Bautin bifurcations of a financial system

This paper is concerned with the qualitative analysis of a financial system. We focus our interest on the stability and cyclicity of the equilibria. Based on some previous results, some notes are given for a class of systems concerning focus quantities, center manifolds and Hopf bifurcations. The analysis of Hopf bifurcations on the center manifolds is carried out based on the computation of focus quantities and other analytical techniques. For each equilibrium, the structure of the bifurcation set is explored in depth. It is proved through the study of Bautin bifurcations that the system can have at most four small limit cycles (on the center manifolds) in two nests and this bound is sharp.


Introduction
Hopf bifurcation is the simplest way in which limit cycles can emerge from an equilibrium point.This phenomenon is an attractive subject of analysis for mathematicians as well as for economists, see [2,5,8,10,13,18,19,21,23,26,31] and the references therein.It occurs when a pair of complex conjugate eigenvalues of an equilibrium point cross the imaginary axis as the bifurcation parameter is varied.We recall that a limit cycle is a periodic orbit isolated in the set of all the periodic oribits of the system.
Hopf bifurcations have been studied in many business models, see, for instance, [14,18,28].For three-dimensional autonomous systems, Asada and Semmler [1] provided rigorous treatments on the analysis of Hopf bifurcations; Makovínyiová [20] proved the existence and stability of business cycles; Guirao, García-Rubio and Vera [7] studied the stability and the Hopf bifurcations of a generalized IS-LM macroeconomic model; Přibylová [23] investigated the Hopf bifurcations in an idealized macroeconomic model with foreign capital investment.
Corresponding author.Email: sangbo_76@163.com The Hopf bifurcations of a 3-dimensional financial system were firstly discussed in a series of two papers by Ma and Chen [16,17], which were far from complete because the focus quantities that characterize the criticality of the bifurcations were not obtained, and the Bautin bifurcations (also known as the generalized Hopf bifurcations) were not taken into account.The same model was later considered in [15] based on computing Lyapunov coefficients (which are equilivalent to focus quantities, see [24, Theorem 6.2.3 (page 261)]) by the method of Kuznetsov [9].However the results in [15] were still far from complete because some parameters were kept fixed.Thus, a more complete mathematical treatment of Hopf bifurcations and Bautin bifurcations in this model is necessary and important.
Consider the model proposed in [16,17], i.e., describing the development of interest rate x, investment demand y and price index z.The parameters a > 0, b > 0 and c > 0 denote the saving amount, the per-investment cost, and the demand elasticity of commercials, respectively.This system is invariant under the transformation (x, y, z) → (−x, y, −z).Despite its simplicity the system exhibits mathematically rich dynamics: from stable equilibria to periodic and even chaotic oscillations depending on the parameter values, see [15][16][17]27].The rest of the paper is organized as follows.In order to acquaint the reader with the focus quantities, center manifolds and Hopf bifurcations in three dimensional systems, in section 2, we gives some notes on these topics based on some works [3,4,6,9,11,12,24,26,[29][30][31].In section 3 the linear stability analysis is performed for the equilibria.In sections 4 and 5, the Hopf and Bautin bifurcations are studied for the equilibrium on the axis and two interior equilibria, respectively.Finally the concluding remarks are presented.

Focus quantities, center manifolds and Hopf bifurcations in R 3
Consider the following 3-dimensional differential system where ω, δ are positive constants, P j , Q are real analytical functions without constant and linear terms, defined in a neighborhood of the origin, j = 1, 2, and is considered as a real parameter.When = 0, the Jacobian matrix at the origin has a pair of purely imaginary eigenvalues λ 1,2 = ±iω and a negative eigenvalue λ 3 = −δ, so the origin is a Hopf point (see [3]) associated to the simple Hopf bifurcation.The simple Hopf bifurcation is a special type of Hopf bifurcations, where a pair of complex conjugate eigenvalues of the Jacobian matrix passes through the imaginary axis while all other eigenvalues have negative real parts, see [11].
For later use, let us write p 1 ,p 2 ,q x p 1 y p 2 z q , where p = (p 1 , p 2 ) and |p| = p 1 + p 2 .By introducing the transformation system (2.1) =0 can be transformed into the following form: where S(u, u, w) is real-valued for all u ∈ C and w ∈ R; and (2.4)

