In this paper, exact wave propagation patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional nonlinear Schrödinger equation are studied. The topological structure of the dynamic system of the equation is studied by the complete discrimination system for the cubic polynomial method, in which the existence conditions of the soliton solutions and periodic solutions are obtained. Then, by the trial equation method, thirteen exact solutions are obtained, including solitary wave solutions, triangular function solutions, rational solutions and the elliptic function double periodic solutions, especially the elliptic function double periodic solutions. Finally, it is found that the system has chaotic behaviors when given the appropriate perturbations.
Citation: Ninghe Yang. Exact wave patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional NLSE[J]. AIMS Mathematics, 2024, 9(11): 31274-31294. doi: 10.3934/math.20241508
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In this paper, exact wave propagation patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional nonlinear Schrödinger equation are studied. The topological structure of the dynamic system of the equation is studied by the complete discrimination system for the cubic polynomial method, in which the existence conditions of the soliton solutions and periodic solutions are obtained. Then, by the trial equation method, thirteen exact solutions are obtained, including solitary wave solutions, triangular function solutions, rational solutions and the elliptic function double periodic solutions, especially the elliptic function double periodic solutions. Finally, it is found that the system has chaotic behaviors when given the appropriate perturbations.
The following differential autonomous systems in a planar vector field
˙x=F(x,y),˙y=G(x,y) | (1) |
have been widely studied and a great deal of attentions have been paid to this problem in many literatures. This activity reflects the breadth of interest in Hilbert's 16th problem and the fact that the above systems are often used as mathematical models to describe real-life problems. Hilbert's 16th problem is to find the maximum number of limit cycles of system (1).
The predator-prey system, the competition system and the cooperation model are the three most basic types of systems in mathematical ecology. Theoretically, many natural predator-prey systems can be discussed and investigated by some kinds of ecological models. The qualitative properties of differential systems are often used to describe the characteristics of ecosystems, as they have been investigated in some literatures, for example, Liénard systems ([2,3,19,21]), Kolmogorov systems ([2,3,4,5,6,7,8,9,10,11,12,16,17,18,19,20,22,23,24,25,26]) and some other differential systems ([13,14,15]) and so on. The Kolmogorov systems (introduced by A. Kolmogorov in 1936 [12]), as a class of significant ecological models, were initially introduced to describe the interaction between two species occupying the same ecological habitat. The form Kolmogorov models are as follows:
{dxdt=xf(x,y),dydt=yg(x,y), | (2) |
in which f(x,y) and g(x,y) are polynomials in x and y. The variables x and y are often described as the number of species in two ecological populations, dxdt and dydt represent the growth rates of x and y. Hence, attention is often restricted to the behavior of orbits in the 'realistic quadrant' {(x,y):x>0,y>0}. Particular significance in applications is the existence of limit cycles and the number of limit cycles that can occur near positive equilibrium points, because a limit cycle corresponds to an equilibrium state of the system. The existence and stability of limit cycles are closely related to the positive equilibrium points. Hence, many references studying Kolmogorov models pay more attention to the limit cycles problem.
On the qualitative analysis and the bifurcation of limit cycles for cubic planar Kolmogorov systems, some papers are as follows: Ref. [1] characterized the center conditions for a cubic Kolmogorov differential system and obtained the condition that the positive equilibrium point can become a fine focus of order five. Ref. [23,24] studied a class of cubic Kolmogorov systems that can bifurcate three limit cycles from the positive equilibrium point (1, 1). Ref. [4] investigated the limit cycles bifurcation problem for a class of cubic Kolmogorov system and showed that this class of the Kolmogorov system could bifurcate five limit cycles including 3 stable cycles. Ref. [22] showed that a class of the cubic Kolmogorov system could bifurcate 6 limit cycles. Ref. [25] obtained the condition of integrability and non-linearizability of weak saddles for a cubic Kolmogorov model. Ref. [6] investigated limit cycles in a class of the quartic Kolmogorov model with three positive equilibrium points. Ref. [7] studied the three-Dimensional Hopf Bifurcation for a class of the cubic Kolmogorov model. Ref. [19] investigated the integrability of a class of 3-dimensional Kolmogorov system and provided the phase portraits.
In addition to cubic Kolmogorov systems, there is also a lot of literature on the bifurcation of limit cycles for some generalized Kolmogorov systems and higher order Kolmogorov systems. For example: Ref. [10,26] studied a general Kolmogorov model and obtained the condition for the existence and uniqueness of limit cycles and it classified a series of models. Ref. [8] investigated the bifurcation of limit cycles for a class of the quartic Kolmogorov model with two symmetrical positive singular points. Ref. [9] investigated the Hopf bifurcation problem about small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1, 1) and proved that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1, 1) was 7. Ref. [20] considered the Kolmogorov system of degree 3 in R2 and R3, and showed it had an equilibrium point in the positive quadrant and octant, and provided sufficient conditions in order that the equilibrium point will be a Hopf point for the planar case and a zero-Hopf point for the spatial one, and studied the limit cycles bifurcating from these equilibria using averaging theory of second and first order.
