymptotics is detected by the method of boundary layer functions [9].
In general, the validity of algebraic asymptotics is restricted in time. An obstacle arises at some instant t0 because of breaking of the roots of functional equations. Since the solutions of differential equations are continuous functions, the discontinuity indicates more complex structure of asymptotics near the critical instant. Only here transition layers with a fast reconfiguration of solutions are detected. At this stage, the matching method [5] is used for asymptotic constructions.
In the study of differential equations the fast reconfigurations of solutions are often explained in terms of catastrophe theory and bifurcation theory [10, 11]. They are associated with sharp changes in the structure of phase portraits when the number of fixed points changes as the parameters cross the bifurcation surface. Strictly speaking, the analysis based on studies of the deformation of the phase portrait makes sense for autonomous systems with an additional bifurcation parameter [1].
In practice, the control of the physical system is dynamical, in other words, the parameters vary with time. If these variations are slow, then for an approximate analysis of the mathematical model one can use the autonomous equations with the frozen parameters. A number of asymptotic methods are based on this idea. However, the approach with frozen equations is not suitable for those time intervals in which the parameters pass through the bifurcation point. There are a lot of papers devoted to the problems with such dynamical bifurcations, see, e. g., the overview [12]. The known results are mainly related to the saddle-center bifurcation which
1 An asymptotic solution is a function which satisfies the equations with a high order of accuracy. A justification theorem for the exact solution is not discussed here.
occurs in nondissipative systems, for example, [13, 14]. Problems with dissipation lead to the dynamical saddle-node bifurcation. The simplest system of this type was considered in [15, 16].
According to its formulation and results, the problem considered here is close to the theory of relaxation oscillations [17-19]. Formally, closeness is detected in using the Riccati equation at the initial stage of the transition layer. However, for the fast stage of the transition layer, the results obtained here are new.
2. Restrictions
1) In the space Rra there is a domain of the parameter a E Rra for which the system of equations (1.1) has an asymptotically stable equilibrium x = p(a). On the smooth surface S C Rra this equilibrium bifurcates in the saddle-node.
2) At the instant t = 0 the line a = A(t) crosses the surface S transversally.
In this section we give a formal description of these restrictions, following [1]. Some constants which determine the structure of the asymptotics are identified.
As the parameter is frozen a = const, the vector equation (1.2) determines a pair of lines on the phase plane (x, y) E R2, the intersection of which gives the fixed points (equilibria) x = p(a). We consider an area where
dx f I =0, \0x9\ =0. (2.1)
In the general case the lines intersect transversally, and the fixed points are simple. The eigenvalues of the Jacobi matrix
d f
M{a) = —(x; a)
x=p(a)
determine the stability properties of the equilibria. The asymptotic stability occurs when the eigenvalues have negative real parts.
When the parameter a is deformed, the fixed points move and it is possible to merge them. This happens if the lines are tangent at the intersection point. The condition of tangency corresponds to parallelism of the gradients; it is formulated as vanishing of the Jacobian:
A(x; a) =
fx fy
9x 9y
0. (2.2)
In the case (2.2), at least one of the eigenvalues of the Jacobi matrix is zero.
Two scalar equations (1.2), (2.2) for the variables (x, y, a) define the (n — 1)-dimensional surface S in the space a £ Rra. The analysis of the functional equations (1.2), (2.2) depends on their specificity, see, for example, [3, 20]. We assume that Eqs. (1.2), (2.2) are solvable and the bifurcation surface exists.
The restrictions under which the bifurcation has a specific structure of saddle-node type are given in what follows. The conditions are imposed on the derivatives of the original vector function f(x; a) at the fixed points x = p(a), i.e., when Eqs. (1.2) and (2.2) hold. For the derivatives under such conditions we use the notation:
di+jf _ di+j g
dxi dyi ' ~ dx^yj
•Ma) = iMa) = x = p(a), a E S.
All these values are functions of a £ S. For these parameters the Jacobi matrix is written as
\A f 10 f01 \ ^
M0 = , a £ S.
\g10 g01)
The zero value of the determinant: f10g01 = f01g10 provides the null eigenvalue. Let L and R denote the corresponding eigenvectors, left and right:
LM0 = 0, M0R = 0, a £ S.
They can be taken as follows:
L = (g10, —f10), R = (-, a £ S.
The nonzero eigenvalue A coincides with the trace of the matrix M0.
