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On the dynamics of a non-linear Duopoly game model
International Journal of Non-Linear Mechanics 57 (2013) 31–38
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On the dynamics of a non-linear Duopoly game model Isabella Torcicollo n Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, Naples, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 24 April 2013 Received in revised form 14 June 2013 Accepted 14 June 2013 Available online 26 June 2013
The Cournot duopoly game modeled by Kopel, with adaptive expectations, is generalized by introducing the self-diffusion and cross-diffusion terms. General properties, such as boundedness and uniqueness, are obtained. Non-linear stability results are reached by the analysis of the stability of a ODE system. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Continuous Cournot–Kopel model Non-linear duopoly game Non-linear stability Adaptive expectations Self-diffusion Cross-diffusion
1. Introduction Oligopoly is the case where the market is controlled by a few number of firms producing similar products. In this paper we will restrict ourselves to the case of two firms, which is called duopoly. The situation in a duopoly is complex, since both firms have to take into account not only the behaviours of the consumers, but actions and reactions of the competitor. The first well-known model which gives a mathematical description of competition in a duopoly market dates back to the French economist Cournot (1838). In this paper, our starting point is the general case of the Cournot duopoly problem, which was modeled by Kopel (see [1–3]); precisely, we will examine the continuous time-scale counterpart of the above-mentioned non-linear Cournot–Kopel duopoly game. Assuming continuous time scales, denoting by u and v the outputs of the two firms X and Y, respectively, a non-linear dynamic system for the evolution of u and v is obtained: (
∂t u ¼ −α1 u þ α1 μ1 vð1−vÞ ∂t v ¼ −α2 v þ α2 μ2 uð1−uÞ
ð1Þ
where, in general, μi and αi with (i ¼1,2) are positive model parameters, ∂t denotes the derivative with respect to time, u : ðx; tÞ∈Ω Rþ -uðx; tÞ∈R, v : ðx; tÞ∈Ω Rþ -vðx; tÞ∈R, Ω being a bounded domain in R3 with smooth boundary ∂Ω.
To be realistic, any dynamical economic model should take into account both the time evolution and the spatial dependence of the characteristic variables. In this paper, in order to generalize the model (1), we take into account the spatial dependence, by introducing the self-diffusion and cross-diffusion terms, by considering the following equations: ( ∂t u ¼ −α1 u þ α1 μ1 vð1−vÞ þ γ 11 Δu þ γ 12 Δv ð2Þ ∂t v ¼ −α2 v þ α2 μ2 uð1−uÞ þ γ 21 Δu þ γ 22 Δv where Δ denotes the Laplacian operator, γ ij ¼ constant for ði; j ¼ 1; 2Þ and 2
∑ γ ij ξi ξj ≥kjξj2
i;j ¼ 1
k 40; ξ ¼ ðξ1 ; ξ2 Þ:
ð3Þ
Already in [4,5] the authors introduce a new class of economic dynamical models, which are called “morphogenetic systems”, which are constructed in order to generalize classical Goodwin's model of business cycle. Non-linear reaction–diffusion equations and systems play an important role in the modeling and study of many phenomena (see, for instance, dynamics of competing species, chemical aggression and convection problems in porous media, non-linear heat conduction, semiconductor devices, in [6–26]). To (2) we append the initial data uðx; 0Þ ¼ u0 ðxÞ;
vðx; 0Þ ¼ v0 ðxÞ for x∈Ω
ð4Þ
and the following boundary conditions: Dirichlet boundary conditions n
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0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.011
u¼u
and v ¼ v on ∂Ω Rþ ;
ð5Þ
32
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
where u, v will be chosen among the constant steady-states solutions of (2), or Robin boundary data ( βu þ ð1−βÞ∇u n ¼ uβ on ∂Ω Rþ ð6Þ βv þ ð1−βÞ∇v n ¼ uβ; where 0 o β o1. The aim of the present paper is to analyse the non-linear L2stability of the constant steady states of (2) by following the procedure introduced by Rionero in [6] and used in [7–11]. Precisely, our aim is to link the stability (instability) of the generic equilibrium to (2) to the stability (instability) of the null solution to a linear system of ordinary differential equations associated to (2). The plan of the paper is the following. Section 2 is dedicated to the derivation of the mathematical model at hand, while, in Section 3, we consider the perturbation equations associated to the generalized model. Section 4 concerns boundedness and uniqueness of perturbations, while Section 5 is devoted to recall some results which allow to obtain conditions guaranteeing the stability (Section 6) of a critical point. Section 7 concerns a simplified case in which the obtained results are exemplified and commented. The paper ends with an Appendix in which the proof of Lemmas 1–2 and of Theorems 3 and 4 is given.
2. The mathematical model Two firms X and Y produce goods which are perfect substitutes and offer them at discrete time periods t ¼0,1,2,…on a common market. In order to determine the quantity of period t+1, the firms X and Y form expectations on the quantity of the other firm yetþ1 and xetþ1 , which might, for example, depend on their own quantity and the quantity of the other firm, both produced in the previous period. If we denote by xt and yt the output of firm X and Y at time t, respectively, the optimization problem through which the firms determine their quantities xtþ1 and ytþ1 are represented by arg maxx Π X ðxt ; yetþ1 Þ and arg maxy Π Y ðxetþ1 ; yt Þ where Π X ð; Þ and Π Y ð; Þ denote the profit of firm X and Y respectively. If we assume that these optimization problems have unique solutions, then xtþ1 ¼ r X ðyetþ1 Þ
ð7Þ
ytþ1 ¼ r Y ðxetþ1 Þ
ð8Þ
where r X ; r Y are called Best Replies (or reaction functions). We will assume that the firms revise their beliefs according to the adaptive expectations rules xetþ1 ¼ xet þ α1 ðxt −xet Þ
ð9Þ
yetþ1 ¼ yet þ α2 ðyt −yet Þ
ð10Þ
where αi 40 are referred to as the adjustment coefficients and we will assume the following well-known type of reaction functions: r x ðyÞ ¼ μ1 yð1−yÞ
ð11Þ
r y ðxÞ ¼ μ2 xð1−xÞ
ð12Þ
where μi ði ¼ 1; 2Þ measure the intensity of the effect that one firm's actions has on the other firm. Many specifications can be found in the literature, but an analytical expression for the Best Replies is complicated. Microeconomic foundations of (11)–(12) can be found in [1]. Then, from (7) to (12) and, in order to simplify the notation, replacing xte, yte with xt, yt, the Cournot–Kopel model is obtained: xtþ1 ¼ ð1−α1 Þxt þ α1 μ1 yt ð1−yt Þ
ð13Þ
ytþ1 ¼ ð1−α2 Þyt þ α2 μ2 xt ð1−xt Þ:
ð14Þ
Firms do not change their productions according to the computed
optimal productions, but they prefer to choose a weighted average between the previous production and the computed one, with weights 1−αi and αi , respectively. The meaning of model implies that the economically relevant case is αi ≤1 ði ¼ 1; 2Þ. The model (13)–(14) is a two-dimensional map, described in [1,27]. Bifurcations of map have been studied in [28,29]; fixed points, their stability and stable cycles have been studied intensively in the literature [27,30,31], in particular under the assumption μ1 ¼ μ2 ¼ μ;
α1 ¼ α2 ¼ α;
ð15Þ
that is the players are homogeneous with regard to the Best Replies and with regard to their expectations, respectively. These two assumptions imply that the two competitors behave identically. In this paper, starting from the continuous time-scale counterpart of the above-mentioned non-linear Cournot duopoly game (1), we will examine the generalized model (2), where we took into account the spatial dependence, by introducing the selfdiffusion and cross-diffusion terms.
3. Perturbation equations associated to the generalized model We denote by ðu; vÞ the generic equilibrium point of (1). Besides the trivial equilibrium ðu ¼ 0; v ¼ 0Þ, system (1) admits other non-trivial constant steady states, which can be found by solving Y 3 þ pY þ q ¼ 0
ð16Þ
where we have set 8 μ μ > Y ¼ μ1 μ2 ð1−uÞ− 1 2 > > 3 > > > < μ21 μ22 2 þ μ1 μ2 p¼− 3 > > > 3 3 3 2 > > > q ¼ −2μ1 μ2 þ μ1 μ2 −μ2 μ : : 1 2 27 3 The solutions to (16) are 8 Y1 ¼ Yþ þ Y− > > > > Yþ þ Y− Y þ −Y − pffiffiffi < þi 3 Y2 ¼ − 2 2 > > p ffiffiffi Y þ Y − Y þ −Y − > > : Y3 ¼ − þ −i 3 2 2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q q2 p3 Yþ ¼ − þ þ ; 2 4 27
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q q2 p3 þ ; Y− ¼ − − 2 4 27
ð17Þ
ð18Þ
ð19Þ
hence ðu; vÞ ¼ ð 23 −Y i =μ1 μ2 ; 29 þ Y i =μ1 −Y 2i =μ21 μ2 Þ, with (i¼1,2,3). Precisely, if Δ ¼ q2 =4 þ p3 =27 is non-positive (positive), which implies −μ21 μ22 þ 4μ1 μ22 −18μ1 μ2 þ 4μ21 μ2 þ 27 non-positive (positive), then three (only one) real solutions are created. Let us set U ¼ u−u;
V ¼ v−v;
ð20Þ
then, the perturbation equations associated to (2) are given by ( ∂t U ¼ −α1 U þ α1 μ1 ð1−2vÞV þ γ 11 ΔU þ γ 12 ΔV−α1 μ1 V 2 ð21Þ ∂t V ¼ α2 μ2 ð1−2uÞU−α2 V þ γ 21 ΔU þ γ 22 ΔV−α2 μ2 U 2 : Denoting by 8 2 > < f ðU; V Þ ¼ −α1 μ1 V ; > :
gðU; VÞ ¼ −α2 μ2 U 2
a11 ¼ −α1 ;
a22 ¼ −α2
a21 ¼ α2 μ2 ð1−2uÞ;
a12 ¼ α1 μ1 ð1−2vÞ
ð22Þ
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
the system (21) becomes ( ∂t U ¼ a11 U þ a12 V þ γ 11 ΔU þ γ 12 ΔV þ f ðU; V Þ ∂t V ¼ a21 U þ a22 V þ γ 21 ΔU þ γ 22 ΔV þ gðU; VÞ: A system of this form has been considered in [6–11]. To (23) we append the initial data ( Uðx; 0Þ ¼ U 0 ðxÞ x∈Ω Vðx; 0Þ ¼ V 0 ðxÞ
ð23Þ
on ∂Ω Rþ ;
ð24Þ
ð26Þ
þ ð∇ϕÞ ∈LðΩÞ; ϕ ¼ 0 on ∂Ωg;
then U, V are bounded.