As far as limit cycles of Kolmogorov models are concerned, many good results have been obtained, especially in lower degree system by analyzing a sole positive equilibrium point's state. However, results for Simultaneous limit cycles bifurcating from several different equilibrium points' is less seen, and perhaps it is difficult to investigate this kind of problem. From an ecological perspective, investigation about multiple positive equilibrium points is meaningful, the equilibrium point (a,b) represents that the ratio of the density about predator and prey is a:b, and so it is possible for several equilibrium points to occur in ecosystem.
In this paper, we study a class of the following quartic Kolmogorov models
{dxdt=16x(y−1)(x2−24ay−6A10x+6A10xy−7y2+30ay2−6a)=P(x,y),dydt=13y(3b+3B10x−x2+6x3+9by−6B10xy−26x2y−2y2−27by2+39xy2+3B10xy2−16y3+15by3)=Q(x,y), | (3) |
in which a,b,A10andB10∈R.
Clearly, model (3) has two positive equilibrium points namely, (1, 1) and (2, 1). We will focus on the limit cycles bifurcations of the two positive equilibrium points. By analyzing and proving carefully, we obtain that each one of the two positive equilibrium points can be a 5th-order fine focus. Furthermore, we find the condition that each positive equilibrium point can bifurcate five limit cycles, of which three limit cycles are stable. Our results are concise (especial that in the expressions of focal values) and the proof about existence of limit cycles is algebraic and symbolic.
This paper includes 4 sections. In Section 2, we introduce the method to compute focal value offered by [18]. In Section 3, we respectively compute the focal values of the two positive equilibrium points of model (3) and obtain the condition that they can be two 5th-order fine focuses. In Section 4, we discuss the bifurcation of limit cycles of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles; we give an example that three stable limit cycles can occur near each positive equilibrium point under a certain condition.
In order to use the algorithm of the singular point value to compute focal values and construct the Poincaré succession function, we need to give some properties of focal values and singular point values.
Consider the following system
{dxdt=δx−y+∞∑k=1Xk(x,y),dydt=x+δy+∞∑k=1Yk(x,y), | (4) |
where Xk(x,y),Yk(x,y) are homogeneous polynomials of degree k on x,y. Under the polar coordinates x=rcosθ,y=rsinθ, system (4) takes the form
drdθ=rδ+∑∞k=2rk−1φk+1(θ)1+∑∞k=2rk−1ψk+1(θ). | (5) |
For sufficiently small h, let
d(h)=r(2π,h)−h,r=r(θ,h)=∞∑m=1v(θ)hm, | (6) |
be the Poincaré succession function and the solution of Eq (6) satisfies the initial value condition r|θ=0=h. It is evident that
v1(θ)=eδθ>0,vm(θ)=0,m=2,3,⋯ | (7) |
Lemma 2.1. For system (4) and any positive integer m, among v2m(2π),vk(2π) andvk(π), there exists expression of the relation
v2m(2π)=11+v1(π)[ξ(0)m(v1(2π)−1)+m−1∑k=1ξ(k)mv2k+1(2π)], | (8) |
where ξ(k)m are all polynomials of v1(π),v2(π),⋯, vm(π) and v1(2π),v2(2π), ⋯,vm(2π) with rational coefficients.
In addition to indicating that v2m=0 under the conditions v1(2π)=1,v2k+1(2π) =0,k=1,2,⋯,m−1. Lemma 2.1 plays an important role in construction of Poincaré succession function.
Definition 2.1. For system (4), in the expression (6), if v1(2π)=1, then the origin is called the rough focus (strong focus); if v1(2π)=1, and v2(2π)=v3(2π)=⋯=v2k(2π)=0,v2k+1(2π)≠0, then the origin is called the fine focus (weak focus) of order k, and the quantity of v2k+1(2π) is called the kth focal values at the origin (k=1,2,⋯); if v1(2π)=1, and for any positive integerk,v2k+1(2π)=0, then the origin is called a center.
By means of transformation
z=x+yi,w=x−yi,T=it,i=√−1, | (9) |
system (4)|δ=0 can be transformed into the following systems
{dzdT=z+∞∑k=2Zk(z,w)=Z(z,w),dwdT=−w−∞∑k=2Wk(z,w)=−W(z,w), | (10) |
where z,w,T are complex variables and
Zk(z,w)=∑α+β=kaαβzαwβ,Wk(z,w)=∑α+β=kbαβwαzβ. |
It is obvious that the coefficients of system (10) satisfy conjugate condition ∀aαβ=¯bαβ, we call that system (4)|δ=0 and (10) are concomitant.