Assumption 1. The condition of saddle - stable node bifurcation is
A = f10 + g01 < 0, a £ S. (2.3)
This provides stability in the nodal sector [1].
Without loss of generality, one can assume that the original equations are written in such a manner that f10 = 0. Then the property det M0 = 0, a £ S with condition (2.1) provides g10 = 0. These values are used to calculate the scale product of the eigenvector: LR =
= Af 10.
Assumption 2. The requirement of nondegeneracy is
H =
fx fy
Ax Ay
= 0, a £ S. (2.4)
Condition (2.4) guarantees the multiplicity of 2 for the fixed point p(a) as a £ S. This condition, taken in the form
V = f 10 (f 10g02 — g10f02) — 2f10(f01g11 — g01 f11) + f01(f01 g20 — g01f20) = 0, is equivalent to the one given in [1, p. 197].
To formulate the condition of transversality of the deformation line L to the surfaces S, we introduce the n-dimensional vectors in the parameter space a £ Rra:
fa = 9J(x; a) ga = dag(x; a) as x = p(a).
They represent the gradients of the functions f, g(x; a) calculated with respect to a and then taken at the fixed point. We use them to write down an expression for a normal vector to the surface S. The transversality condition is given in terms of these vectors.
Lemma 1. Let conditions (2.1), (2.3), (2.4) hold. If the vector composed by gradients
na = /10(a)5a — gio^^ a £ S (2.5)
is not zero (|na| = 0), then the surface S at this point a is smooth and the vector na is directed along the normal. In this case, the condition of transversality of the line L to the bifurcation surface S is
v = (na, A'(0)) =0, a = A(0) £ S. (2.6)
Proof. We consider the equations g(x; a) = 0 and A(x; a) = 0. For a £ S, they have the solution corresponding to the fixed point x = p(a). Because of properties (2.1) this solution is continued x = x(a) in the neighborhood of the surface S. Consider the scalar function F(a) = = /(x(a); a). The level surface F(a) = 0, which is obtained for a £ S, coincides with the bifurcation surface. The gradient vector daF(a) is orthogonal to the level surface and is calculated from the formula
daF(a) = dx f dax(a) + dyf day + daf.
The identity g(x(a); a) = 0 gives the equality dxg dax(a) + dyg day + dag = 0. By combining these equalities, we get
dxg daF(a) = [dx f dyg — dyf dxg\day + dxg daf — dxf dag.
The determinant A is here obtained as the multiplier. Since A = 0 on the bifurcation surface a £ S, we have
gio daF(a) = gio daf — fio dag, a £ S.
So the vector (2.5) differs from the gradient daF(a) by a nonzero factor g1o. Hence, this vector is orthogonal to the bifurcation surface at the points of smoothness. The lemma is proved. □
Equations (1.1) are considered on the finite fixed interval t £ [—5, 5], which includes the instant of bifurcation t = 0. It is assumed that there are no equilibria after passing through the bifurcation, at least in some neighborhood of the limit point x = p(A(0)). In addition to the transversality condition we have to specify the appropriate direction of deformation of the parameter a along the line L so that the loss of equilibrium occurs as t grows. The direction is identified by the vector A'(0) as a condition on the sign of the scalar product (2.6). We give this condition in the form
It guarantees the existence of a stable equilibrium for t < 0 and its absence at t > 0 in a neighborhood of the point p(A(0)), as will be shown in the subsequent section.
3. Statement of results
The asymptotic solution constructed below is defined uniquely. It represents asymptotics of the two-parameter families of the exact solutions which are different in the initial layer near the left boundary t & —5. The results for the initial boundary layer are well known [9] and are therefore not discussed here. We mainly deal with construction of the leading-order terms of asymptotics in the transition layer near the instant of bifurcation t = 0. Formulas with the infinite series are given only for orientation in the structure of the high-order terms. Their detailed analysis does not contain any fundamental novelty and is not given here.
There are three time scales in the construction of the asymptotics: the slow time t, the intermediate time 0 = £-2/3T and the fast time n = £ 1t — s(e). The value s, named the time shift, is calculated.