Focus quantities
Before we introduce the concept of focus quantities, we need a theorem, which is a generalization of [26,Theorem 3.1].
For (p 1 , p 2 , q) = (p 1 , p 1 , 0), where p 1 ≥ 2, we set The V n in (2.6) are determined by Proof.By direct computation, we find that Comparing the above power series with the right side of (2.6), we can obtain the recursive formulas (2.7) and (2.9).This completes the proof.
Remark 2.2.From (2.6), we can see that in order to compute V n , we only need to find a polynomial in the following form which is an approximation of (2.5) up to (2n + 2)-th order.
The following result can be proved using an argument similar to the proof of Theorem 2.1.
Corollary 2.4.F(u, u, w) is real-valued for u ∈ C and w ∈ R.
Proof.In order to prove the conclusion, we only need to show that We use induction on |p| + q = p 1 + p 2 + q.The statement is obviously true for |p| + q = 2, because we have already set Assume that the statement holds true for (p 1 , p 2 , q) : 2 ≤ |p| + q < N.
By the induction hypothesis and in view of (2.4), (2.7), (2.8) and (2.10), the statement holds true for |p| + q = N.This completes the proof of Corollary 2.4.
be the inverse of the transformation (2.2) and F be the formal series in Theorem 2.1, then G := F(u, v, w) is in the following form: (2.12) Definition 2.5.The functions V n in (2.12), which can be expressed as polynomials in the coefficients of (2.1) =0 , i.e., c p 1 ,p 2 ,q , d p 1 ,p 2 ,q , are called the n-th order focus quantities of system (2.1) =0 .
Remark 2.6.The definition is a natural extension of the focus quantities for two-dimensional systems.For the latter case, see [12, Definition 2.2.3] and [24,Definition 3.3.3].
In Theorem 2.1, if we try any other choice of C p 1 ,p 1 ,0 for p 1 ≥ 2, we may get different focus quantities V n .However using the same idea (based on normal form theory) as in [6,24], we can prove that: for any s ≥ 1, we have i.e., these two ideals are the same.Thus our definition for focus quantities is well-defined.

Focus quantities, center manifolds and Hopf bifurcations
Returning to system (2.1) =0 , for every r ∈ N, according to the center manifold theorem [4, Theorem 1, Theorem 2, Theorem 3], there exists, in a sufficiently small neighborhood of the origin, a C r−1 center manifold z = h(x, y) (which need not to be unique) such that (2.13) Moreover, system (2.1) =0 is locally topologically equivalent near the origin to the system In general the closed-form solution h(x, y) of (2.13) is very difficult to be found.However using formal Taylor series method, we can compute an approximate cener manifold to any desired degree of accuracy.Let where the function h is the center manifold of system (2.1) =0 .In view of (2.13), we obtain by direct computation that which implies that w = h(u, v) is the center manifold of system (2.3).Using similar arguments, we can prove that: Now we consider the restriction of system (2.1) =0 to the center manifold, i.e., (2.14) From (2.11), we can construct In view of (2.12), (2.13), we obtain by direct computation that From the identity above and [12, Definition 2.2.3] we know that V n are also the n-th focus quantities of the restriction system (2.14).
Remark 2.7.From the above discussion, we know that the focus quantities of system (2.1) =0 and system (2.14) are the same.This conclusion is of great importance: on the one hand, we can compute the focus quantities without recourse to center manifold reduction; on the other hand, just as in the 2-dimensional case, we can use focus quantities to analysis the Hopf bifurcations occurring on the center manifolds.
Focus quantities indicate the level of degeneration of the system (2.1) =0 .When V 1 = 0, on a two-dimensional center manifold of the origin, the Hopf bifurcation occurring at = 0 is non-degenerate.If V 1 < 0 then there is a stable limit cycle on the center manifold for > 0; the Hopf bifurcation is then called supercritical.If V 1 > 0 then there is an unstable limit cycle on the center manifold for < 0; the Hopf bifurcation is then called subcritical.
In order to describe the occurrence of Bautin bifurcation (the co-dimensional two Hopf bifurcation), we need to consider a special type of system (2.1), i.e., where ω, δ are positive constants, P j,a , Q a (a is a parameter) are real analytical functions without constant and linear terms, defined in a neighborhood of the origin, j = 1, 2, and , a ∈ R are considered as two parameters.Let V 1 (a), V 2 (a) be the first two focus quantities of system (2.15) ) is regular (see [9,31]), then a Bautin bifurcation occurs at = 0, a = a 0 on a two-dimensional center manifold of the origin.Moreover, if V 2 (a 0 ) < 0 then the Bautin bifurcation is supercritical; if V 2 (a 0 ) > 0 then the Bautin bifurcation is subcritical.In both cases, at most two limit cycles (on the local center manifold of the origin) can be found for the system by varying the parameters.