Lemma 2.2. (See [16]) For system (10), we can derive successively the terms of the following formal series
M(z,w)=∞∑α+β=0cαβzαwβ, | (11) |
such that
∂(MZ)∂z−∂(MW)∂w=∞∑m=1(m+1)μm(zw)k, | (12) |
where ckk∈R,k=1,2,⋯, and to any integer m, μm is determined by following recursion formulas
c0,0=1, |
when(α=β>0)orα<0,orβ<0,cα,β=0, |
else
cα,β=1β−αα+β+2∑k+j=3[(α+1)ak,j−1−(β+1)bj,k−1]cα−k+1,β−j+1, | (13) |
μm=2m+2∑k+j=3(ak,j−1−bj,k−1)cm−k+1,m−j+1. | (14) |
Lemma 2.3. (See[18]) For systems (5)|δ=0, (10) and any positive integer m, the following assertion holds
v2m+1(2π)=iπ(μm+m−1∑k=1ξ(k)mμk), | (15) |
where ξ(k)m(k=1,2,⋯,m−1) be polynomial functions of coefficients of system (10).
For the linearized system of model (3), its coefficient matrix in point (x0,y0) is as follows:
A(x0,y0)=[∂P(x,y)∂x∂P(x,y)∂y∂Q(x,y)∂x∂Q(x,y)∂y](x0,y0), |
in which
∂P(x,y)∂x=16(y−1)(−6a−12A10x+3x2−24ay+12A10xy−7y2+30ay2),∂P(x,y)∂y=16x(18a−12A10x+x2+14y−108ay+12A10xy−21y2+90ay2),∂Q(x,y)∂x=13y(3B10−2x+18x2−6B10y−52xy+39y2+3B10y2),∂Q(x,y)∂y=13(3b+3B10x−x2+6x3+18by−12B10xy−52x2y−6y2−81by2+117xy2+9B10xy2−64y3+60by3). |
For the two positive equilibrium points (1, 1) and (2, 1) of model (3), their coefficient matrixes of the linearized system of model (3) become
A(1,1)=A(2,1)=[0−110]. |
Clearly, A(1,1) and A(2,1) have the same two eigenvalues ±i. Hence, model (3) can be changed into the system (4) by making some appropriate transformations.
For convenience, we will respectively compute the focal values of each positive equilibrium point of model (3).
By means of transformation
x=u+1,y=v+1, | (16) |
model (3) takes the following form
{dudt=−v−uv+16(u+1)v(2u+u2−14v+36av+6A10v+6A10uv−7v2+30av2),dvdt=u+uv+13(v+1)(6u3−9u2+26uv−26u2v−11v2+18bv2+3B10v2+39uv2+3B10uv2−16v3+15bv3), | (17) |
and the equilibrium point (1,1) of model (3) becomes the origin of (17) correspondingly.
Under the transformation
z=u+iv,w=u−iv,T=it,i=√−1, | (18) |
system (17) becomes the following complex system
{dzdT=z+Z2(z,w)+Z3(z,w)+Z4(z,w),dwdT=−w−W2(z,w)−W3(z,w)−W4(z,w), | (19) |
in which
Z2(z,w)=112i(22+18a+3A10+18ib+3iB10)w2+16(−20+7i−18ia−3iA10+18b+3B10)wz+112i(−36−4i+18a+3A10+18ib+3iB10)z2,Z3(z,w)=i8[−5+18i+(6+5i)a+2A10−11b−(1−i)B10]w3+124[74−109i+(45−18i)a−6iA10+99ib+(3+9i)B10]w2z−124[−92−123i+(45+18i)a+6iA10+99ib−(3−9i)B10]wz2+124[−64+i+(15+18i)a+6iA10+33ib−(3−3i)B10]z3,Z4(z,w)=−148(−14+33i+15a−3iA10−15b+3iB10)w4+124(29+45i+15a−30b+3iB10)w3z+124(−74−3iA10+45b)w2z2−124(−35+45i+15a+30b+3iB10)wz3+116(2+11i+5a+iA10+5b+iB10)z4,W2(z,w)=112(4+36i−18ia−3iA10−18b−3B10)w2+16(−20−7i+18ia+3iA10+18b+3B10)wz−112i(22+18a+3A10−18ib−3iB10)z2,W3(z,w)=−124i[1−64i+(18+15i)a+6A10+33b+(3−3i)B10]w3+124[92−123i−(45−18i)a+6iA10+99ib+(3+9i)B10]w2z+124[74+109i+(45+18i)a+6iA10−99ib+(3−9i)B10]wz2−18[18−5i+(5+6i)a+2iA10−11ib+(1−i)B10]z3,W4(z,w)=116(2−11i+5a−iA10+5b−iB10)w4−124(−35−45i+15a+30b−3iB10)w3z+124(−74+3iA10+45b)w2z2+124(29−45i+15a−30b−3iB10)wz3−148(−14−33i+15a+3iA10−15b−3iB10)z4. |
Obviously, system (19) belongs to the type of system (10). Then, we can compute the focal values of the origin of (19) (namely the focal values of the equilibrium point (1,1) of model (3)) by using the method of Section 2. According to the recursion formulas (13) and (14) offered by Lemma 2.2, the following theorem holds. For convenience, here we note that B10=λ−6b.