Theorem 1. Let conditions (2.1), (2.3), (2.4), (2.7) hold. Then for Eqs. (1.1) there exists an asymptotic solution as £ ^ 0 which is represented in the form differing on the different time intervals:
x(t; £)
p(A(t)), as 5 > —t » £2/3;
0 _a
p(A(0)) + eWr{0)R, as - 9 « e"1/3, —f-— » e1/3;
Iln(0* — 0)|
X(n), as \nI < £-1/3.
The different time intervals overlap. The asymptotics are matching in the overlap domains.
Explanation. The first formula gives the asymptotics in the form of the root of the functional equation (1.2). It is valid until the time of bifurcation is reached: t = 0. The second formula describes the asymptotics near the instant of bifurcation. The scalar function r(d) is defined from the Riccati equation and one has a pole at the point 0^ > 0. This approximation is valid until the instant d^. The third formula represents the asymptotics near the pole d^. The vector function x(n) is a solution of Eqs. (1.1) under the bifurcation value of the parameter a = a0. At this stage, the solution leaves the neighborhood of the point p(a0). The variable n = £-1 t + s(£) corresponds to the fast time t = £-1t with the additional shift s(£) = e^£-1/3 + /3 In £. The constant / is determined from the second correction in the intermediate layer. The shift s(£) is interpreted as the delay of the loss of stability. The comparison of the exact solution with asymptotic solution is given in Fig. 2.
z z
Fig. 2. Asymptotic approximation in the transient layers for the system (1.3) under £ = 0.03. Left: the component z (t) near the instant of bifurcation t = 0 is approximated by the solution of the Riccati equation. Right: the asymptotic approximation of the component z(t) near the pole 0 = 0^. There is the initial boundary layer on the left edge of the exact solution
The proof of Theorem 1 is given in the following sections.
4. Asymptotics of the slow motion 4.1. Formalism
Equations (1.1) are written in the slow time t. They contain a small factor e at the derivatives. An asymptotic solution, usually called external expansion, [5], is constructed as the series in powers of the small parameter:
x(t; e) = xo(t) +
<x
y.ek Xk (t ). k=i
(4.1)
The leading-order term corresponds to the stable equilibrium of the frozen system: x0 (t) = = p(A(t)). The higher-order corrections are uniquely determined from the linear algebraic equations
Mxk = fk.
They are obtained from (1.1) by equating expressions at the same degrees ek. Here M = M(t) is the matrix of the linearized system. It is the Jacobi matrix with the parameter a taken at the line L:
df (x, a)
M(t ) =
dx
x=p(a), a=A(r)
The determinant is the Jacobian introduced earlier:
det M = A(X a)|x=p(a), a=A(T) = A(t).
The right-hand sides are given by the recurrent formulas using the previous approximations:
„ _ dxk-i , n , \
rfc — h rfc-Ux0) • • •) xfc-l> a>-
For example, there is f1 = x0(t) on the first step.
4.2. Features of the leading-order term
The formal construction is based on the root p(a) taken for the leading-order term of the asymptotic solution. We have taken into account that the interval on T of existence of the smooth root depends on the position of the curve L in the space Rra. If the curve L does not intersect the bifurcation surface, then the asymptotic solution with the leading-order term x0(t) = p(A(t)) is defined for all t. If the curve L intersects the bifurcation surface at the instant t = 0, then the solution in the form (4.1) is defined as t < 0 and does not exist further.
As long as the fixed point remains simple, the vector function p(a) is smooth with respect to a £ L. This function has a weak singularity as a —y ag £ S. The singularity structure is determined by both the type of multiple fixed point ao and the direction of approach. The saddle-node bifurcation corresponds to the simplest fold catastrophe of the map x = p(a). Therefore, the function p(a) has the square root singularity as a — ao £ S. This singularity determines the asymptotics of p(A(t)) as t — —0.
Theorem 2. Let conditions (2.3), (2.4), (2.7) hold. Then the leading-order term of the asymptotic solution (4.1) has the power asymptotic expansion as t — —0:
<x
x0(r) EE p(A(r)) = Po + + £(-r)fc/2Pfc, T -0. (4.2)
k=2
The coefficients Pk S R2 are constant vectors; p0 = p(A(0)). The first correction is determined by the eigenvector p1 = cR with the factor c = — sgn(¿t) • \/2u/f10fi.
Proof. The equation for the fixed point is considered: f(x0; A(t)) = 0. In this relation the desired asymptotics (4.2) is inserted. Known functions are expanded in asymptotic series, in particular,
<x
A(t) = a0 + tA'(0) + y^ tkak, t — 0, ak G Rra
k=2
The expressions with the same powers (—t)k/2 are equated. This results in the recurrent system of equalities, from which the vector coefficients pk are defined.