1d ð‖U‖2 þ ‖V‖2 Þ 2 dt
Ω
ρ ¼ ρðΩÞ 40 the positive constant, appearing in the Poincaré inequality ‖∇ϕ‖2 ≥ρ‖ϕ‖2
ð27Þ
is the lowest eigenvalue ρ of the spectral problem Δϕ þ ρϕ ¼ 0
β ‖ϕ‖2∂Ω ≥α‖ϕ‖2 ; 1−β
ð29Þ
is the lowest eigenvalue α of the spectral problem Δϕ þ αϕ ¼ 0
Ω
ð35Þ
Ω
Ω
Ω
Ω
β β ‖U‖2∂Ω −γ 11 ‖∇U‖2 −γ 22 ‖V‖2∂Ω −γ 22 ‖∇V‖2 1−β 1−β Z Z β −ðγ 12 þ γ 21 Þ UV dΣ−ðγ 12 þ γ 21 Þ ∇U∇V dΩ 1−β ∂Ω Ω β 2 2 2 ð‖U‖∂Ω þ ‖V ‖2∂Ω Þ ≤−λmin ‖∇U‖ þ ‖∇V‖ þ ð1−βÞ ¼ −γ 11
≤−λmin αð‖U‖2 þ ‖V‖2 Þ:
ð36Þ
Then (35) becomes ð28Þ
in H 10 ðΩÞ; α ¼ αðΩ; βÞ 4 0 the positive constant, appearing in the following inequality:
ð30Þ
1d ð‖U‖2 þ ‖V‖2 Þ ≤ −α1 ‖U‖2 −α2 ‖V‖2 −λmin αð‖U‖2 þ ‖V‖2 Þ 2 dt Z Z Z þ ða12 þ a21 Þ
in H ðΩ; βÞ. 2
We study the stability of U ¼ V ¼ 0 in the L ðΩÞnorm with respect to the perturbations (U, V) belonging, ∀t∈Rþ , to ½H 10 ðΩÞ2 in the case (25) and to ½H 1 ðΩ; βÞ2 in the case (26).
Ω
UVdΩ−α2 μ2
Ω
U 2 V dΩ−α1 μ1
UV 2 dΩ:
Let us denote by Ωþ ¼ Ω Rþ and C 21 ðΩþ Þ the set of the functions belonging to CðΩþ Þ with their first time derivative and first and second spatial derivatives. In order to prove the boundedness of the perturbations and a uniqueness theorem for (21), let us remark that, from (3), it follows that ð31Þ
ð38Þ
Ω
is satisfied for 8 < λ oα1 þ λmin α
2 : ðλ−αλmin Þ2 −ðα1 þ α2 Þðλ−αλmin Þ þ α1 α2 − a12 þ a21 ≥0 2 which is verified when (34) holds and for λ o λ1 with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α þ α 2 a þ a 2 α1 þ α2 1 2 12 21 − −α1 α2 þ λ1 ¼ αλmin þ 40 2 2 2
4. Boundedness and uniqueness
ð37Þ
Ω
The relation Z f−ðα1 þ λmin αÞU 2 þ ða12 þ a21 ÞUV−ðα2 þ λmin αÞV 2 g dΩ Ω Z ≤−λ ðU 2 þ V 2 Þ dΩ
1
γ 11 X 2 þ ðγ 12 þ γ 21 ÞXY þ γ 22 Y 2 ≥kðX 2 þ Y 2 Þ
ð34Þ
By using (26), the divergence theorem and (31), then the Poincaré inequality (29) leads to Z Z Z Z γ 11 UΔU þ γ 22 V ΔV dΩ þ γ 21 V ΔU dΩ þ γ 12 UΔV dΩ
2
ϕ∈H 1 ðΩ; βÞ-fϕ2 þ ð∇ϕÞ2 ∈LðΩÞ; βϕ þ ð1−βÞ∇ϕ n ¼ 0 on ∂Ωg;
‖∇ϕ‖2 þ
with k1 ¼ α1 μ1 þ α2 μ2 and λ given by (39), and a þ a 2 12 21 −α1 α2 ≤0 2
Ω
H1 ðΩ; βÞ the functional space such that
ð33Þ
Z ¼ −α1 ‖U‖2 −α2 ‖V‖2 þ a12 UV dΩ Ω Z Z Z Z þa21 UV dΩ þ γ 22 VΔV dΩ þ γ 11 UΔU þ γ 12 UΔV Ω Ω Ω ZΩ Z Z þγ 21 VΔU dΩ−α2 μ2 U 2 V dΩ−α1 μ1 UV 2 dΩ:
∥ ∥ the L2 ðΩÞnorm; ‖ ‖∂Ω the L2 ð∂ΩÞnorm; 〈; 〉 the scalar product in L2 ðΩÞ; H 10 ðΩÞ the functional space such that ϕ∈H 10 ðΩÞ-fϕ2
λ k1
ðjU 0 j2 þ jV 0 j2 Þ1=2 o
Proof. Multiplying (21)1 by U and (21)2 by V, summing and integrating on Ω and by using (22)3, it follows that
where 0 o β o1: In the following, we denote by:
ð32Þ
Theorem 1. Let (U, V) with fU; V∈C 21 ðΩþ g be a solution to (21) under conditions (24), (26). If
ð25Þ
or the Robin boundary conditions 8 ð1−βÞ > > > < U þ β ∇U n ¼ 0 on ∂Ω Rþ ð1−βÞ > > > : V þ β ∇V n ¼ 0;
∀X; Y and k 4 0. We can choose k ¼ λmin with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ þ γ 2 γ þ γ 2 γ 11 þ γ 22 11 22 21 − − γ 11 γ 22 − 12 λmin ¼ 2 2 2
the minimum eigenvalue of the symmetric part of the matrix ðγ ij Þ.
with U 0 ðxÞ ¼ u0 ðxÞ−u and V 0 ðxÞ ¼ v0 ðxÞ−v. Choosing the constants u, v in the boundary conditions (5), (6) equal to the steady state under consideration, we obtain the Dirichlet boundary conditions U¼V ¼0
33
ð39Þ
ð40Þ
In addition, from Z Z −α2 μ2 U 2 V dΩ−α1 μ1 UV 2 dΩ ≤ðα1 μ1 þ α2 μ2 Þð‖U‖2 þ ‖V‖2 Þ3=2 Ω
Ω
Eq. (31) leads to 1d ð‖U‖2 þ ‖V‖2 Þ ≤ −λð‖U‖2 þ ‖V‖2 Þ þ k1 ð‖U‖2 þ ‖V‖2 Þ3=2 2 dt
ð41Þ
34
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
with k1 ¼ ðα1 μ1 þ α2 μ2 Þ. From
d k1 ð‖U‖2 þ ‖V‖2 Þ ≤ −2λð‖U‖2 þ ‖V ‖2 Þ 1− ð‖U‖2 þ ‖V‖2 Þ1=2 dt λ
ð42Þ
by (33) it follows that ð‖U 0 ‖2 þ ‖V 0 ‖2 Þe−2λt
‖U‖2 þ ‖V‖2 ≤
k1 1− ð‖U 0 ‖2 þ ‖V 0 ‖2 Þ1=2 ð1−e−λt Þ λ
2 ;
t≥0:
ð43Þ
On taking into account (49)–(51), (48) becomes
Theorem 2. The problem (21) under conditions (24), (26) can admit only one solution in the class of perturbations (U, V) such that fU; V ∈C 21 ðΩþ Þ verifying (34)g. Proof. Let ðU 1 ; V 1 Þ and ðU 2 ; V 2 Þ be two solutions of (21), (24), (26). On setting W ¼ U 1 −U 2 ;
Z ¼ V 1 −V 2
ð44Þ
it follows that ( ∂t W ¼ −α1 W þ α1 μ1 ð1−2vÞZ þ γ 11 ΔW þ γ 12 ΔZ−α1 μ1 ZðV 1 þ V 2 Þ ∂t Z ¼ α2 μ2 ð1−2uÞW−α2 Z þ γ 21 ΔW þ γ 22 ΔZ−α2 μ2 WðU 1 þ U 2 Þ ð45Þ with (W, Z) satisfying the boundary conditions 8 dW > > ¼0 < βW þ ð1−βÞ dn on ∂Ω Rþ dZ > > : βZ þ ð1−βÞ ¼ 0; dn
ð46Þ
Ω
ð52Þ
and (31) leads to d ð‖W‖2 þ ‖Z‖2 Þ dt ≤ðA−2α1 þ MÞ‖W‖2 þ ðA−2α2 þ MÞ‖Z‖2 β ð‖W‖2∂Ω þ ‖Z‖2∂Ω Þ : −2λmin ‖∇W‖2 þ ‖∇Z‖2 þ ð1−βÞ
ð53Þ
By using the Poincaré inequality (29), it follows that ð54Þ
with k≥maxðA−2α1 þ M−2λmin α; A−2α2 þ M−2λmin αÞ
‖W‖2 þ ‖Z‖2 ≤ð‖W 0 ‖2 þ ‖Z 0 ‖2 Þekt ¼ 0; ð47Þ
ð55Þ
Ω
þγ 12
WΔZ dΩ þ γ 21
Z
−α2 μ2
Ω
ZΔW dΩ
Ω
WZðU 1 þ U 2 Þ dΩ WZðV 1 þ V 2 Þ dΩ:
ð48Þ
By virtue of (46), the divergence theorem leads to Z Z γ 11 WΔW þ γ 22 ZΔZ dΩ Ω
β ‖W‖2∂Ω −γ 11 ‖∇W‖2 ¼ −γ 11 1−β β ‖Z‖2∂Ω −γ 22 ‖∇Z‖2 : −γ 22 1−β From Schwartz inequality, it follows that Z ða12 þ a21 Þ ð‖W‖2 þ ‖Z‖2 Þ: ða12 þ a21 Þ WZ dΩ ≤ 2 Ω
ð49Þ
ð50Þ
Since (34) is verified, the solutions of (21) are bounded, and there exist positive constants Mi, Ni such that jU i j ≤M i , jV i j ≤N i , i ¼1,2. Denoting by M ¼ α2 μ2 ðM 1 þ M 2 Þ þ α1 μ1 ðN 1 þ N2 Þ it follows that Z Z −α2 μ2 WZðU 1 þ U 2 Þ dΩ−α1 μ1 WZðV 1 þ V 2 Þ dΩ Ω
ð57Þ □
5. Stability conditions
Ω
Z −α1 μ1
ð56Þ
Remark 1. Theorems 1 and 2, under conditions (24), (25), continue to value. In particular, the terms evaluated on ∂Ω are zero and the Poincaré constant α is replaced with ρ given by (27).
Z
Ω
ZðtÞ ¼ 0 ∀t≥0
and uniqueness is proved.
Ω
Z
t≥0
i.e. WðtÞ ¼ 0;
Z 1d ð‖W‖2 þ ‖Z‖2 Þ ¼ −α1 ‖W‖2 −α2 ‖Z‖2 þ ða12 þ a21 Þ WZ dΩ 2 dt Ω Z Z þγ 22 ZΔZ dΩ þ γ 11 WΔW
M ≤ ð‖W‖2 þ ‖Z‖2 Þ: 2
ðγ 11 j∇Wj2 þ ðγ 12 þ γ 21 Þ∇W∇Z þ γ 22 j∇Zj2 Þ dΩ Z 2β ðγ jWj2 þ ðγ 12 þ γ 21 ÞWZ þ γ 22 jZj2 Þ dΣ − ð1−βÞ ∂Ω 11
−2
d ð‖W‖2 þ ‖Z‖2 Þ ≤kð‖W‖2 þ ‖Z‖2 Þ dt
From (45) it follows the following relation:
Ω
d ð‖W‖2 þ ‖Z‖2 Þ ≤ðA−2α1 þ MÞ‖W‖2 þ ðA−2α2 þ MÞ‖Z‖2 dt Z
and hence
and the initial conditions ( Wðx; 0Þ ¼ 0 x∈Ω Zðx; 0Þ ¼ 0:
Ω
By virtue of (46), the divergence theorem and Schwartz inequality lead to Z Z γ 21 ZΔW dΩ þ γ 12 WΔZ dΩ Ω Ω Z Z β WZ dΣ−ðγ 12 þ γ 21 Þ ∇W∇Z dΩ ð51Þ ¼ −ðγ 12 þ γ 21 Þ 1−β ∂Ω Ω
In the absence of diffusion, the study of the linear stability of system (1) is based on the linearization of the following perturbation problem: ( ∂t U ¼ −α1 U þ α1 μ1 ð1−2vÞV−α1 μ1 V 2 ð58Þ ∂t V ¼ α2 μ2 ð1−2uÞU−α2 V−α2 μ2 U 2 : In the equilibrium point ðu; vÞ the eigenvalues are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 0 7 I 20 −4A0 λ¼ 2 with ( I 0 ¼ −α1 −α2 A0 ¼ α1 α2 ½1−μ1 μ2 ð1−2vÞð1−2uÞ
ð59Þ
hence, since I 0 o 0, the condition A0 4 0, that is a12 a21 −α1 α2 o 0;
ð60Þ
guarantees the stability of the zero solution (U¼V ¼0) of the linearized system. Remark 2. Eq. (34) can be written as ðða12 −a21 Þ=2Þ2 þ a12 a21 −α1 α2 ≤0 and it implies (60); in particular, if the system (21) were symmetrical, a12 −a21 ¼ 0, then (34) coincides with (60).