Theorem 3.1. The first five singular point values at the origin of system (19) are as follows:
μ1=−136i(379−378a−54A10+135b−102λ+ 108aλ+18A10λ); μ2=19720iA10f1+129160if2; μ3=141990400ig2+ 14665600iA10(9A10g3+g4); μ4=−1226748160000i(486A410g5+ 3A10g6+g7+9A210g8+81A310g9+7290A510g10+3280500A610g11); μ5=−12203992115200000i(f3+3A10f4+27A10f5λ+27A10f6λ2+81A10f7λ3+ 243A10f8λ4+2187A10f9λ5+98415A10f10λ6 +98415A10f11λ6);
in which fi,gi,(i∈{1,2,⋯11}) are the expressions about A10,a,λ, their expressions can be obtained via computing by reader.
According to the relation between model (3) and system (17) and system (19), from Theorem 3.1 and Lemma 2.3, the following theorem is visible.
Theorem 3.2. The simplified expressions of the first five focal values in the equilibrium (1,1) of model (3) (or the first five focal values at the origin of system (19)) are as follows:
v3=136π(379−378a−54A10+135b−102λ+108aλ+ 18A10λ); v5=−19720πA10f1− 129160πf2; v7=−141990400πg2−14665600 πA10(9A10g3+g4); v9=1226748160000π(486A410g5+ 3A10g6+g7+9A210g8+81A310g9+7290A510g10+ 3280500A610g11); v11=12203992115200000π(f3+3A10f4+ 27A10f5λ+27A10f6λ2+ 81A10f7λ3+243A10f8λ4+2187A10f9λ5+ 98415A10f10λ6+98415A10f11λ6). From Theorem 3.2, we have the following theorems.
Theorem 3.3. The equilibrium point (1,1) of model (3) can be a 5th-order fine focus at most.
Proof. According to Definition 2.1, we need to prove that there exits a group of real values about λ,A10,a,b such that v3=v5=v7=v9=0,v11≠0.
At first, we prove v3=v5=v7=v9=0 have real solutions. Let v3=0, then we have
b=1135(−379+378a+54A10+102λ−108aλ−18A10λ). | (20) |
From the quality of resultant, if f(x,y)=0,g(x,y)=0 have solutions, then the resultant of f(x,y),g(x,y) with respect to x or y will vanish. When computer soft Mathematica 6.0 is used, f(x,y),g(x,y) with respect to x is shown as Resultant[f,g,x]. Hence, v3=v5=v7=v9=0 hold if and only if Eq (20) holds and
{r57=Resultant[v5,v7,A10]=0,r59=Resultant[v5,v9,A10]=0. | (21) |
While Eq (21) hold if and only if
r579=Resultant[r57,r59,a]=0. | (22) |
By computing, we obtain
r579=Resultant[r57,r59,a]=2401(53−65λ+10λ2)2(4261−1966λ+200λ2)2g(λ), |
in which g(λ) is a 242 degrees function on λ. It can be seen that Eq (22) has some real solutions such as λ=120(65±√2105) et al. Hence, v3=v5=v7=v9=0 have real solutions.
In fact, we can find 13 groups of real number solutions such that v3=v5=v7=v9=0, namely
1)A10≈0.459518722796388637074,a≈0.20757605414524699732908423106,
λ≈−187.1523806676270387309,b≈−100.90105602889687249731131061;
2)A10≈−51.84584918202522022025,a≈7.9357437035495313851150889631,
b≈7.006788358368472491711,λ≈6.31370574128761387341294162;
3)A10≈25.494503512525795111362,a≈−3.07139640695089280406061269,
b≈−1.532179840913880774192,λ≈1.72922556020556141044796397;
4)A10≈−25.070512253424586307122,a≈3.33783060983953401572921236,
b≈3.263720382764871832166,λ≈4.72919037845355221552810321;
5)A10≈−14.449047061417717199600,a≈3.08815966599950902620342510,
b≈−0.629369782476961469600,λ≈−3.25754589079607174405111000;
6)A10≈8.111924106836898022434,a≈−1.53372638845377554695210010,
b≈0.126446067636650042023,λ≈0.53142294384083763082100010;
7)A10≈5.621924167862105789322,a≈0.00165824258524943110012000,
b≈−0.514536038089562571070,λ≈8.50571390132676346023508968;
8)A10≈3.0566036738431139791551,a≈−0.98949281181308481001231800,
b≈1.3119509650856638974769,λ≈4.97304633639086336473699900;
9)A10≈6.6897266686712602090401,a≈−0.25129280165975847841010671,
b≈−1.7878517956323023614398,λ≈−14.74192519032294176080864107;
10)A10≈1.6274879010489748450201,a≈0.18419310366272218518672210,
b≈0.4716932080651445815482,λ≈5.39966835957640020711011671;
11)A10≈0.1394463380540504608910,a≈−0.08831387583182001769245266,
b≈1.6869344282004140469154,λ≈5.80208273810907688645907382;
12)A10≈−8.0555918046326485797027,a≈0.68203278918102018306728290,
b≈0.8147669629127921204300,λ≈3.84321467839847163723015678;
13)A10≈−7.2153236483212033683720,a≈1.12357380119512738273882722,
b≈2.6297408958156117722512,λ≈6.32346437261518950112432918.