The leading-order term p0 corresponds to the multiple fixed point p(a0) as a0 = A(0) e S. For the first correction, a homogeneous system of algebraic equations is obtained: M0p1 = 0, with the matrix M0 = M(0). The determinant of the matrix is zero detM = A(0) = 0. Therefore, the solution is defined via the right eigenvector
p,=^ R = (—y
up to the factor c = const. This constant is included in the right-hand side on the next step. If we take into account (\/—'r)2 = — r > 0, then for p2 the equation
M0p2 = —c2M,RR + daf A'(0) (4.3)
is obtained. Here the vector M,RR represents the second coefficient in Taylor's expansion of f (x, a0) at the multiple fixed point x ^ p(a0):
M RR — - (^2^ /02/10 \ 2 Vg20/01 — 2gii /10/01 + #02/10)
The specific multipliers which are quadratic in the first derivatives of f are liable to the structure of the first correction p1, which is expressed by the vector R. The second right-hand term in (4.3) comes from the expansion f with parameter a:
dafA(0)= (d:/((x; a))A//((0))N|, x = p(a), a = a0 e S.
Vda#(x; a) A(0) J
The solvability condition for the inhomogeneous system (4.3) with the matrix M0 consists of orthogonality of the right-hand side to the left eigenvector L = (g10, — /10). This requirement yields the quadratic equation for the constant c:
c2L^RR = L daf A'(0).
The coefficients are expressed here by means of constants from the original data
LA^RR = L0af A'(0) = -1/.
So the equation for c is reduced to the form c2 = 2v/figi and it is solvable under condition (2.7). Just the requirement of positivity of the right-hand side in this ratio identifies a suitable direction of deformation of parameter a along the line L.
Two values of the resulting factors c = ±\f2v/ fw/j, correspond to two fixed points (saddle and node) which merge to the point p(a0) as r —> —0. Thus, the first correction \f^TY>\ = C\/tR with two values of c detects the splitting of the multiple root when the parameter is moving from the bifurcation surface.
To complete the proof of the theorem, one has to choose one of two possible values of c. We take the value of c at which the first correction of the asymptotics at the fixed point agrees with the branch of node. The choice is based on an asymptotic analysis of the eigenvalues for the matrix of the linearized system. The formula for the eigenvalues of the Jacobi matrix reads
= \Ux+9y) ±\\]{fx+ 9y)2 - 4{fxgy - fygx).
The Jacobian (fxgy — fygx) = A(x, y; A(t)) — 0 as t — -0. Since (fx + gy)|t=g = A < 0, the asymptotics for the eigenvalue that corresponds to zero at t = 0 is described by the formula
\fx+9y\
A± = — —-r + 0{A2), A —> 0.
In order to calculate the leading-order term of the Jacobian, the first-order correction of the asymptotics
x0(r) = Q = p(a0) + v^c ^ J + 0(r), t -0
is used. Then the asymptotics of Jacobian leads to the formula with the factor i from condition (2.4): A = -y/^rcfj, + 0(t), t ->■ -0. Hence, A+ = V^Tcp/|A| + O(r), r ->■ -0. The sign of c determines the sign for A+ under small \/—t. In order for A+ to match the eigenvalue for the stable node, the sign c must match the sign so that ci < 0. Hence, the coefficient p1 for the first correction in the expansion (4.2) is determined.
The vector coefficient p2 in the second correction is taken as a solution of the inhomogeneous system (4.3). It is determined up to a factor when solving the homogeneous system. This factor is found uniquely on the next step from the solvability of the equation for p3. In the general step, the proof is obtained by induction.
The constructed series (4.2) is an expansion of the exact solution which corresponds to the stable equilibrium p(A(t)). This fact is established in a standard way by using the successive approximation technique [21]. The theorem 2 is proved. □
4.3. Features of higher approximations
In the higher-order corrections of the asymptotic solution (4.1) the order of singularity at the point t = 0 increases by 3/2 in each step.
Theorem 3. In the asymptotic solution (4.1) the coefficients have power singularities near the critical instant, the order of which grows with the number:
xfc(t)= t(1-3k)/2[xfc + o(1)j, t ^ —0, k = 1, 2,...; xfc = const = 0.