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
Following the procedure of Rionero [6–11], we associate to (23) the binary system of ODE's 8 dξ > > < ¼ b11 ξ þ b12 η dt ð61Þ dη > > : ¼ b21 ξ þ b22 η dt where b11 ¼ a11 −γ 11 ρ;
b12 ¼ a12 −γ 12 ρ;
b21 ¼ a21 −γ 21 ρ;
b22 ¼ a22 −γ 22 ρ:
Denoting by a and b two positive constants and setting U n ¼ aU;
V n ¼ bV ;
I ¼ b11 þ b22 ¼ −ðα1 þ γ 11 ρÞ−ðα2 þ γ 22 ρÞ o 0;
ð62Þ
μn ¼
a ; b
−1
f ¼ a−1 f ðaU; bVÞ;
g ¼ b gðaU; bVÞ
n
n
f ¼ γ 11 ðΔU þ ρUÞ;
g ¼ γ 22 ðΔV þ ρV Þ
f~ ¼ μn γ 21 ðΔU þ ρUÞ;
−1 g~ ¼ μn γ 12 ðΔV þ ρVÞ G ¼ g þ g n þ f~
n ~ F ¼ f þ f þ g;
pffiffiffiffiffiffiffiffiffiffiffiffiffi The eigenvalues of (61) are given by λ1;2 ¼ ðI 7 I 2 −4AÞ=2, then, since
35
the system (23) becomes ( ∂t U n ¼ b11 U n þ μn −1 b12 V n þ F
ð66Þ
ð67Þ
∂t V n ¼ μn b21 U n þ b22 V n þ G:
if
Remark 6. Let us remark that, the system (67) has the same eigenvalues as system (61).
A ¼ b11 b22 −b12 b21 ¼ λ1 λ2 ¼ A0 þ ρ½α1 γ 22 þ α2 γ 11 þ ργ 11 γ 22 þγ 12 α2 μ2 ð1−2uÞ þ γ 21 α1 μ1 ð1−2vÞ 4 0
The analysis is based on the functional introduced by Rionero [7–11]
ð63Þ
the asymptotic stability of equilibrium ðξ ¼ η ¼ 0Þ of (61) is guaranteed. Our aim is to obtain the stability (instability) of the economic equilibrium state of (23) via the stability (instability) of the zero solution of (61). Since I ¼ b11 þ b22 o 0, in the self-diffusion case, the condition for asymptotic stability is simply def
A ¼ A0 þ ρ½α1 γ 22 þ α2 γ 11 þ ργ 11 γ 22 ¼ A1 4 0;
ð64Þ
V ¼ 12½Að‖U n ‖2 þ ‖V n ‖2 Þ þ ‖b11 V n −μn b21 U n ‖2 þ‖μn−1 b12 V n −b21 U n ‖2
ð68Þ
whose derivative, along the solutions of the system (67), is given by dV ¼ AIð‖U n ‖2 þ ‖V n ‖2 Þ þ Φ dt
ð69Þ
where Φ ¼ Φ1 þ Φ2
then, A0 4 0 assures A1 4 0.
n ~ þ 〈a2 V n −a3 U n ; g n þ f~ 〉 Φ1 ¼ 〈a1 U n −a3 V n ; f þ g〉
Remark 3. Turing instability cannot occur if γ 12 ¼ γ 21 ¼ 0.
Φ2 ¼ 〈a1 U n −a3 V n ; f 〉 þ 〈a2 V n −a3 U n ; g〉
Remark 4. If A0 o0 then, in the absence of diffusion, the zero solution of (23) is unstable and the self-diffusion stability is driven according to (64).
a1 ¼ A þ μn2 b21 þ b22 ;
Remark 5. If an economic equilibrium state is stable in the selfdiffusion case, that is (64) is verified, then, it can become unstable, by introducing cross-diffusion in the equations. Precisely, each of the following conditions:
2
a3 ¼ μn b11 b21 þ
2
2
a2 ¼ A þ b11 þ
1 b12 b22 : μn
1 2 b ; μn2 12 ð70Þ
5.1. Self-diffusion case: γ 12 ¼ γ 21 ¼ 0 Under the assumption
(1) ð1−2uÞ 4 0 and ð1−2vÞγ 21 4 0 and γ 12 o −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ o0; (2) ð1−2uÞ 4 0 and ð1−2vÞγ 21 o0 and 0 o γ 12 o −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ; (3) ð1−2uÞ o 0 and ð1−2vÞγ 21 4 0 and γ 12 4 −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ 40; (4) ð1−2uÞ o 0 and ð1−2vÞγ 21 o 0 and 0 4 γ 12 4 −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ; (5) ð1−2uÞγ 12 o 0 and ð1−2vÞγ 21 o 0 is a necessary condition for ( A1 4 0 A ¼ A1 þ γ 12 α2 μ2 ð1−2uÞ þ γ 21 α1 μ1 ð1−2vÞ o 0:
γ 12 ¼ γ 21 ¼ 0
ð71Þ
Lemmas 1 and 2 hold for system (67), both for Dirichlet boundary conditions (25) and Robin boundary conditions (26). The proof of these lemmas can be deduced from the analogous results in [9]. However, for the sake of completeness, we give a sketch of the proof in the Appendix. Lemma 1. Let ( γ 11 ¼ γ 22 ¼ γ A1 4 0: ð65Þ
In order to study the non-linear stability of the zero solution ðU ¼ V ¼ 0Þ of the perturbation equations (21) with the Dirichlet boundary conditions (25) or the Robin boundary conditions (26) by a Lyapunov functional, we follow the procedure introduced by Rionero in [6] and used in [7–11]. Let us recall some results, proved in [6,10], useful in the following sections, rewritten by using the parameters introduced in the model. These results provide conditions guaranteeing the linear stability of the zero solution.
ð72Þ
Then Φ1 ≤0: Lemma 2. Let ( b11 b22 a12 a21 o 0 A1 4 0:
ð73Þ
ð74Þ
Then (73) holds. Remark 7. On taking into account that b11 b22 4 0, the condition (74)1 reduces itself to a12 a21 o 0, that is ð1−2vÞð1−2uÞ ≤ 0.
36
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
In addition, in the model at hand, the critical case ( b11 b22 a12 a21 4 0 a12 a21 o0 cannot occur, while the case ( b11 b22 a12 a21 4 0 a12 a21 40
the stability of the critical point of (21). In fact, on taking into account (62), Theorems 3 and 4 hold. ð75Þ
ð76Þ
is easily embraced in the proof of general stability condition (see Appendix, Proof of Theorem 4). 5.2. Cross-diffusion case: γ 12 ≠0 or γ 21 ≠0 Under the assumption γ 12 ≠0 or γ 21 ≠0 and when to system (67) we append (25), the following lemmas hold, proved by Flavin and Rionero (compare with Lemmas 1–3 in [10]). For the crossdiffusion case in the presence of Robin boundary conditions we refer to [8]. Lemma 3. Let inf ða1 ; a2 ; γ 11 ; γ 22 Þ 4 0 with γ 11 ¼ γ 11 −μn ða3 =a1 Þγ 21 and γ 22 ¼ γ 22 −μn−1 ða3 =a2 Þγ 12 : Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðγ 11 þ γ 22 Þa3 −ðμn−1 a1 γ 12 þ μn a2 γ 21 Þj ≤ 2 a1 a2 γ 11 γ 22 implies (73). Lemma 4. Let inf ða1 ; a2 Þ 40 8 2 > γ n11 ¼ γ 11 −b12 b22 ðA þ b22 Þ−1 γ 21 4 0 > > > > > b22 > > < γ n22 ¼ γ 22 − γ 21 4 0 b12 2 2 > > jðγ þ γ Þb > 11 22 12 b22 −½ðA þ b22 Þγ 12 þ b12 γ 21 j > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : ≤2jb12 j ðA þ b222 Þγ n11 γ n22 or 8 b11 > > γ n11 ¼ γ 11 − γ 21 4 0 > > > b21 > > > < γ n ¼ γ −b b ðA þ b2 Þγ 4 0 11 21 22 22 12 22 2 2 > > jðγ þ γ Þb b −½b γ > 11 22 11 21 21 12 ðA þ b11 Þγ 21 j > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : ≤jb21 j ðA þ b211 Þγ n11 γ n22 : Then (73) holds. Lemma 5. Let inf ða1 ; a2 Þ 40 and b11 b12 b21 b22 o 0. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ja 1 γ 12 þ μ 2 a 2 γ 21 jo 2μ a 1 a 2 γ 11 γ 22 where b b 1=2 12 22 μ¼ b11 b21 b b b 12 21 22 2 a 1 ¼ a1 ðμÞ ¼ A þ þ b22 4 0 b11 b b b 11 12 21 2 a 2 ¼ a2 ðμÞ ¼ A þ b11 þ 40 b22 implies (73). 6. Stability analysis 6.1. Self-diffusion case: γ 12 ¼ γ 21 ¼ 0 Under the assumption (71), Lemmas 1 and 2 hold. In this hypothesis, Lemmas 1 and 2 allow to obtain conditions guaranteeing
Theorem 3. Under the assumption (71) the linear asymptotic exponential L2 stability of the zero solution to (67) is guaranteed by (64). Proof. The proof can be found in [7] in the case (25) and in [9] in the case (26). However we give a sketch of the proof in the Appendix. □ Theorem 4. Let Theorem3 hold. Then Φ2 ≤Cð‖U n ‖2 þ ‖V n ‖2 Þ1þε
ε; C ¼ pos:const:
ð77Þ
2
guarantees the (local) L ðΩÞnonlinear asymptotic exponential stability of the zero solution of (67). Proof. The proof can be found in [7] in the case (25) and in [9] in the case (26). However, we give a sketch of the proof in the Appendix. □ Remark 8. In the problem at hand Φ2 ¼ 〈a1 U n −a3 V n ; f 〉 þ 〈a2 V n −a3 U n ; g〉 ≤C〈jU n j þ jV n j; jU n j2 þ jV n j2 〉
ð78Þ
where C ¼ maxðja1 j; ja2 j; ja3 jÞ maxðα1 μ1 b=a; α2 μ2 a=bÞ. By using, pffiffiffi jU n j þ jV n j ≤ 2ðjU n j2 þ jV n j2 Þ1=2 it follows that pffiffiffi Φ2 ≤C 2ð‖U n ‖2 þ ‖V n ‖2 Þ3=2 :