Linear stability of the equilibria
For convenience we define some test functions of the three positive parameters by which are needed hereafter.
Proof.The Jacobian matrix evaluated at E 1 is Let us denote the corresponding characteristic polynomial by If k 1 > 0, k 2 > 0, then this polynomial has three roots with negative real parts, which implies that the equilibrium is asymptotically stable.Because k 1 > 0, the system has a unique equilibrium at E 1 = (0, 1/b, 0), and thus the presence of E 2 and E 3 is impossible.This completes the proof.
Proof.Due to the symmetry, we only consider the stability of E 2 .
The Jacobian matrix evaluated at this equilibrium is Let us denote the corresponding characteristic polynomial by where the coefficients are defined by By the Routh-Hurwitz criteria, this polynomial has three roots with negative real parts if and only if It can be easily checked that these inequalities are equivalent to k 3 > 0, k 4 > 0. Thus if k 3 > 0, k 4 > 0, then E 2 is asymptotically stable.This completes the proof.

Hopf and Bautin bifurcations of the system at E 1
In this section we study the Hopf and Bautin bifurcations at E 1 .Taking a as the bifurcation parameter, that is, the coefficients in (3.2) can be rewritten as follows: .
According to the criterion [1, Proposition], a Hopf bifurcation occurs at a certain value of a, say a = a 0 > 0, if More specifically, by solving this semi-algebraic system, we can concluded that a Hopf bifurcation occurs at E 1 for a = a 0 , where For a near a 0 , the Jacobian matrix at E 1 has a pair of complex conjugate eigenvalues λ 1,2 (a) = δ(a) ± ω(a)i and a negative eigenvalue λ 3 (a) = −b, where δ(a 0 ) = 0, ω(a 0 ) > 0 and dδ da (a 0 ) = 0.This claim implies that the bifurcation at a = a 0 is the simple Hopf bifurcation, see [11] or Section 2 of this paper for the definition.
In order to determine the sign of dδ da (a 0 ) and the value of ω(a 0 ), we consider the quadratic factor of (3.2), i.e., Since g 1,1 (λ 1,2 ) = 0, we have δ(a) = −(ab + bc − 1)/(2b), so and Proposition 4.1.For a near a 0 , where a has a unique equilibrium E 1 , implying that Hopf bifurcation occurs at E 1 in the absence of any other equilibrium.
Proof.From the discussion above, we know that a Hopf bifurcation occurs at E 1 for Let us think of k 1 as a function of a, which is continuous for all a > 0. From the continuity of this function at a = a 0 , we have k 1 > 0 for a near a 0 .In this case, system (1.1) has a unique equilibrium E 1 , and thus the presence of the other equilibria E 2 and E 3 is impossible for a near a 0 .
This completes the proof.
By introducing the transformation the system (1.1) with a = a 0 becomes where ω 0 := ω(a 0 ) = √ 1 − c 2 .By performing computation on the first focus quantity V 1 (b, c) at (u, v, w) = (0, 0, 0) of system (4.3),we get Let S = 3 j=1 S j be a subset of {(b, c) : b > 0, 0 < c < 1}, where < c < 1 .Before we discuss the Hopf and Bautin bifurcations of system (1.1) at E 1 , we should know that these bifurcations occur on a center manifold of E 1 .Due to the complexity of the expression, we only give the approximate center manifold of system (4.3) up to third order, i.e., where the cubic terms are all zero.
Theorem 4.2.On a center manifold of system (1.1) at E 1 , a supercritical Hopf bifurcation occurs at a = a 0 = −c + 1/b with (b, c) ∈ S, leading to a stable limit cycle on the center manifold for a < a 0 and near a 0 ; and a subcritical Hopf bifurcation occurs at a = a 0 with (b, c) ∈ U, leading to an unstable limit cycle on the center manifold for a > a 0 and near a 0 .
Proof.Since we have assumed that b > 0 and 0 < c < 1, the denominator of V 1 (b, c) is positive, thus the sign of V 1 (b, c) is only determined by its numerator.Under the constraints the two sets of parameters: S and U, respectively.Thus the conclusion of this theorem follows from the Hopf bifurcation theorem [22,Theorem 3.15] along with the transversality condition (4.1).This completes the proof.
By performing the computation on the second focus quantity V 2 (b, c), we get where the numerator of V 2 (b, c) is reduced w.r.t. that of V 1 (b, c).For further simplification of this quantity, we solve V 1 (b, c) = 0 for c and obtain a unique solution c = c 0 := −b/8 Hence a supercritical Bautin bifurcation may occur at E 1 for (a, c) = (a (1) 0 , c 0 ), where a (1) Theorem 4.3.On a center manifold of system (1.1) at E 1 , a supercritical Bautin bifurcation occurs at (a, c) = (a (1) 0 , c 0 ) with 0 < b < 2/3.This bifurcation generates two small amplitude limit cycles on the center manifold for the fixed parameters in the set {(a, b, c) : 0 < a − a (1) 0 c − c 0 1, 0 < b < 2/3}, with the outermost cycle stable and the inner cycle unstable.Moreover in this case the equilibria E 2 , E 3 don't exist.
A simple computation gives δ(a Thus all the conditions of Bautin bifurcation are satisfied (see [9,Theorem 8.2] or [31,Theorem 2.3]), so that the conclusion on the limit cycles is proved.
For any b ∈ (0, 2/3), we have k 1 (a , which implies that the equilibria E 2 and E 3 don't exist.In summary, we complete the proof.