Proof. The corrections in the asymptotic solution are found from solutions of a linear algebraic system with a matrix M(t), whose determinant has order 0(t 1/2) as t ^ —0. The vectors of the right-hand sides contain derivatives of the previous approximation. So in every step the order of the singularities at t ^—0 increases by 3/2. For example, the right-hand side in the first step f1 contains singularity in the form
f1(r) = -i(-r)-1/2p1[l + 0(y^)], t —>■ —0, Pl = c( /o; V c/o.
2 1 " " {—/
10
Therefore we get asymptotics x1(r) = r 1x1 [l + O (■>/—r)], t —> — 0 with a nonzero coefficient x1. In the general step the asymptotics (4.2) is constructed by induction. □
Conclusion 1. The series (4.1) is asymptotic as e ^ 0 under the time restriction e2/3 ^ ^ —t ^ 5. The asymptotic solution in the form (4.1) is not suitable near the critical point t = 0.
5. The initial stage of rearrangement. The Riccati layer
In this section we construct the asymptotic solution as e ^ 0 for time near the critical instant t = 0. The formal construction is similar to the previous stage. The Riccati differential equation is detected instead of the square algebraic equation.
5.1. Formalism of the asymptotic solutions
The structure of singularities in the asymptotic solution (4.1) suggests an appropriate variable scale in the next step [5]. We introduce the intermediate variable 9 = e-2/3 t . The leadingorder term of the asymptotic solution is taken in the form of a fixed point p0 = p(A(0)) and the change of variables x(t; e) = p0 + e1/3X(9; e) is performed. For the vector function X(9; e) the following system of differential equations with a small parameter is obtained:
dX
t-2/337r = F(X, 0- e)- XgR2, 6 g R1. (5.1)
d9
The equations are considered on the semiaxis < 9 < 9^. The upper boundary 9^ will be specified below.
The right-hand side F(X, 9; e) = f(p0 + e1/3X; a), a = A(e2/39) with the property of the fixed point F(X, 9; e)|£=0 = f(p0; a0) = 0, a0 = A(0) has a fractional power expansion as e ^ 0:
F(X, 9; e) = e1/3M0X + e2/3[M1XX + 9daf A/(0)] + O(e). The asymptotic solution is constructed in the form of the fractional power series:
<x
X(9; e) = X1(9) + £efc/3Xfc+1(9). (5.2)
k=1
It is easy to see that the asymptotic construction as e — 0 again reduces to the algebraic equations, but now with the constant matrix Mo. For the first term a homogeneous system is obtained:
MoX1 = 0.
The solution is defined via the right eigenvector
Xi (9) = r(9)R, R = f^j
up to the scalar factor r(9). This function is included in the right-hand side in the next step
M0X2 (9) = F2(9); (5.3)
117'
F2 = -~-B.-r2M1KR-6dJA'{0), a = a0 e S. (5.4)
d9
The solvability condition for the inhomogeneous system with the matrix Mo consists of orthogonality of the right-hand side to the left eigenvector L = (glo, — flo). This requirement leads to a differential equation for the function r(9)
dr
— LR = r^LA^RR + 9 Ldaf A'(0). d9
It is the Riccati equation with nonzero coefficients taked from the original conditions:
ho\fe=-\ho^:2-v9. (5.5)
The differential equation is supplemented by the condition at minus infinity, which is obtained from the matching considerations:
r(9) = cV^9[l + o(l)}, 00; c = -sgn(At)y/2^-. (5.6)
The Riccati equation can be reduced to the standard form with unit coefficients
Î = r2 + 9, ¥(9) =-V^9[l + o(l)}, 9^-oo (5.7)
d9
by the scale transformations 9 = 909, r = r09 with
2^2/iq\ 1/3 ( , 4v x1/3
-A-
>2/:
1/3
V J \ V2/IQ)
5.2. Properties of the Riccati solution
Theorem 4. The Riccati equation (5.5) under the condition (5.6) has a unique solution. This solution is smooth on the half-axis : r(9) G 9^), 9^ = const > 0. One has the
first-order pole at the point 9^. The asymptotics near the pole reads
^ / 2A\ r(d) = (dt-6)-1a + '£(dt-d)nan, 9^9,-0 (an = const, a =--.
n=1 V V J
The proofs are given in [16]. For the solution fixed by asymptotic as 9 — the pole position is not calculated analytically. The dependence of the pole on the input parameters can be extracted from the scale transformations. It gives
V IU
Here 90 & 2 is the absolute constant, its approximate value is obtained from the numerical experiments with the Riccati equation (5.7) [22].