ð79Þ
6.2. Cross-diffusion case: γ 12 ≠0 or γ 21 ≠0 Under the assumption γ 12 ≠0
or
γ 21 ≠0;
ð80Þ
and if Lemmas 3–5 hold, Theorem 5 is true. The proof can be found in [10] (see Theorem 3 within). Theorem 5. Let j〈U; f 〉j þ j〈V ; g〉j ≤ kð‖U‖2 þ ‖V‖2 Þ1þε (with ε and k positive constants), and let one of Lemmas3–5 (ensuring Φ1 ≤ 0) hold. Then the stability of the zero solution of (61) implies the conditional non-linear asymptotic exponential stability of the zero solution of (67) with respect to the L2 ðΩÞnorm. Remark 9. Recalling that, in the considered model j〈U; f 〉j þ j〈V; g〉j ¼ 〈U; −α1 μ1 V 2 〉 þ 〈V; −α2 μ2 U 2 〉 ≤C 1 〈jUj þ jVj; jUj2 þ jVj2 〉
ð81Þ
where C 1 ¼ maxðα1 μ1 ; α2 μ2 Þ, since pffiffiffi jUj þ jVj ≤ 2ðjUj2 þ jVj2 Þ1=2 from (81) it follows that pffiffiffi j〈U; f 〉j þ j〈V; g〉j ≤C 1 2ð‖U‖2 þ ‖V‖2 Þ3=2 : 7. A simplified case For the sake of simplicity, in order to obtain a simple analytical solution, in this section we assume (15). Under these assumptions, for α≠0, besides E0 ¼ ð0; 0Þ, the following positive equilibria exist: μ−1 μ−1 ; for μ 4 1 E1 ¼ μ μ E2 ¼ ðx; yÞ E3 ¼ ðy; xÞ
for μ4 3 ð82Þ
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
where
pffiffiffiffi μþ1þ Ψ x¼ ; 2μ
pffiffiffiffi μ þ 1− Ψ y¼ ; 2μ
Ψ ¼ ðμ−3Þðμ þ 1Þ: We notice that in equilibrium E2, the beliefs are such that the firm 1 will dominate the market. In equilibrium E3 firm 2 dominates. Remark 10. We summarize that, by the analysis of eigenvalues pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ¼ −1 7 μ ð1−2uÞð1−2vÞ, it follows that E1 (stable node) is asymptotically stable for 1 o μ o 3 and it is unstable (saddle point) for μ 43; E2 ; E3 are asymptotically stable for μ 4 3, precisely, stable pffiffiffi pffiffiffi node for 3 o μ ≤ 1 þ 5 and spiral sink for μ 4 1 þ 5. In the self-diffusion case, the condition for asymptotic stability of the economic equilibrium ðu; vÞ is given by A1 ¼ α2 ½1−μ2 ð1−2vÞð1−2uÞ þ ρ½αγ 22 þ αγ 11 þ γ 11 γ 22 ρ 40: By introducing the cross-diffusion, (e.g. γ 12 ≠0 and γ 21 ¼ 0), the stable equilibrium ðu; vÞ becomes unstable if
37
On taking into account (72), it turns out that β Φ1 ¼ −Aγa1 ð‖u‖2∂Ω þ ‖∇u‖2 ‖∇u‖2 þ ‖∇v‖2 Þ−αð‖u‖2 þ ‖v‖2 1−β β ‖b1 1u−μn a12 v‖2 þ ‖∇ðb11 u−μn a12 vÞ‖2 −γ 1−β β β þ γ 22 〈u; v〉∂Ω þ a3 ðγ 11 þ γ 22 Þ〈∇u; ∇v〉 þa3 γ 11 1−β 1−β þγ 11 a1 α‖u‖2 þ γ 22 a2 α‖v‖2 −a3 ðγ 11 α þ γ 22 αÞ〈u; v〉:
one obtains a3 ¼ 0. Then by virtue of (85) it follows that β ‖u‖2∂Ω þ ‖∇u‖2 −α‖u‖2 Φ1 ¼ −γ 11 a1 1−β β ‖v‖2∂Ω þ ‖∇v‖2 −α‖v‖2 : −γ 22 a2 1−β Proof of Theorem 3. In the case (74), in view of
For instance, if we consider the stable equilibrium E2 ¼ ðx; yÞ, since ð1−2uÞ o 0, it follows that E2, in the presence of cross-diffusion, becomes unstable if
k1 k2 ð‖U n ‖2 þ ‖V n ‖2 Þ ≤V ≤ ð‖U n ‖2 þ ‖V n ‖2 Þ 2 2
ð83Þ
On the other hand, ifpffiffiffiffi (83) holds, the equilibrium pffiffiffiffiE3 ¼ ðy; xÞ remains stable if ð Ψ −1Þ 4 0 or fγ oA =αj Ψ −1j and 1 12 pffiffiffiffi ð Ψ −1Þ o 0g hold.
2
This paper has been performed under the auspices of G.N.F.M. of I.N.D.A.M. The author thanks gratefully Prof. Salvatore Rionero for his helpful suggestions and for his continuous encouragement and support. The accuracy and the comments of an unknown referee are gratefully acknowledged.
Appendix A
2
ð87Þ
ð88Þ 2
2
with k 1 ¼ A, k 2 ¼ A þ 2ðb11 þ ð1=μn 2 Þb12 þ μn 2 b21 þ b22 Þ, it turns out that dV ≤ −dV dt
ð89Þ
with d ¼ 2AjIj=k 2 , which, setting E ¼ 12 ð‖U n ‖2 þ ‖V n ‖2 Þ, implies E≤
Acknowledgments
ð86Þ
Proof of Lemma 2. On choosing pffi b12 a12 μn ¼ b11 a21
A ¼ A1 þ γ 12 αμð1−2uÞ o 0:
A1 pffiffiffiffi : γ 12 4 αð1 þ Ψ Þ
□
k2 k1
E0 e−dt :
ð90Þ
In the case (76), on choosing rffiffiffiffiffiffiffi a12 μn ¼ μ1 ; μ1 ¼ const: a21 the linearized version of (67) becomes 8 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂U a12 a21 n > n n > > V þf < ∂t ¼ b11 U 7 μ 1 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂V n > > ¼ 7μ1 a12 a21 U n þ b22 V n þ g n : : ∂t
ð91Þ
and easily follows that Proof of Lemma 1. By virtue of (72)1, (70)4, (70)5 it follows that ai 4 0 (i¼1,2). Since 8 β > > ‖u‖2∂Ω þ ‖∇u‖2 > 〈u; Δu〉 ¼ 〈u∇u; n〉∂Ω −‖∇u‖2 ¼ − > 1−β > > > > > β > 2 2 > > < 〈v; Δv〉 ¼ − 1−β‖v‖∂Ω þ ‖∇v‖ ð84Þ β > > > 〈u; v〉∂Ω −〈∇v; ∇u〉 〈v; Δu〉 ¼ 〈v∇u; n〉∂Ω −〈∇v; ∇u〉 ¼ − > > 1−β > > > > β > > > : 〈u; Δv〉 ¼ −1−β〈u; v〉∂Ω −〈∇v; ∇u〉
n
þb22 ‖V n ‖2 þ 〈U n ; f 〉 þ 〈V n ; g n 〉: n
Since b11 o0; b22 o 0, 〈U n ; f 〉 ≤0, 〈V n ; g n 〉 ≤ 0, for μ1 ¼ 1, it turns out that pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE ≤−b11 ‖U n ‖2 þ 2 a12 a21 〈U n ; V n 〉−b22 ‖V n ‖2 : dt implies On the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi other hand A ¼ b11 b22 −a12 a21 4 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a12 a21 ¼ ε b11 b22 , with 0 o ε o1, then one easily obtains dE ≤−ð1−εÞ½b11 ‖U n ‖2 þ b22 ‖V n ‖2 dt
one obtains β β Φ1 ¼ −γ 11 a1 ‖u‖2∂Ω þ ‖∇u‖2 −γ 22 a2 ‖v‖2∂Ω þ ‖∇v‖2 1−β 1−β β β þ γ 22 〈u; v〉∂Ω þ a3 ðγ 11 þ γ 22 Þ〈∇u; ∇v〉 þa3 γ 11 1−β 1−β þγ 11 a1 α‖u‖2 þ γ 22 a2 α‖v‖2 −a3 ðγ 11 α þ γ 22 αÞ〈u; v〉:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dE ¼ b11 ‖U n ‖2 7 a12 a21 þ μ1 〈U n ; V n 〉 dt μ1
which implies dE n ≤−d E dt ð85Þ
n
with d ¼ 2ð1−εÞinf ðjb11 j; jb22 jÞ.
38
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
Proof of Theorem 4. In the presence of non-linear terms one obtains dV ¼ AIð‖U n ‖2 þ ‖V n ‖2 Þ þ Φ1 þ Φ2 : dt
ð92Þ
In the case (74) it follows that 1þε dV 2 ≤−cV þ CV 1þε dt k1 and hence " # dU 2 1þε ε ≤− c− CU U: dt k1 Then c−η k 1 V0 o C 2
!1þε ;
ηoc
implies V ≤V 0 e−ηt : In the case (76) one obtains dE n ≤−d E þ ð2Þ1þε CE1þε dt n
with d ¼ 2ð1−εÞinf ðjb11 j; jb22 jÞÞ and hence Eε0 o
n
d −η Cð2Þ1þε
;
ηod
n
implies E ≤E0 e−ηt : References [1] M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly models, Chaos, Solitons and Fractals 12 (1996) 2031–2048. [2] G.I. Bischi, M. Kopel, Equilibrium selection in a nonlinear duopoly game with adaptive expectations, Journal of Economic Behavior and Organization 46 (1) (2001) 73–100. [3] G.I. Bischi, L. Gardini, M. Kopel, Noninvertible maps and complex basin boundaries in dynamic economic models with coexisting attractors, Chaos and Complexity Letters 2 (1) (2006) 43–74. [4] E. Pessa, B. Rizzi, Una possibile generalizzazione dei modelli economici dinamici, Rivista di Matematica per le Scienze economiche e sociali. Anno 10 (1& 2) (1988) 53. [5] E. Pessa, B. Rizzi, Sulla formulazione matematica dei modelli economici di tipo morfogenetico, Rivista di Matematica per le Scienze economiche e sociali. Anno 10 (1& 2) (1988) 65. [6] S. Rionero, A rigorous reduction of the L2-stability of the solutions to a nonlinear binary reaction-diffusion system of PDE's to the stability of the solutions to a linear system of ODE's, Journal of Mathematical Analysis and Application 319 (2006) 377–397. [7] S. Rionero, A nonlinear L2-stability analysis for two species dynamics with dispersal, Mathematical Biosciences and Engineering 3 (1) (2006) 189–204.