The Hopf and Bautin bifurcations at E 2
Let us choose a as the bifurcation parameter, that is, the coefficients of (3.3) can be rewritten as follows: According to the criterion [1, Proposition], a Hopf bifurcation occurs at E 2 for a certain value of a, say a = a 1 > 0, if More specifically, by solving this semi-algebraic system for the critical value a 1 , we can concluded that a Hopf bifurcation occurs at E 2 for a = a 1 with h 1 > 0, h 2 > 0, h 3 > 0, where and 2) The condition h 1 > 0 follows from p 2,2 (a 1 ) > 0. The condition h 2 > 0 follows from the presence of E 2 .The condition h 3 > 0 follows from the fact that the critical value a 1 must be positive.
Proposition 5.1.The Hopf bifurcation occurs at E 2 for a = a 1 is a simple Hopf bifurcation.
With the same notations for the roots as in the proof of Proposition 5.1.If we set a = a 1 in (3.3), then the characteristic polynomial becomes Thus which is a factor of h 3 , seen in (5.2).Before checking the sign of δ 1 (a 1 ), we need the following lemma.

Lemma 5.2. If h
According to the Taylor expansion formula, we rewrite h 3,0 in the following way: which makes h 3,0 < 0. Thus we have reached a contradiction and so that c > 1.
Since c > 1 and h 1 > 0, bc < 1.Thus it follows from h 3 > 0 that h 3,0 < 0. In summary, we end the proof of this lemma.
As a direct consequence of Corollary 5.3, we have the following result.
Corollary 5.4.The Hopf bifurcation set is With the same notations for the roots as in the proof of Proposition 5.1, we have the following result.
Proof.Suppose that h 1 > 0, h 3 > 0.Then, by Lemma 5.2 and Corallary 5.4, we know that c > 1 and a Hopf bifurcation occurs at a = a 1 .
Recalling (3.3), the occurrence of Hopf bifurcation implies that Since c > 1 and h 1 > 0, we have 1 − bc > 0 and (5.5) is positive.From the proof of [1, Proposition], we know that the sign of δ 1 (a 1 ) is different from that of (5.5), so we complete the proof.The following result is the converse of Lemma 5.2.
As a direct consequence of Lemma 5.2, Lemma 5.7 and Corallary 5.4, we have the following result.
Corollary 5.8.The Hopf bifurcation set S defined by (5.4) can be implicitly rewritten as follows: (5.6) For later use, let We now seek to find the explicit representation of the bifurcation set S, which is described by (5.6).To achieve this goal, we need the following lemmas, which are related to the roots of polynomial h 3,0 in b, seen in (5.3).
Proof.This inequality can be proved by noting that c 2 > 1 and (5.8) can be rewritten as follows: which is positive when c > 1.So that ∆ > 0. Thus we complete the proof of this lemma.
For c > 1 and c = √ 3, according to Lemma 5.9, the quadratic polynomial h 3,0 has two distinct roots for b.By the direct computations, these roots can be represented by It can be easily checked that τ 1 > 0 for c > 1 and c = √ 3. The sign of τ 2 is positive for 1 < c < √ 3 and negative for c > √ 3.