5.3. Properties of the higher corrections
The first approximation as the solution of the homogeneous system in the form X1(9) = = r(9)R is determined for 9 <9^ and one has the first-order pole at 9 = 9^.
In the next step the second-order pole at 9 = 9^ arises both on the right-hand side (5.4) and in the partial solution X2(9) of the inhomogeneous system (5.3). The general solution X2(9) = = r2(9)R + X2(9) includes the solution of the homogeneous system with an undetermined factor r2 (9). This function is found from the solvability requirement for X3(9). The inhomogeneous linearized Riccati equation is obtained for r2(9). An additional condition as 9 — is taken from matching with an external asymptotic solution. It provides the uniqueness of the solution r2(9).
The right-hand side of the linearized Riccati equation has the third-order pole. As a result of integration, the order decreases by unity. Moreover, the following logarithm [16] is detected in the asymptotics near the pole for the function r2(9):
r2(0) = y(0* - 0)-2 in(0* - 0) [1 + O ((0* - 0)2)] + Yi(0* - 0)-2 [1 + c(0* - 0)] So we get the asymptotics near the pole:
0 ^ 0, - 0.
X2(0) = Y(0* - 0)-2 ln(0* -0) [1 + O ((0* - 0)2)] R + (0* -
\-2
X 2 + O0-
0 ^ 0„ - 0.
The constant y, Y1 and the vector X2 are nonzero in the general case, as shown in [16]. They depend on the cubic terms in the Taylor expansion of f(x; a) with x — p0. We do not discuss it here.
For the general step, the coefficients Xk (9) have power and logarithmic singularities whose order increases with increasing number
xfc(0) = (0* - 0)-k lnk-1(0* - 0) Xfc + 0(1) , 0 ^ 0* - 0, Xfc = const
where Xk G R2. The proof by induction is similar to that given in [16].
Due to matching the validity domains of the external expansion (4.1) and the Riccati layer (5.2) overlap. Since the pole is positive (0* > 0), the asymptotic solution reaches the bifurcations point t = 0. However, there is a limitation on the right half-axis 0 > 0. Because of the growth of the singularity, the asymptotic solution in the form of the series (5.2) is not valid near the pole. This is a common situation in various problems with bisingular perturbation [5].
Conclusion 2. On the right half-axis 0 ^ 0 the series (5.2) represents an asymptotic solution of Eq. (5.1) under the condition el/3(0* - 0)-1| ln(0* - 0)| ^ 1.
6. Fast rearrangement stage
The Riccati equation permits one to continue the asymptotic solution through point t = 0. However, this advance is not large and does not reach the pole: t ^ e1/39*; and more importantly, the solution remains near the equilibrium p0 = p(A(0)). In this section we construct the asymptotic solution as e — 0 in a neighborhood of the pole 9 = 9*. This stage describes the process of departure of the solution from the neighborhood of the equilibrium p0 for a distance of the order of unity. It is the instant corresponding to the pole t* = e1/3 9, that can be associated with the instant of loss of stability. The fact that t* > 0 does not coincide with the instant of bifurcation t = 0 is interpreted as the effect of delay of the loss of stability. The following evolution of the solution depends on the global structure of the original system and is not discussed here: an example is considered in [16].
6.1. Formalism
The structure of the singularities of the constructed asymptotic solution near the pole indicates the scale which is suitable for constructing an asymptotic solution at the next stage. The fast variable is introduced as a stretching in the neighborhood of the pole 9*. It is given by the formula n = (9 — 9*)e-1/3 — a(e). This variable corresponds to the fast time t = T/e with a shift: s = 9*e-1/3 + a(e). The essential part of the shift has the order O(e-1/3) and is determined by the position of the pole in the Riccati solution. The additional shift a(e) has to be determined in asymptotic constructions. In a naive approach, it should be taken to be equal to zero. However, when the asymptotics is constructed, it can be shown that the additional shift significantly depends on the small parameter a(e) & ln e and there is no asymptotic solution in a simpler form [16]. Note that logarithmic shifts in the phase often occur in asymptotics for nonlinear equations.