Next, we prove v11≠0 if v3=v5=v7=v9=0.
Let r1=Resultant[v5,v11,A10],r2=Resultant[v7,v11,A10],r3= Resultant[v9,v11,A10], and r12=Resultant[r1,r2,a],r13=Resultant[r1,r3,a].
If v3=v5=v7=v9=v11=0, then r123=Resultant[r12,r13,λ]=0. In fact, by computing we obtain r123=Resultant[r12,r13,λ]=3076590098860238833065040⋯≠0.
Hence, v11≠0 if v3=v5=v7=v9=0, then the equilibrium point (1,1) of model (3) can be a 5th-order fine focus at most.
Next, we compute the focal values of the positive equilibrium point (2, 1) of model (3). By means of transformation
x=˜u+2,y=˜v+1, | (23) |
model (3) takes the following form:
{d˜udt=−˜v−˜u˜v2+˜v6(˜u+2)(4˜u+˜u2−14˜v+36a˜v+12A10˜v+6A10˜u˜v−7˜v2+30a˜v2),d˜vdt=˜u+˜u˜v+13(˜v+1)(6˜u3+9˜u2−26˜u˜v−26˜u2˜v+28˜v2+18b˜v2+6B10˜v2+39˜u˜v2+3B10˜u˜v2−16˜v3+15b˜v3), | (24) |
and the equilibrium point (2,1) of model (3) becomes the origin of system (24) correspondingly. Under the transformation
˜z=˜u+i˜v,˜w=˜u−i˜v,T=it,i=√−1, | (25) |
system (24) becomes its concomitant complex system, i.e.,
{d˜zdT=˜z+Z2(˜z,˜w)+Z3(˜z,˜w)+Z4(˜z,˜w),d˜wdT=−˜w−W2(˜z,˜w)−W3(˜z,˜w)−W4(˜z,˜w), | (26) |
in which
Z2(˜z,˜w)=124i(−74+33i+72a+24A10+36ib+12iB10)˜w2+16(37+14i−36ia−12iA10+18b+6B10)˜w˜z+124i(18+43i+72a+24A10+36ib+12iB10)˜z2,Z3(˜z,˜w)=18i[−12−i+(6+10i)a+4A10−11b−(2−i)B10)˜w3+124[13+26i+(90−18i)a−12iA10+99ib+(3+18i)B10]˜w2˜z−124[−49+12i+(90+18i)a+12iA10+99ib−(3−18i)B10]˜w˜z2+124[−17+22i+(30+18i)a+12iA10+33ib−(3−6i)B10]˜z3,Z4(˜z,˜w)=148(14−33i−15a+3iA10+15b−3iB10)˜w4+124(29+45i+15a−30b+3iB10)˜w3˜z+124(−74−3iA10+45b)˜w2˜z2−124(−35+45i+15a+30b+3iB10)˜w˜z3+116(2+11i+5a+iA10+5b+iB10)˜z4,W2(˜z,˜w)=−124i(18−43i+72a+24A10−36ib−12iB10)˜w2+16(37−14i+36ia+12iA10+18b+6B10)˜w˜z−124i(−74−33i+72a+24A10−36ib−12iB10)˜z2,W3(˜z,˜w)=−124i[22−17i+(18+30i)a+12A10+33b+(6−3i)B10]˜w3+124[49+12i−(90−18i)a+12iA10+99ib+(3+18i)B10]˜w2˜z+124[13−26i+(90+18i)a+12iA10−99ib+(3−18i)B10]˜w˜z2−18[−1−12i+(10+6i)a+4iA10−11ib+(1−2i)B10]˜z3,W4(˜z,˜w)=116(2−11i+5a−iA10+5b−iB10)˜w4−124(−35−45i+15a+30b−3iB10)˜w3˜z+124(−74+3iA10+45b)˜w2˜z2+124(29−45i+15a−30b−3iB10)˜w˜z3−148(−14−33i+15a+3iA10−15b−3iB10)˜z4. |
Obviously, system (26) belongs to the class of system (10). Next, we begin to compute the focal values of the origin of system (24) (namely the focal values of the equilibrium point (2,1) of model (3)). According to the recursion formulas of Lemma 2.2, we have the following result. For convenience, we note that B10=λ1−3b−176.