[8] S. Rionero, L2-energy stability via new dependent variables for circumventing strongly nonlinear reaction terms, Nonlinear Analysis 70 (2009) 2530–2541. [9] S. Rionero, Problemi di evoluzione nei Continui, Lecture Notes given at the PHD School in Mathematics of University “Federico II”, 2008–2009. [10] J.N. Flavin, S. Rionero, Cross-diffusion influence on the nonlinear L2-stability analysis for a Lotka–Volterra reaction–diffusion system of PDEs, IMA Journal of Applied Mathematics 72 (2007) 540–555. [11] S. Rionero, Longtime behavior of three competing species and mutualistic communities, in: T. Ruggeri, M. Sammartino (Eds.), Asymptotic Methods in Nonlinear Wave Phenomena. In Honour of the 65th Birthday of Antonio Greco, vol. 171, World Scientific Publishing, Singapore, 2007, pp. 171–186. [12] I. Torcicollo, Su alcuni problemi di diffusione non lineare, Bollettino della Unione Matematica Italiana A 3 (3) (2000) 407–410. [13] S. Rionero, I. Torcicollo, On an ill-posed problem in nonlinear heat conduction, Transport Theory and Statistical Physics 29 (1& 2) (2000) 173–186. [14] A. D'Anna, M. De Angelis, G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta Applicandae Mathematicae 122 (1) (2012) 255–267. [15] G. Alì, I. Torcicollo, Nonlinear stability of smooth solutions of the energytransport model for semiconductors, ZAMM—Zeitschrift fur Angewandte Mathematik und Mechanik 85 (4) (2005) 267–276. [16] G. Alì, PDAE models of integrated circuits, Mathematical and Computer Modelling 51 (2010) 915–926. [17] G. Alì, A. Bartel, M. Günther, Parabolic differential-algebraic models in electrical network design, SIAM Multiscale Modeling and Simulation 4 (3) (2005) 813–838. [18] G. Alì, M. Bisi, G. Spiga, I. Torcicollo, Kinetic approach to sulphite chemical aggression in porous media, International Journal of Non-Linear Mechanics 47 (2012) 769–776. [19] G. Alì, V. Furuholt, R. Natalini, I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I: modeling and qualitative analysis, Transport in Porous Media 69 (2007) 109–122. [20] G. Alì, V. Furuholt, R. Natalini, I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II: numerical approximation, Transport in Porous Media 69 (2007) 175–188. [21] G. Alì, I. Torcicollo, S. Vessella, Inverse doping problems for a P–N junction, Journal of Inverse and Ill-posed Problems 14 (6) (2006) 537–546. [22] F. Capone, R. De Luca, I. Torcicollo, Longtime behavior of vertical throughflows for binary mixtures in porous layers, International Journal of Non-Linear Mechanics 52 (2013) 1–7. [23] F. Capone, R. De Luca, S. Rionero, On the stability of non-autonomous perturbed Lotka–Volterra models, Applied Mathematics and Computation 219 (12) (2013) 6868–6881. [24] M. Gentile, B. Straughan, Structural stability in resonant penetrative convection in a Forchheimer porous material, Nonlinear Analysis: Real World Applications 14 (2013) 397–401. [25] F. Capone, M. Gentile, A.A. Hill, Penetrative convection in anisotropic porous media with variable permeability, Acta Mechanica 216 (2011) 49–58. [26] F. Capone, On the dynamics of predator–prey models with the Beddington–De Angelis functional response under Robin boundary conditions, Ricerche di Matematica 57 (2008) 137–157. [27] H.N. Agiza, On the analysis of stability, bifurcation, chaos, and chaos control of Kopel map, Chaos, Solitons and Fractals 10 (1999) 1909–1916. [28] J.P. Carcasses, Singularities of the parametric plane of an n-dimensional map, Determination of different configurations of fold and flip bifurcation curves, International Journal of Bifurcation and Chaos 5 (2) (1995) 419–447. [29] J.P. Carcasses, C. Mira, C. Simó, M. Bosch, J.C. Tatjer, “Cross-road area-spring area”, transition (I) Parameter plane representation, International Journal of Bifurcation and Chaos 1 (2) (1991) 339–348. [30] H.N. Agiza, A.A. Elsadany, M. Kopel, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A 320 (2003) 512–524. [31] W. Govaerts, R. Khoshsiar Ghaziani, Stable cycles in a Cournot duopoly model of Kopel, Journal of Computational and Applied Mathematics 218 (2008) 247–258.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
On the dynamics of a non-linear Duopoly game model Isabella Torcicollo n Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, Naples, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 24 April 2013 Received in revised form 14 June 2013 Accepted 14 June 2013 Available online 26 June 2013
The Cournot duopoly game modeled by Kopel, with adaptive expectations, is generalized by introducing the self-diffusion and cross-diffusion terms. General properties, such as boundedness and uniqueness, are obtained. Non-linear stability results are reached by the analysis of the stability of a ODE system. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Continuous Cournot–Kopel model Non-linear duopoly game Non-linear stability Adaptive expectations Self-diffusion Cross-diffusion
1. Introduction Oligopoly is the case where the market is controlled by a few number of firms producing similar products. In this paper we will restrict ourselves to the case of two firms, which is called duopoly. The situation in a duopoly is complex, since both firms have to take into account not only the behaviours of the consumers, but actions and reactions of the competitor. The first well-known model which gives a mathematical description of competition in a duopoly market dates back to the French economist Cournot (1838). In this paper, our starting point is the general case of the Cournot duopoly problem, which was modeled by Kopel (see [1–3]); precisely, we will examine the continuous time-scale counterpart of the above-mentioned non-linear Cournot–Kopel duopoly game. Assuming continuous time scales, denoting by u and v the outputs of the two firms X and Y, respectively, a non-linear dynamic system for the evolution of u and v is obtained: (
∂t u ¼ −α1 u þ α1 μ1 vð1−vÞ ∂t v ¼ −α2 v þ α2 μ2 uð1−uÞ
ð1Þ
where, in general, μi and αi with (i ¼1,2) are positive model parameters, ∂t denotes the derivative with respect to time, u : ðx; tÞ∈Ω Rþ -uðx; tÞ∈R, v : ðx; tÞ∈Ω Rþ -vðx; tÞ∈R, Ω being a bounded domain in R3 with smooth boundary ∂Ω.
To be realistic, any dynamical economic model should take into account both the time evolution and the spatial dependence of the characteristic variables. In this paper, in order to generalize the model (1), we take into account the spatial dependence, by introducing the self-diffusion and cross-diffusion terms, by considering the following equations: ( ∂t u ¼ −α1 u þ α1 μ1 vð1−vÞ þ γ 11 Δu þ γ 12 Δv ð2Þ ∂t v ¼ −α2 v þ α2 μ2 uð1−uÞ þ γ 21 Δu þ γ 22 Δv where Δ denotes the Laplacian operator, γ ij ¼ constant for ði; j ¼ 1; 2Þ and 2
∑ γ ij ξi ξj ≥kjξj2
i;j ¼ 1
k 40; ξ ¼ ðξ1 ; ξ2 Þ:
ð3Þ
Already in [4,5] the authors introduce a new class of economic dynamical models, which are called “morphogenetic systems”, which are constructed in order to generalize classical Goodwin's model of business cycle. Non-linear reaction–diffusion equations and systems play an important role in the modeling and study of many phenomena (see, for instance, dynamics of competing species, chemical aggression and convection problems in porous media, non-linear heat conduction, semiconductor devices, in [6–26]). To (2) we append the initial data uðx; 0Þ ¼ u0 ðxÞ;
vðx; 0Þ ¼ v0 ðxÞ for x∈Ω
ð4Þ
and the following boundary conditions: Dirichlet boundary conditions n
Tel.: +39 816132388; fax: +39 816132597. E-mail address: [email protected]
0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.011
u¼u
and v ¼ v on ∂Ω Rþ ;
ð5Þ
32
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
where u, v will be chosen among the constant steady-states solutions of (2), or Robin boundary data ( βu þ ð1−βÞ∇u n ¼ uβ on ∂Ω Rþ ð6Þ βv þ ð1−βÞ∇v n ¼ uβ; where 0 o β o1. The aim of the present paper is to analyse the non-linear L2stability of the constant steady states of (2) by following the procedure introduced by Rionero in [6] and used in [7–11]. Precisely, our aim is to link the stability (instability) of the generic equilibrium to (2) to the stability (instability) of the null solution to a linear system of ordinary differential equations associated to (2). The plan of the paper is the following. Section 2 is dedicated to the derivation of the mathematical model at hand, while, in Section 3, we consider the perturbation equations associated to the generalized model. Section 4 concerns boundedness and uniqueness of perturbations, while Section 5 is devoted to recall some results which allow to obtain conditions guaranteeing the stability (Section 6) of a critical point. Section 7 concerns a simplified case in which the obtained results are exemplified and commented. The paper ends with an Appendix in which the proof of Lemmas 1–2 and of Theorems 3 and 4 is given.
2. The mathematical model Two firms X and Y produce goods which are perfect substitutes and offer them at discrete time periods t ¼0,1,2,…on a common market. In order to determine the quantity of period t+1, the firms X and Y form expectations on the quantity of the other firm yetþ1 and xetþ1 , which might, for example, depend on their own quantity and the quantity of the other firm, both produced in the previous period. If we denote by xt and yt the output of firm X and Y at time t, respectively, the optimization problem through which the firms determine their quantities xtþ1 and ytþ1 are represented by arg maxx Π X ðxt ; yetþ1 Þ and arg maxy Π Y ðxetþ1 ; yt Þ where Π X ð; Þ and Π Y ð; Þ denote the profit of firm X and Y respectively. If we assume that these optimization problems have unique solutions, then xtþ1 ¼ r X ðyetþ1 Þ
ð7Þ
ytþ1 ¼ r Y ðxetþ1 Þ
ð8Þ
where r X ; r Y are called Best Replies (or reaction functions). We will assume that the firms revise their beliefs according to the adaptive expectations rules xetþ1 ¼ xet þ α1 ðxt −xet Þ
ð9Þ
yetþ1 ¼ yet þ α2 ðyt −yet Þ
ð10Þ
where αi 40 are referred to as the adjustment coefficients and we will assume the following well-known type of reaction functions: r x ðyÞ ¼ μ1 yð1−yÞ
ð11Þ
r y ðxÞ ¼ μ2 xð1−xÞ
ð12Þ
where μi ði ¼ 1; 2Þ measure the intensity of the effect that one firm's actions has on the other firm. Many specifications can be found in the literature, but an analytical expression for the Best Replies is complicated. Microeconomic foundations of (11)–(12) can be found in [1]. Then, from (7) to (12) and, in order to simplify the notation, replacing xte, yte with xt, yt, the Cournot–Kopel model is obtained: xtþ1 ¼ ð1−α1 Þxt þ α1 μ1 yt ð1−yt Þ
ð13Þ
ytþ1 ¼ ð1−α2 Þyt þ α2 μ2 xt ð1−xt Þ:
ð14Þ
Firms do not change their productions according to the computed
optimal productions, but they prefer to choose a weighted average between the previous production and the computed one, with weights 1−αi and αi , respectively. The meaning of model implies that the economically relevant case is αi ≤1 ði ¼ 1; 2Þ. The model (13)–(14) is a two-dimensional map, described in [1,27]. Bifurcations of map have been studied in [28,29]; fixed points, their stability and stable cycles have been studied intensively in the literature [27,30,31], in particular under the assumption μ1 ¼ μ2 ¼ μ;
α1 ¼ α2 ¼ α;
ð15Þ
that is the players are homogeneous with regard to the Best Replies and with regard to their expectations, respectively. These two assumptions imply that the two competitors behave identically. In this paper, starting from the continuous time-scale counterpart of the above-mentioned non-linear Cournot duopoly game (1), we will examine the generalized model (2), where we took into account the spatial dependence, by introducing the selfdiffusion and cross-diffusion terms.