The asymptotic solution is constructed as a series in fractional powers:
X
x(t; e) = *(n) + £ek/3Xk(n, e). k=1
The original equations (1.1) written in fast time n contain the small parameter in the function a = = A(en + e2/3 9, + ea(e)) whose value at e = 0 is constant A(0) = a0 e S. Therefore, the autonomous system is obtained for the leading-order term
dX
— = {(X;a0); X e R2, r G R1. (6.1)
The equation is considered on the axis < n <
6.2. Matching
The system (6.1) has a fixed point p(a0) of saddle-stable node type. The differential equations are supplemented with the condition at minus infinity from considerations of matching with the solution in the Riccati layer near the pole. In the leading-order term of asymptotics, the condition corresponds to the exit trajectory from the fixed point:
X(n) — p(a0) as n — —<x>. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(1), 119 135_
However, such a condition defines only the phase trajectory of the autonomous equations, while the solution is determined up to shift in the independent variable n + const. For identification the unique solution needs more detailed asymptotics at infinity. A suitable condition is obtained from matching with the asymptotics in the Riccati layer near the pole. Just the matching requirement leads to logarithms ln e in the additional time shift a(e).
The first step in the matching procedure is to rewrite the solution, obtained in the Riccati layer, in the new fast time. If we restrict ourselves to two corrections of the asymptotic solution (5.2), then we obtain near the pole the relation
x(t; e) = p(ag) +
a
R +
1
-n + o (—n + o)2
71
Ine1/3R + 1n(—n + o)R + X2] • [1 + o(1)].
In order to eliminate the logarithm 1n e, one has to choose the time shift
7
<r(t) = — Int. a
The resulting expression is taken as the asymptotic condition
a
7
1
X(ri) = P(ao) " -R + -2 ln(-r?)R + -^X2 + 0{rfà ln(-r?)), rj -»■ -oo
n
n
n2
(6.2)
Here a = —2A/^.
For the next corrections, nonhomogeneous linear equations are obtained. The right-hand sides are written using previous approximations. The matching requirement results in specifying the phase shift a(e) in the higher order of asymptotics. Their analysis is not discussed here.
Remark 1. In the asymptotics (6.2) the parameters a, y, X2 are determined by the first two corrections from the asymptotic solution in the Riccati layer (5.2). They do not depend on the higherorder corrections.
6.3. Autonomous system
The purpose of this section is to prove the solvability of the problem (6.1)-(6.2) for the leading-order term.
It is well known [1, p. 88], that the unique phase trajectory leaves the fixed point of the saddle-stable node type p(a0). This is a separatrix trajectory in the saddle sector. For the autonomous system, the phase trajectory defines the solution up to an arbitrary constant shift in the independent variable n. The shift is determined by the asymptotic conditions (6.2). Whether this condition can be satisfied becomes clear after detecting the asymptotic structure of the solution that corresponds to the separatrix trajectory.
Theorem 5. For Eqs. (6.1) there is a unique solution which satisfies the matching condition (6.2). It has an asymptotic expansion
X(V) = P(ao) " + 3 ln(-r?)R + 4x2 + ^ rTraR„-i(ln(-r?))
.
(6.3)
n=3
Coefficients of the series Rn-1({) are vector polynomials of degree not higher than n — 1.
The proof consists in calculating the coefficients from the relations which are obtained by substituting the series (6.3) into Eqs. (6.1) and by equating expressions with the same degrees
of variable n and logarithms ln(-n). Justification of asymptotics (with a proof of the existence of an exact solution) follows from the well-known general results [23, 24]. All other solutions which leave the fixed point differ by time shift. Their asymptotics as n — differ from (6.3) in the members of order n-2. Hence, the solution under condition (6.2) is unique.
This statement completes the proof of the main theorem (Theorem 1).
7. Conclusion
The asymptotic description of the dynamical saddle-node bifurcation was earlier performed for the model system [15, 16]. The analysis of the specific applied problem (for example, [3, 4, 20]) involves lengthy and cumbersome computations. The results obtained here for the general systems (1.1) make it much easier. Essentially they reduce to analysis of the bifurcation surfaces in the parameter space. In fact, such an analysis has been performed for a number of specific applications in magnetodynamics [4, 20]. So it remains to calculate the specific constants A, v in order to get the dynamics of the loss of stability under bifurcations.
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