Theorem 3.4. The first five singular point values at the origin of system (24) are as follows:
˜μ1=−112i(312a+110A10+45b+8λ1+144aλ1+48A10λ1);
˜μ2=14860i(h1+λ1h2+108λ21h3);
˜μ3=127993600i(−15240960λ41h4−1440λ31h5+12λ21h6+λ1h7+h8);
˜μ4=−1604661760000i(−10484051097600λ61h9−207360λ51h10−41472λ41h11 +192λ31h12+4λ21h13−λ1h14+h15);
˜μ5=−17836416409600000i(6216291333365760000λ81h16+149299200λ71h17+1244160 ×λ61h18−6912λ51h19−1728λ41h20+48λ31h21+4λ21h22+λ1h23+h24);
in which hi,(i∈{1,2,⋯24}) are the expressions about A10,a.
Considering the relation between model (3) and system (24) and system (26), from Theorem 3.4 and Lemma 2.3, the following theorem is visible.
Theorem 3.5. The first five focal values of the equilibrium point (2,1) of model (3) (or the first five focal values at the origin of system (24)) are as follows:
˜v3=112π(312a+110A10+45b+8λ1+144aλ1+48A10λ1);
˜v5=−14860π(h1+λ1h2+108λ21h3);
˜v7=−127993600π(−15240960λ41h4−1440λ31h5+12λ21h6+λ1h7+h8);
˜v9=1604661760000π(−10484051097600λ61h9−207360λ51h10−41472λ41h11 +192λ31h12+4λ21h13−λ1h14+h15);
˜v11=17836416409600000π(6216291333365760000λ81h16+149299200λ71h17+1244160 ×λ61h18−6912λ51h19−1728λ41h20+48λ31h21+4λ21h22+λ1h23+h24).
From Theorem 3.5, we have
Theorem 3.6. The equilibrium point (2,1) of model (3) can be a 5th-order fine focus at most.
Proof. According to Definition 2.1, we need to prove that there exits a group of real values about λ1,A10,a,b such that ˜v3=˜v5=˜v7=˜v9=0,˜v11≠0.
At first, we prove that ˜v3=˜v5=˜v7=˜v9=0 have real number solutions.
Let ˜v3=0, we have
b=−245(156a+55A10+4λ1+72aλ1+24A10λ1). | (27) |
By using computer soft Mathematica 6.0 to compute, ˜v3=˜v5=˜v7=˜v9=0 hold if and only if Eq (27) holds and
{˜r57=Resultant[˜v5,˜v7,A10]=0,˜r59=Resultant[˜v5,˜v9,A10]=0. | (28) |
While Eq (28) hold if and only if
˜r579=Resultant[˜r57,˜r59,a]=0. | (29) |
By computing, we obtain
˜r579=Resultant[r57,r59,a]=(3107+3210λ1+720λ21)2(13964+10293λ1+1800λ21)2˜g(λ1), |
in which ˜g(λ) is a 242 degrees function on λ1. It can be seen that Eq (29) has some real solutions such as λ1=1240(−535±9√465) et al. Hence, ˜v3=˜v5=˜v7=˜v9=0 have real number solutions.
In fact, we can find 15 groups of real number solutions such that ˜v3=˜v5=˜v7=˜v9=0, namely
1)A10≈−4643.3018425501515263,a≈1550.8973397671423233,
b≈1036.3362485482110284,λ1≈−43.0579290316570603;
2)A10≈0.9959258074047267,a≈−0.0496867911910842,
b≈−215.6922502839121132,λ1≈197.5784671350668527;
3)A10≈17.6221151647949041,a≈−5.8674840306887030,
b≈−5.1961268584644113,λ1≈14.0933015765297504;
4)A10≈−22.4133844512194665,a≈7.6232290779662400,
b≈−1.4651057037980100,λ1≈5.1151041647768187;
5)A10≈6.0154299290528854,a≈−3.1637820182235810,
b≈−7.3080323729222197,λ1≈−4.1189107630353050;
6)A10≈−5.3540843597677715,a≈2.210452369910683,
b≈1.9959489233153501,λ1≈−2.748983659910321;
7)A10≈−3.2879140223160507,a≈1.636028779039390,
b≈1.4395307705732038,λ1≈−2.489840572873740;
8)A10≈−4.1024413004886361,a≈2.121698633718532,
b≈−1.3456036840775022,λ1≈−1.287647593274435;
9)A10≈−2.7095465588712933,a≈1.607999835283884,
b≈0.1502206861914896,λ1≈−1.921623601914863;
10)A10≈0.0859310460513803,a≈0.219273746634790,
b≈1.1721730119425250,λ1≈−2.988862274797442;
11)A10≈0.3976086548746663,a≈0.119165192985334,
b≈1.0985939724503711,λ1≈−2.946168239121486;
12)A10≈0.3522522540516692,a≈0.044767143636023,
b≈0.9813497899428434,λ1≈−3.089686791404887;
13)A10≈0.5942698666771841,a≈0.294891555364813,
b≈1.1887051809847815,λ1≈−2.669570213107159;
14)A10≈1.0727163914454935,a≈0.107785966220006,
b≈1.1631832585231690,λ1≈−2.719197614164823;
15)A10≈1.5505080546244919,a≈−1.101693440557772,
b≈−2.0337942792540713,λ1≈−3.472776931203230. .