3. Perturbation equations associated to the generalized model We denote by ðu; vÞ the generic equilibrium point of (1). Besides the trivial equilibrium ðu ¼ 0; v ¼ 0Þ, system (1) admits other non-trivial constant steady states, which can be found by solving Y 3 þ pY þ q ¼ 0
ð16Þ
where we have set 8 μ μ > Y ¼ μ1 μ2 ð1−uÞ− 1 2 > > 3 > > > < μ21 μ22 2 þ μ1 μ2 p¼− 3 > > > 3 3 3 2 > > > q ¼ −2μ1 μ2 þ μ1 μ2 −μ2 μ : : 1 2 27 3 The solutions to (16) are 8 Y1 ¼ Yþ þ Y− > > > > Yþ þ Y− Y þ −Y − pffiffiffi < þi 3 Y2 ¼ − 2 2 > > p ffiffiffi Y þ Y − Y þ −Y − > > : Y3 ¼ − þ −i 3 2 2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q q2 p3 Yþ ¼ − þ þ ; 2 4 27
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 q q2 p3 þ ; Y− ¼ − − 2 4 27
ð17Þ
ð18Þ
ð19Þ
hence ðu; vÞ ¼ ð 23 −Y i =μ1 μ2 ; 29 þ Y i =μ1 −Y 2i =μ21 μ2 Þ, with (i¼1,2,3). Precisely, if Δ ¼ q2 =4 þ p3 =27 is non-positive (positive), which implies −μ21 μ22 þ 4μ1 μ22 −18μ1 μ2 þ 4μ21 μ2 þ 27 non-positive (positive), then three (only one) real solutions are created. Let us set U ¼ u−u;
V ¼ v−v;
ð20Þ
then, the perturbation equations associated to (2) are given by ( ∂t U ¼ −α1 U þ α1 μ1 ð1−2vÞV þ γ 11 ΔU þ γ 12 ΔV−α1 μ1 V 2 ð21Þ ∂t V ¼ α2 μ2 ð1−2uÞU−α2 V þ γ 21 ΔU þ γ 22 ΔV−α2 μ2 U 2 : Denoting by 8 2 > < f ðU; V Þ ¼ −α1 μ1 V ; > :
gðU; VÞ ¼ −α2 μ2 U 2
a11 ¼ −α1 ;
a22 ¼ −α2
a21 ¼ α2 μ2 ð1−2uÞ;
a12 ¼ α1 μ1 ð1−2vÞ
ð22Þ
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
the system (21) becomes ( ∂t U ¼ a11 U þ a12 V þ γ 11 ΔU þ γ 12 ΔV þ f ðU; V Þ ∂t V ¼ a21 U þ a22 V þ γ 21 ΔU þ γ 22 ΔV þ gðU; VÞ: A system of this form has been considered in [6–11]. To (23) we append the initial data ( Uðx; 0Þ ¼ U 0 ðxÞ x∈Ω Vðx; 0Þ ¼ V 0 ðxÞ
ð23Þ
on ∂Ω Rþ ;
ð24Þ
ð26Þ
þ ð∇ϕÞ ∈LðΩÞ; ϕ ¼ 0 on ∂Ωg;
then U, V are bounded.
1d ð‖U‖2 þ ‖V‖2 Þ 2 dt
Ω
ρ ¼ ρðΩÞ 40 the positive constant, appearing in the Poincaré inequality ‖∇ϕ‖2 ≥ρ‖ϕ‖2
ð27Þ
is the lowest eigenvalue ρ of the spectral problem Δϕ þ ρϕ ¼ 0
β ‖ϕ‖2∂Ω ≥α‖ϕ‖2 ; 1−β
ð29Þ
is the lowest eigenvalue α of the spectral problem Δϕ þ αϕ ¼ 0
Ω
ð35Þ
Ω
Ω
Ω
Ω
β β ‖U‖2∂Ω −γ 11 ‖∇U‖2 −γ 22 ‖V‖2∂Ω −γ 22 ‖∇V‖2 1−β 1−β Z Z β −ðγ 12 þ γ 21 Þ UV dΣ−ðγ 12 þ γ 21 Þ ∇U∇V dΩ 1−β ∂Ω Ω β 2 2 2 ð‖U‖∂Ω þ ‖V ‖2∂Ω Þ ≤−λmin ‖∇U‖ þ ‖∇V‖ þ ð1−βÞ ¼ −γ 11
≤−λmin αð‖U‖2 þ ‖V‖2 Þ:
ð36Þ
Then (35) becomes ð28Þ
in H 10 ðΩÞ; α ¼ αðΩ; βÞ 4 0 the positive constant, appearing in the following inequality:
ð30Þ
1d ð‖U‖2 þ ‖V‖2 Þ ≤ −α1 ‖U‖2 −α2 ‖V‖2 −λmin αð‖U‖2 þ ‖V‖2 Þ 2 dt Z Z Z þ ða12 þ a21 Þ
in H ðΩ; βÞ. 2
We study the stability of U ¼ V ¼ 0 in the L ðΩÞnorm with respect to the perturbations (U, V) belonging, ∀t∈Rþ , to ½H 10 ðΩÞ2 in the case (25) and to ½H 1 ðΩ; βÞ2 in the case (26).
Ω
UVdΩ−α2 μ2
Ω
U 2 V dΩ−α1 μ1
UV 2 dΩ:
Let us denote by Ωþ ¼ Ω Rþ and C 21 ðΩþ Þ the set of the functions belonging to CðΩþ Þ with their first time derivative and first and second spatial derivatives. In order to prove the boundedness of the perturbations and a uniqueness theorem for (21), let us remark that, from (3), it follows that ð31Þ
ð38Þ
Ω
is satisfied for 8 < λ oα1 þ λmin α
2 : ðλ−αλmin Þ2 −ðα1 þ α2 Þðλ−αλmin Þ þ α1 α2 − a12 þ a21 ≥0 2 which is verified when (34) holds and for λ o λ1 with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α þ α 2 a þ a 2 α1 þ α2 1 2 12 21 − −α1 α2 þ λ1 ¼ αλmin þ 40 2 2 2
4. Boundedness and uniqueness
ð37Þ
Ω
The relation Z f−ðα1 þ λmin αÞU 2 þ ða12 þ a21 ÞUV−ðα2 þ λmin αÞV 2 g dΩ Ω Z ≤−λ ðU 2 þ V 2 Þ dΩ
1
γ 11 X 2 þ ðγ 12 þ γ 21 ÞXY þ γ 22 Y 2 ≥kðX 2 þ Y 2 Þ
ð34Þ
By using (26), the divergence theorem and (31), then the Poincaré inequality (29) leads to Z Z Z Z γ 11 UΔU þ γ 22 V ΔV dΩ þ γ 21 V ΔU dΩ þ γ 12 UΔV dΩ
2
ϕ∈H 1 ðΩ; βÞ-fϕ2 þ ð∇ϕÞ2 ∈LðΩÞ; βϕ þ ð1−βÞ∇ϕ n ¼ 0 on ∂Ωg;
‖∇ϕ‖2 þ
with k1 ¼ α1 μ1 þ α2 μ2 and λ given by (39), and a þ a 2 12 21 −α1 α2 ≤0 2
Ω
H1 ðΩ; βÞ the functional space such that
ð33Þ
Z ¼ −α1 ‖U‖2 −α2 ‖V‖2 þ a12 UV dΩ Ω Z Z Z Z þa21 UV dΩ þ γ 22 VΔV dΩ þ γ 11 UΔU þ γ 12 UΔV Ω Ω Ω ZΩ Z Z þγ 21 VΔU dΩ−α2 μ2 U 2 V dΩ−α1 μ1 UV 2 dΩ:
∥ ∥ the L2 ðΩÞnorm; ‖ ‖∂Ω the L2 ð∂ΩÞnorm; 〈; 〉 the scalar product in L2 ðΩÞ; H 10 ðΩÞ the functional space such that ϕ∈H 10 ðΩÞ-fϕ2
λ k1
ðjU 0 j2 þ jV 0 j2 Þ1=2 o
Proof. Multiplying (21)1 by U and (21)2 by V, summing and integrating on Ω and by using (22)3, it follows that
where 0 o β o1: In the following, we denote by:
ð32Þ
Theorem 1. Let (U, V) with fU; V∈C 21 ðΩþ g be a solution to (21) under conditions (24), (26). If
ð25Þ
or the Robin boundary conditions 8 ð1−βÞ > > > < U þ β ∇U n ¼ 0 on ∂Ω Rþ ð1−βÞ > > > : V þ β ∇V n ¼ 0;
∀X; Y and k 4 0. We can choose k ¼ λmin with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ þ γ 2 γ þ γ 2 γ 11 þ γ 22 11 22 21 − − γ 11 γ 22 − 12 λmin ¼ 2 2 2
the minimum eigenvalue of the symmetric part of the matrix ðγ ij Þ.
with U 0 ðxÞ ¼ u0 ðxÞ−u and V 0 ðxÞ ¼ v0 ðxÞ−v. Choosing the constants u, v in the boundary conditions (5), (6) equal to the steady state under consideration, we obtain the Dirichlet boundary conditions U¼V ¼0
33
ð39Þ
ð40Þ
In addition, from Z Z −α2 μ2 U 2 V dΩ−α1 μ1 UV 2 dΩ ≤ðα1 μ1 þ α2 μ2 Þð‖U‖2 þ ‖V‖2 Þ3=2 Ω
Ω
Eq. (31) leads to 1d ð‖U‖2 þ ‖V‖2 Þ ≤ −λð‖U‖2 þ ‖V‖2 Þ þ k1 ð‖U‖2 þ ‖V‖2 Þ3=2 2 dt
ð41Þ
34
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
with k1 ¼ ðα1 μ1 þ α2 μ2 Þ. From
d k1 ð‖U‖2 þ ‖V‖2 Þ ≤ −2λð‖U‖2 þ ‖V ‖2 Þ 1− ð‖U‖2 þ ‖V‖2 Þ1=2 dt λ
ð42Þ
by (33) it follows that ð‖U 0 ‖2 þ ‖V 0 ‖2 Þe−2λt
‖U‖2 þ ‖V‖2 ≤
k1 1− ð‖U 0 ‖2 þ ‖V 0 ‖2 Þ1=2 ð1−e−λt Þ λ
2 ;
t≥0:
ð43Þ
On taking into account (49)–(51), (48) becomes
Theorem 2. The problem (21) under conditions (24), (26) can admit only one solution in the class of perturbations (U, V) such that fU; V ∈C 21 ðΩþ Þ verifying (34)g. Proof. Let ðU 1 ; V 1 Þ and ðU 2 ; V 2 Þ be two solutions of (21), (24), (26). On setting W ¼ U 1 −U 2 ;
Z ¼ V 1 −V 2
ð44Þ
it follows that ( ∂t W ¼ −α1 W þ α1 μ1 ð1−2vÞZ þ γ 11 ΔW þ γ 12 ΔZ−α1 μ1 ZðV 1 þ V 2 Þ ∂t Z ¼ α2 μ2 ð1−2uÞW−α2 Z þ γ 21 ΔW þ γ 22 ΔZ−α2 μ2 WðU 1 þ U 2 Þ ð45Þ with (W, Z) satisfying the boundary conditions 8 dW > > ¼0 < βW þ ð1−βÞ dn on ∂Ω Rþ dZ > > : βZ þ ð1−βÞ ¼ 0; dn
ð46Þ
Ω
ð52Þ
and (31) leads to d ð‖W‖2 þ ‖Z‖2 Þ dt ≤ðA−2α1 þ MÞ‖W‖2 þ ðA−2α2 þ MÞ‖Z‖2 β ð‖W‖2∂Ω þ ‖Z‖2∂Ω Þ : −2λmin ‖∇W‖2 þ ‖∇Z‖2 þ ð1−βÞ
ð53Þ
By using the Poincaré inequality (29), it follows that ð54Þ
with k≥maxðA−2α1 þ M−2λmin α; A−2α2 þ M−2λmin αÞ
‖W‖2 þ ‖Z‖2 ≤ð‖W 0 ‖2 þ ‖Z 0 ‖2 Þekt ¼ 0; ð47Þ
ð55Þ
Ω
þγ 12
WΔZ dΩ þ γ 21
Z
−α2 μ2
Ω
ZΔW dΩ
Ω
WZðU 1 þ U 2 Þ dΩ WZðV 1 þ V 2 Þ dΩ:
ð48Þ
By virtue of (46), the divergence theorem leads to Z Z γ 11 WΔW þ γ 22 ZΔZ dΩ Ω
β ‖W‖2∂Ω −γ 11 ‖∇W‖2 ¼ −γ 11 1−β β ‖Z‖2∂Ω −γ 22 ‖∇Z‖2 : −γ 22 1−β From Schwartz inequality, it follows that Z ða12 þ a21 Þ ð‖W‖2 þ ‖Z‖2 Þ: ða12 þ a21 Þ WZ dΩ ≤ 2 Ω
ð49Þ
ð50Þ
Since (34) is verified, the solutions of (21) are bounded, and there exist positive constants Mi, Ni such that jU i j ≤M i , jV i j ≤N i , i ¼1,2. Denoting by M ¼ α2 μ2 ðM 1 þ M 2 Þ þ α1 μ1 ðN 1 þ N2 Þ it follows that Z Z −α2 μ2 WZðU 1 þ U 2 Þ dΩ−α1 μ1 WZðV 1 þ V 2 Þ dΩ Ω
ð57Þ □
5. Stability conditions
Ω
Z −α1 μ1
ð56Þ
Remark 1. Theorems 1 and 2, under conditions (24), (25), continue to value. In particular, the terms evaluated on ∂Ω are zero and the Poincaré constant α is replaced with ρ given by (27).