Lemma 5.10.For 1 < c < √ 3, we have Proof.Since 1 < c < √ 3, the left inequality is equivalent to (5.9) By squaring and rearranging, the desired inequality (5.9) can be reduced to c 2 > 1, which is obviously true.Since 1 < c < √ 3, the right inequality is equivalent to This is obviously true because the right hand side equals to (−c 2 + 1)(c 2 + 3), which is negative for 1 < c < √ 3.In summary, we complete the proof of this lemma.Lemma 5.11.For c > √ 3, we have Proof.Since c > √ 3, the inequality is equivalent to Both sides are positive.By squaring and rearranging, the desired inequality (5.10) can be reduced to c 2 > 1 which is obviously true.This completes the proof.
Theorem 5.12.The Hopf bifurcation set S defined by (5.4) can be rewritten as follows: where By performing computation on the first two focus quantities for system (5.13),we get where and the expression of n 21 is somewhat complicated and can be found in the Appendix.Before we discuss the Hopf and Bautin bifurcations of system (1.1) at E 2 , we should know that these bifurcations occur on a center manifold of E 2 .Due to the complexity of the quadratic approximation of center manifold of system (5.13),we will not present here.
According to the Hopf bifurcation theorem [22, Theorem 3.15] and Remark 5.6, we have the following theorem.Theorem 5.13.Assume that (b, c) ∈ S * , where S * is described by (5.7) and V 1 (b, c) = 0. On a center manifold of system (1.1) at E 2 , a Hopf bifurcation occurs at a = a 1 .More precisely, the bifurcation is supercritical for V 1 (b, c) < 0, giving rise to a stable limit cycle on the center manifold for a < a 1 ; and subcritical for V 1 (b, c) > 0 giving rise to an unstable limit cycle on the center manifold for a > a 1 .
In Maple 2016 (a computer algebra system), the command RootFinding[Isolate] isolates the real roots of univariate polynomials and polynomial systems with a finite number of solutions.By default it computes isolating intervals for each of the roots and numerically evaluates the midpoints of those intervals at the current setting of digits.All significant digits returned by the program are correct, and unlike purely numerical methods no roots are ever lost.Now we consider the semi-algebraic system where S * is described by (5.7).
Using the command RootFinding[Isolate], we find there is no solution to (5.16) with (b, c) ∈ S * .So there is no need to calculate V 3 (b, c), and thus the system can have at most two small limit cycles in some neighborhood of E 2 .Due to the symmetry, at most four small limit cycles can be found on the center manifolds that spiral around the equilibria E 2 and E 3 .

Concluding remarks
We investigated a financial system that describes the development of interest rate, investment demand and price index.By performing computations on focus quantities using the recursive formula, we derived the conditions at which limit cycles can bifurcate from the equilibria E 1 and E 2,3 , respectively.The stabilities of the bifurcated limit cycles were also investigated in detail.Based on the analysis of Bautin bifurcations, it was proved that the system have at most four small limit cycles on the center manifolds and this bound is sharp.

Remark 5 . 6 .
If we treat δ 1 as a function of a, b and c, then by Corollary 5.5, we have ∂δ 1 ∂a (a 1 , b, c) < 0. This fact will be used in the future.