Next, we prove ˜v11≠0 if ˜v3=˜v5=˜v7=˜v9=0
Let
˜r1=Resultant[˜v5,˜v11,A10],˜r2=Resultant[˜v7,˜v11,A10],˜r3=Resultant[˜v9,˜v11,A10], and ˜r12=Resultant[˜r1,˜r2,a],˜r13=Resultant[˜r1,˜r3,a].
If ˜v3=˜v5=˜v7=˜v9=0, then ˜r123=Resultant[˜r12,˜r13,λ1]=0. By computing, we obtain: ˜r123=Resultant[˜r12,˜r13,λ1]=54655887129325749812881285470922⋯≠0.
Hence, ˜v11≠0 if ˜v3=˜v5=˜v7=˜v9=0, then the equilibrium point (2,1) of model (3) can be a 5th-order fine focus at most. Proof end.
After finding focal values of two positive equilibrium points of model (3), we will consider the limit cycle bifurcation near (1,1) and (2,1) of perturbed model (3).
Lemma 4.1. Let J1 be the Jacobin of the function group (v3,v5,v7,v9) with respect to the variables (a,λ,A10,b), if the equilibrium (1,1) of model (3) is a 5th-order fine focus, then J1≠0.
Proof. Suppose that J1=0, next we deduce a contradictory result. The Jacobin of the function group (v3,v5,v7,v9) with respect to the variables (a,λ,A10,b) has the following form
J1=|∂v3∂a∂v3∂A10∂v3∂λ∂v3∂b∂v5∂a∂v5∂A10∂v5∂λ∂v5∂b∂v7∂a∂v7∂A10∂v7∂λ∂v7∂b∂v9∂a∂v9∂A10∂v9∂λ∂v9∂b|. |
Obviously, J1 is a function with respect to the variables (a,λ,A10,b). Because the equilibrium point (1,1) of model (3) is a 5th-order fine focus, then v3=v5=v7=v9=0. Suppose that J1=0, then the resultant of J1,vi,(i∈{3,5,7,9}) with respect to the variable b will become 0.
Let R1=Resultant[J1,v3,b],R2=Resultant[J1,v5,b],R3=Resultant[J1,v7,b], R4=Resultant[J1,v9,b], then Ri=0,i∈{1,2,3,4}.
While Ri=0,i∈{1,2,3,4} will deduce that R12=Resultant[R1,R2,A10]=0, R13=Resultant[R1,R3,A10]=0, R14=Resultant[R1,R4,A10]=0. Similarly, R12=R13=R14=0 deduce that R23=Resultant[R12,R13,a]=0 and R24=Resultant[R12,R14,a]=0. In the same way, R23=R24=0 deduce that R34=Resultant[R23,R24,λ]=0. In fact, with help of computer, we obtain that
R34=2423141861632306631749624885799862490490775362310417⋯≠0.
R34≠0 pushes J1≠0. Proof end.
Similarly, we can obtain the following lemma.
Lemma 4.2. Let J2 be the Jacobin of the function group (˜v3,˜v5,˜v7,˜v9) with respect to the variables (a,λ1,A10,b), if the equilibrium point (2,1) of model (3) is a 5th-order fine focus, then J2≠0.
Theorem 4.1. Suppose that (1,1) is a 5-th order fine focus of system (3), then by small perturbations of the parameter group (a,λ,A10,b), the point (1,1) of perturbed model (3) can bifurcate at least 5 small amplitude limit cycles.
Proof. From lemma 4.1, J1≠0, while (1,1) is a 5-th order fine focus of system (3), according to the theory of reference [14], the conclusion of Theorem 4.1 holds. Proof end.
Similarly, we have the following theorem.
Theorem 4.2. Suppose that (2,1) is a 5-th fine focus of system (3), then by small perturbations of the parameter group (a,λ1,A10,b), the point (2,1) of perturbed model (3) can bifurcate at least 5 small amplitude limit cycles.
Next we will give a case that (1,1) of the model (3) can bifurcate 5 limit cycles of which 3 limit cycles are stable.
Obviously, vi,(i=3,5,7,9,11) are functions about (a,λ,A10,b). The course of Theorem 3.3 shows there exists a group solutions (a,λ,A10,b) = (˜a,˜λ,~A10,˜b) such that vi=0,i∈{3,5,7,9},v11≠0. Hence, we may as well let
v3=ϵ1,v5=ϵ2,v7=ϵ3,v9=ϵ4, | (30) |
in which ϵi,i∈{1,2,3,4} are a group of arbitrary given real small parameters.
According to existence theorem of implicit function and the result of Lemma 4.1, Eq (30) has a group of solutions as follows:
a=a(ϵ1,ϵ2,ϵ3,ϵ4),λ=λ(ϵ1,ϵ2,ϵ3,ϵ4),A10=A10(ϵ1,ϵ2,ϵ3,ϵ4),b=b(ϵ1,ϵ2,ϵ3,ϵ4). | (31) |
From (30) and (31), clearly the following theorem holds.