Z
Ω
ZðtÞ ¼ 0 ∀t≥0
and uniqueness is proved.
Ω
Z
t≥0
i.e. WðtÞ ¼ 0;
Z 1d ð‖W‖2 þ ‖Z‖2 Þ ¼ −α1 ‖W‖2 −α2 ‖Z‖2 þ ða12 þ a21 Þ WZ dΩ 2 dt Ω Z Z þγ 22 ZΔZ dΩ þ γ 11 WΔW
M ≤ ð‖W‖2 þ ‖Z‖2 Þ: 2
ðγ 11 j∇Wj2 þ ðγ 12 þ γ 21 Þ∇W∇Z þ γ 22 j∇Zj2 Þ dΩ Z 2β ðγ jWj2 þ ðγ 12 þ γ 21 ÞWZ þ γ 22 jZj2 Þ dΣ − ð1−βÞ ∂Ω 11
−2
d ð‖W‖2 þ ‖Z‖2 Þ ≤kð‖W‖2 þ ‖Z‖2 Þ dt
From (45) it follows the following relation:
Ω
d ð‖W‖2 þ ‖Z‖2 Þ ≤ðA−2α1 þ MÞ‖W‖2 þ ðA−2α2 þ MÞ‖Z‖2 dt Z
and hence
and the initial conditions ( Wðx; 0Þ ¼ 0 x∈Ω Zðx; 0Þ ¼ 0:
Ω
By virtue of (46), the divergence theorem and Schwartz inequality lead to Z Z γ 21 ZΔW dΩ þ γ 12 WΔZ dΩ Ω Ω Z Z β WZ dΣ−ðγ 12 þ γ 21 Þ ∇W∇Z dΩ ð51Þ ¼ −ðγ 12 þ γ 21 Þ 1−β ∂Ω Ω
In the absence of diffusion, the study of the linear stability of system (1) is based on the linearization of the following perturbation problem: ( ∂t U ¼ −α1 U þ α1 μ1 ð1−2vÞV−α1 μ1 V 2 ð58Þ ∂t V ¼ α2 μ2 ð1−2uÞU−α2 V−α2 μ2 U 2 : In the equilibrium point ðu; vÞ the eigenvalues are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 0 7 I 20 −4A0 λ¼ 2 with ( I 0 ¼ −α1 −α2 A0 ¼ α1 α2 ½1−μ1 μ2 ð1−2vÞð1−2uÞ
ð59Þ
hence, since I 0 o 0, the condition A0 4 0, that is a12 a21 −α1 α2 o 0;
ð60Þ
guarantees the stability of the zero solution (U¼V ¼0) of the linearized system. Remark 2. Eq. (34) can be written as ðða12 −a21 Þ=2Þ2 þ a12 a21 −α1 α2 ≤0 and it implies (60); in particular, if the system (21) were symmetrical, a12 −a21 ¼ 0, then (34) coincides with (60).
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
Following the procedure of Rionero [6–11], we associate to (23) the binary system of ODE's 8 dξ > > < ¼ b11 ξ þ b12 η dt ð61Þ dη > > : ¼ b21 ξ þ b22 η dt where b11 ¼ a11 −γ 11 ρ;
b12 ¼ a12 −γ 12 ρ;
b21 ¼ a21 −γ 21 ρ;
b22 ¼ a22 −γ 22 ρ:
Denoting by a and b two positive constants and setting U n ¼ aU;
V n ¼ bV ;
I ¼ b11 þ b22 ¼ −ðα1 þ γ 11 ρÞ−ðα2 þ γ 22 ρÞ o 0;
ð62Þ
μn ¼
a ; b
−1
f ¼ a−1 f ðaU; bVÞ;
g ¼ b gðaU; bVÞ
n
n
f ¼ γ 11 ðΔU þ ρUÞ;
g ¼ γ 22 ðΔV þ ρV Þ
f~ ¼ μn γ 21 ðΔU þ ρUÞ;
−1 g~ ¼ μn γ 12 ðΔV þ ρVÞ G ¼ g þ g n þ f~
n ~ F ¼ f þ f þ g;
pffiffiffiffiffiffiffiffiffiffiffiffiffi The eigenvalues of (61) are given by λ1;2 ¼ ðI 7 I 2 −4AÞ=2, then, since
35
the system (23) becomes ( ∂t U n ¼ b11 U n þ μn −1 b12 V n þ F
ð66Þ
ð67Þ
∂t V n ¼ μn b21 U n þ b22 V n þ G:
if
Remark 6. Let us remark that, the system (67) has the same eigenvalues as system (61).
A ¼ b11 b22 −b12 b21 ¼ λ1 λ2 ¼ A0 þ ρ½α1 γ 22 þ α2 γ 11 þ ργ 11 γ 22 þγ 12 α2 μ2 ð1−2uÞ þ γ 21 α1 μ1 ð1−2vÞ 4 0
The analysis is based on the functional introduced by Rionero [7–11]
ð63Þ
the asymptotic stability of equilibrium ðξ ¼ η ¼ 0Þ of (61) is guaranteed. Our aim is to obtain the stability (instability) of the economic equilibrium state of (23) via the stability (instability) of the zero solution of (61). Since I ¼ b11 þ b22 o 0, in the self-diffusion case, the condition for asymptotic stability is simply def
A ¼ A0 þ ρ½α1 γ 22 þ α2 γ 11 þ ργ 11 γ 22 ¼ A1 4 0;
ð64Þ
V ¼ 12½Að‖U n ‖2 þ ‖V n ‖2 Þ þ ‖b11 V n −μn b21 U n ‖2 þ‖μn−1 b12 V n −b21 U n ‖2
ð68Þ
whose derivative, along the solutions of the system (67), is given by dV ¼ AIð‖U n ‖2 þ ‖V n ‖2 Þ þ Φ dt
ð69Þ
where Φ ¼ Φ1 þ Φ2
then, A0 4 0 assures A1 4 0.
n ~ þ 〈a2 V n −a3 U n ; g n þ f~ 〉 Φ1 ¼ 〈a1 U n −a3 V n ; f þ g〉
Remark 3. Turing instability cannot occur if γ 12 ¼ γ 21 ¼ 0.
Φ2 ¼ 〈a1 U n −a3 V n ; f 〉 þ 〈a2 V n −a3 U n ; g〉
Remark 4. If A0 o0 then, in the absence of diffusion, the zero solution of (23) is unstable and the self-diffusion stability is driven according to (64).
a1 ¼ A þ μn2 b21 þ b22 ;
Remark 5. If an economic equilibrium state is stable in the selfdiffusion case, that is (64) is verified, then, it can become unstable, by introducing cross-diffusion in the equations. Precisely, each of the following conditions:
2
a3 ¼ μn b11 b21 þ
2
2
a2 ¼ A þ b11 þ
1 b12 b22 : μn
1 2 b ; μn2 12 ð70Þ
5.1. Self-diffusion case: γ 12 ¼ γ 21 ¼ 0 Under the assumption
(1) ð1−2uÞ 4 0 and ð1−2vÞγ 21 4 0 and γ 12 o −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ o0; (2) ð1−2uÞ 4 0 and ð1−2vÞγ 21 o0 and 0 o γ 12 o −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ; (3) ð1−2uÞ o 0 and ð1−2vÞγ 21 4 0 and γ 12 4 −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ 40; (4) ð1−2uÞ o 0 and ð1−2vÞγ 21 o 0 and 0 4 γ 12 4 −γ 21 α1 μ1 ð1−2vÞ=α2 μ2 ð1−2uÞ; (5) ð1−2uÞγ 12 o 0 and ð1−2vÞγ 21 o 0 is a necessary condition for ( A1 4 0 A ¼ A1 þ γ 12 α2 μ2 ð1−2uÞ þ γ 21 α1 μ1 ð1−2vÞ o 0:
γ 12 ¼ γ 21 ¼ 0
ð71Þ
Lemmas 1 and 2 hold for system (67), both for Dirichlet boundary conditions (25) and Robin boundary conditions (26). The proof of these lemmas can be deduced from the analogous results in [9]. However, for the sake of completeness, we give a sketch of the proof in the Appendix. Lemma 1. Let ( γ 11 ¼ γ 22 ¼ γ A1 4 0: ð65Þ
In order to study the non-linear stability of the zero solution ðU ¼ V ¼ 0Þ of the perturbation equations (21) with the Dirichlet boundary conditions (25) or the Robin boundary conditions (26) by a Lyapunov functional, we follow the procedure introduced by Rionero in [6] and used in [7–11]. Let us recall some results, proved in [6,10], useful in the following sections, rewritten by using the parameters introduced in the model. These results provide conditions guaranteeing the linear stability of the zero solution.
ð72Þ
Then Φ1 ≤0: Lemma 2. Let ( b11 b22 a12 a21 o 0 A1 4 0:
ð73Þ
ð74Þ
Then (73) holds. Remark 7. On taking into account that b11 b22 4 0, the condition (74)1 reduces itself to a12 a21 o 0, that is ð1−2vÞð1−2uÞ ≤ 0.