Theorem 4.3. Suppose that (a,λ,A10,b) disturb by way of (31), then v3=ϵ1,v5=ϵ2,v7=ϵ3,v9=ϵ4, in which ϵi,i∈{1,2,3,4} are a group of arbitrary given real small parameters.
Next, we let ϵi,i∈{1,2,3,4} be some special values, we have the following theorem.
Theorem 4.4. Suppose that the coefficients of model (3) disturb via v3=21076c5πϵ8+o(ϵ9),v5=−7645c5πϵ6+o(ϵ7),v7=1023c5πϵ4+o(ϵ5), v9=−55c5πϵ2+o(ϵ3),c5=v11, then the point (1, 1) of model (3) can bifurcate 5 small limit cycles which are near to circles (x−1)2+(y−1)2=k2ϵ2,k=1,2,3,4,5 in which 3 limit cycles can be stable.
Proof. Suppose that v3=21076c5πϵ8+o(ϵ9),v5=−7645c5πϵ6+o(ϵ7),v7=1023c5πϵ4+o(ϵ5),v9=−55c5πϵ2+o(ϵ3),c5=v11, then we have
v1(2π,ϵ,δ)=e2πδ=1+c0πϵ10+o(ϵ11),v3(2π,ϵ,δ)=c1πϵ8+o(ϵ9),v5(2π,ϵ,δ)=c2πϵ6+o(ϵ7),v7(2π,ϵ,δ)=c3πϵ4+o(ϵ5),v9(2π,ϵ,δ)=c4πϵ2+o(ϵ3),v11(2π,ϵ,δ)=c5+o(ϵ), |
in which
c5=v11|ϵ=0, and c0=−14400c5,c1=21076c5,c2=−7645c5,c3=1023c5,c4=−55c5.
At this time, according to (6), Poincarˊe succession function for the point (1, 1) of model (3) is as follows:
d(ϵh)=r(2π,ϵh)−ϵh=(v1(2π,ϵ,δ)−1)ϵh+v2(2π,ϵ,δ)(ϵh)2+v3(2π,ϵ,δ)(ϵh)3+…+v11(2π,ϵ,δ)(ϵh)11+…=πϵ11h[g(h)+ϵhG(h,ϵ)], |
in which
g(h)=c0+c1h2+c2h4+c3h6+c4h8+j0h10=c6(h2−1)(h2−4)(h2−9)(h2−16)(h2−25)=−14400c5+21076c5h2−7645c5h4+1023c5h6−55c5h8+c5h10, |
and G(h,ϵ) is analytic at (0,0).
Obviously, g(h)=0 has 5 simple positive zero points 1,2,3,4,5. From implicit function theorem, the number of positive zero points of equation d(ϵh)=0 is equal to one of g(h)=0, and these positive zero points are close to 1,2,3,4,5 when 0<|ϵ|≪1. The above analysis shows there are 5 small limit cycles in a small enough neighborhood of (1,1) of model (3), which are near to circles (x−1)2+(y−1)2=k2ϵ2,k=1,2,3,4,5.
Obviously, if v11<0, then the point (1, 1) of model (3) can bifurcate 3 stable cycles which are near circles (x−1)2+(y−1)2=k2ϵ2,k=1,3,5. While by analyzing the 13 groups of solutions showed in the proof course of Theorem 3.3, this kind of solutions exist such as the second group of solution. Hence, the result of Theorem 4.4 holds. Proof end.
Remark: We can also find the solution such that v3=v5=v7=v9=0,v11>0 among the 13 groups of solutions showed in the proof course of Theorem 3.3. Hence, the point (1, 1) of model (3) can also bifurcate 2 stable cycles which are near circles (x−1)2+(y−1)2=k2ϵ2,k=2,4.
Similarly, we can obtain the following theorem.
Theorem 4.5. The point (2, 1) of model (3) can bifurcate 3 stable limit cycles under certain condition.
The work of this paper focuses on investigating the limit cycle bifurcation of a class of the quartic Kolmogorov model, which is an interesting and significant ecological model both in theory and applications. We used the singular values method to compute focal value. First, we give the relation between focal value and singular point value at the origin, which is necessary for us to investigate bifurcations of limit cycles. Sceond, we give the singular point values' recursive formulas and respectively compute the focal values of the two positive equilibrium points of model (3) and obtain the condition that they can be two 5th-order fine focuses. Next, we discuss the bifurcation of limit cycle of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles. Then, we show a case that each positive equilibrium point can bifurcate 3 stable limit cycles at most by making use of algebraic and symbolic proof.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of our paper. This research is partially supported by the Research Fund of Hunan provincial education department (22A719) and the National Natural Science Foundation of China (12061016).
The authors declare that they have no competing interest.
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