36
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
In addition, in the model at hand, the critical case ( b11 b22 a12 a21 4 0 a12 a21 o0 cannot occur, while the case ( b11 b22 a12 a21 4 0 a12 a21 40
the stability of the critical point of (21). In fact, on taking into account (62), Theorems 3 and 4 hold. ð75Þ
ð76Þ
is easily embraced in the proof of general stability condition (see Appendix, Proof of Theorem 4). 5.2. Cross-diffusion case: γ 12 ≠0 or γ 21 ≠0 Under the assumption γ 12 ≠0 or γ 21 ≠0 and when to system (67) we append (25), the following lemmas hold, proved by Flavin and Rionero (compare with Lemmas 1–3 in [10]). For the crossdiffusion case in the presence of Robin boundary conditions we refer to [8]. Lemma 3. Let inf ða1 ; a2 ; γ 11 ; γ 22 Þ 4 0 with γ 11 ¼ γ 11 −μn ða3 =a1 Þγ 21 and γ 22 ¼ γ 22 −μn−1 ða3 =a2 Þγ 12 : Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðγ 11 þ γ 22 Þa3 −ðμn−1 a1 γ 12 þ μn a2 γ 21 Þj ≤ 2 a1 a2 γ 11 γ 22 implies (73). Lemma 4. Let inf ða1 ; a2 Þ 40 8 2 > γ n11 ¼ γ 11 −b12 b22 ðA þ b22 Þ−1 γ 21 4 0 > > > > > b22 > > < γ n22 ¼ γ 22 − γ 21 4 0 b12 2 2 > > jðγ þ γ Þb > 11 22 12 b22 −½ðA þ b22 Þγ 12 þ b12 γ 21 j > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : ≤2jb12 j ðA þ b222 Þγ n11 γ n22 or 8 b11 > > γ n11 ¼ γ 11 − γ 21 4 0 > > > b21 > > > < γ n ¼ γ −b b ðA þ b2 Þγ 4 0 11 21 22 22 12 22 2 2 > > jðγ þ γ Þb b −½b γ > 11 22 11 21 21 12 ðA þ b11 Þγ 21 j > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : ≤jb21 j ðA þ b211 Þγ n11 γ n22 : Then (73) holds. Lemma 5. Let inf ða1 ; a2 Þ 40 and b11 b12 b21 b22 o 0. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ja 1 γ 12 þ μ 2 a 2 γ 21 jo 2μ a 1 a 2 γ 11 γ 22 where b b 1=2 12 22 μ¼ b11 b21 b b b 12 21 22 2 a 1 ¼ a1 ðμÞ ¼ A þ þ b22 4 0 b11 b b b 11 12 21 2 a 2 ¼ a2 ðμÞ ¼ A þ b11 þ 40 b22 implies (73). 6. Stability analysis 6.1. Self-diffusion case: γ 12 ¼ γ 21 ¼ 0 Under the assumption (71), Lemmas 1 and 2 hold. In this hypothesis, Lemmas 1 and 2 allow to obtain conditions guaranteeing
Theorem 3. Under the assumption (71) the linear asymptotic exponential L2 stability of the zero solution to (67) is guaranteed by (64). Proof. The proof can be found in [7] in the case (25) and in [9] in the case (26). However we give a sketch of the proof in the Appendix. □ Theorem 4. Let Theorem3 hold. Then Φ2 ≤Cð‖U n ‖2 þ ‖V n ‖2 Þ1þε
ε; C ¼ pos:const:
ð77Þ
2
guarantees the (local) L ðΩÞnonlinear asymptotic exponential stability of the zero solution of (67). Proof. The proof can be found in [7] in the case (25) and in [9] in the case (26). However, we give a sketch of the proof in the Appendix. □ Remark 8. In the problem at hand Φ2 ¼ 〈a1 U n −a3 V n ; f 〉 þ 〈a2 V n −a3 U n ; g〉 ≤C〈jU n j þ jV n j; jU n j2 þ jV n j2 〉
ð78Þ
where C ¼ maxðja1 j; ja2 j; ja3 jÞ maxðα1 μ1 b=a; α2 μ2 a=bÞ. By using, pffiffiffi jU n j þ jV n j ≤ 2ðjU n j2 þ jV n j2 Þ1=2 it follows that pffiffiffi Φ2 ≤C 2ð‖U n ‖2 þ ‖V n ‖2 Þ3=2 :
ð79Þ
6.2. Cross-diffusion case: γ 12 ≠0 or γ 21 ≠0 Under the assumption γ 12 ≠0
or
γ 21 ≠0;
ð80Þ
and if Lemmas 3–5 hold, Theorem 5 is true. The proof can be found in [10] (see Theorem 3 within). Theorem 5. Let j〈U; f 〉j þ j〈V ; g〉j ≤ kð‖U‖2 þ ‖V‖2 Þ1þε (with ε and k positive constants), and let one of Lemmas3–5 (ensuring Φ1 ≤ 0) hold. Then the stability of the zero solution of (61) implies the conditional non-linear asymptotic exponential stability of the zero solution of (67) with respect to the L2 ðΩÞnorm. Remark 9. Recalling that, in the considered model j〈U; f 〉j þ j〈V; g〉j ¼ 〈U; −α1 μ1 V 2 〉 þ 〈V; −α2 μ2 U 2 〉 ≤C 1 〈jUj þ jVj; jUj2 þ jVj2 〉
ð81Þ
where C 1 ¼ maxðα1 μ1 ; α2 μ2 Þ, since pffiffiffi jUj þ jVj ≤ 2ðjUj2 þ jVj2 Þ1=2 from (81) it follows that pffiffiffi j〈U; f 〉j þ j〈V; g〉j ≤C 1 2ð‖U‖2 þ ‖V‖2 Þ3=2 : 7. A simplified case For the sake of simplicity, in order to obtain a simple analytical solution, in this section we assume (15). Under these assumptions, for α≠0, besides E0 ¼ ð0; 0Þ, the following positive equilibria exist: μ−1 μ−1 ; for μ 4 1 E1 ¼ μ μ E2 ¼ ðx; yÞ E3 ¼ ðy; xÞ
for μ4 3 ð82Þ
I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
where
pffiffiffiffi μþ1þ Ψ x¼ ; 2μ
pffiffiffiffi μ þ 1− Ψ y¼ ; 2μ
Ψ ¼ ðμ−3Þðμ þ 1Þ: We notice that in equilibrium E2, the beliefs are such that the firm 1 will dominate the market. In equilibrium E3 firm 2 dominates. Remark 10. We summarize that, by the analysis of eigenvalues pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ¼ −1 7 μ ð1−2uÞð1−2vÞ, it follows that E1 (stable node) is asymptotically stable for 1 o μ o 3 and it is unstable (saddle point) for μ 43; E2 ; E3 are asymptotically stable for μ 4 3, precisely, stable pffiffiffi pffiffiffi node for 3 o μ ≤ 1 þ 5 and spiral sink for μ 4 1 þ 5. In the self-diffusion case, the condition for asymptotic stability of the economic equilibrium ðu; vÞ is given by A1 ¼ α2 ½1−μ2 ð1−2vÞð1−2uÞ þ ρ½αγ 22 þ αγ 11 þ γ 11 γ 22 ρ 40: By introducing the cross-diffusion, (e.g. γ 12 ≠0 and γ 21 ¼ 0), the stable equilibrium ðu; vÞ becomes unstable if
37
On taking into account (72), it turns out that β Φ1 ¼ −Aγa1 ð‖u‖2∂Ω þ ‖∇u‖2 ‖∇u‖2 þ ‖∇v‖2 Þ−αð‖u‖2 þ ‖v‖2 1−β β ‖b1 1u−μn a12 v‖2 þ ‖∇ðb11 u−μn a12 vÞ‖2 −γ 1−β β β þ γ 22 〈u; v〉∂Ω þ a3 ðγ 11 þ γ 22 Þ〈∇u; ∇v〉 þa3 γ 11 1−β 1−β þγ 11 a1 α‖u‖2 þ γ 22 a2 α‖v‖2 −a3 ðγ 11 α þ γ 22 αÞ〈u; v〉:
one obtains a3 ¼ 0. Then by virtue of (85) it follows that β ‖u‖2∂Ω þ ‖∇u‖2 −α‖u‖2 Φ1 ¼ −γ 11 a1 1−β β ‖v‖2∂Ω þ ‖∇v‖2 −α‖v‖2 : −γ 22 a2 1−β Proof of Theorem 3. In the case (74), in view of
For instance, if we consider the stable equilibrium E2 ¼ ðx; yÞ, since ð1−2uÞ o 0, it follows that E2, in the presence of cross-diffusion, becomes unstable if
k1 k2 ð‖U n ‖2 þ ‖V n ‖2 Þ ≤V ≤ ð‖U n ‖2 þ ‖V n ‖2 Þ 2 2
ð83Þ
On the other hand, ifpffiffiffiffi (83) holds, the equilibrium pffiffiffiffiE3 ¼ ðy; xÞ remains stable if ð Ψ −1Þ 4 0 or fγ oA =αj Ψ −1j and 1 12 pffiffiffiffi ð Ψ −1Þ o 0g hold.
2
This paper has been performed under the auspices of G.N.F.M. of I.N.D.A.M. The author thanks gratefully Prof. Salvatore Rionero for his helpful suggestions and for his continuous encouragement and support. The accuracy and the comments of an unknown referee are gratefully acknowledged.
Appendix A
2
ð87Þ
ð88Þ 2
2
with k 1 ¼ A, k 2 ¼ A þ 2ðb11 þ ð1=μn 2 Þb12 þ μn 2 b21 þ b22 Þ, it turns out that dV ≤ −dV dt
ð89Þ
with d ¼ 2AjIj=k 2 , which, setting E ¼ 12 ð‖U n ‖2 þ ‖V n ‖2 Þ, implies E≤
Acknowledgments
ð86Þ
Proof of Lemma 2. On choosing pffi b12 a12 μn ¼ b11 a21
A ¼ A1 þ γ 12 αμð1−2uÞ o 0:
A1 pffiffiffiffi : γ 12 4 αð1 þ Ψ Þ
□
k2 k1
E0 e−dt :
ð90Þ
In the case (76), on choosing rffiffiffiffiffiffiffi a12 μn ¼ μ1 ; μ1 ¼ const: a21 the linearized version of (67) becomes 8 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂U a12 a21 n > n n > > V þf < ∂t ¼ b11 U 7 μ 1 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂V n > > ¼ 7μ1 a12 a21 U n þ b22 V n þ g n : : ∂t
ð91Þ
and easily follows that Proof of Lemma 1. By virtue of (72)1, (70)4, (70)5 it follows that ai 4 0 (i¼1,2). Since 8 β > > ‖u‖2∂Ω þ ‖∇u‖2 > 〈u; Δu〉 ¼ 〈u∇u; n〉∂Ω −‖∇u‖2 ¼ − > 1−β > > > > > β > 2 2 > > < 〈v; Δv〉 ¼ − 1−β‖v‖∂Ω þ ‖∇v‖ ð84Þ β > > > 〈u; v〉∂Ω −〈∇v; ∇u〉 〈v; Δu〉 ¼ 〈v∇u; n〉∂Ω −〈∇v; ∇u〉 ¼ − > > 1−β > > > > β > > > : 〈u; Δv〉 ¼ −1−β〈u; v〉∂Ω −〈∇v; ∇u〉
n
þb22 ‖V n ‖2 þ 〈U n ; f 〉 þ 〈V n ; g n 〉: n
Since b11 o0; b22 o 0, 〈U n ; f 〉 ≤0, 〈V n ; g n 〉 ≤ 0, for μ1 ¼ 1, it turns out that pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE ≤−b11 ‖U n ‖2 þ 2 a12 a21 〈U n ; V n 〉−b22 ‖V n ‖2 : dt implies On the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi other hand A ¼ b11 b22 −a12 a21 4 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a12 a21 ¼ ε b11 b22 , with 0 o ε o1, then one easily obtains dE ≤−ð1−εÞ½b11 ‖U n ‖2 þ b22 ‖V n ‖2 dt
one obtains β β Φ1 ¼ −γ 11 a1 ‖u‖2∂Ω þ ‖∇u‖2 −γ 22 a2 ‖v‖2∂Ω þ ‖∇v‖2 1−β 1−β β β þ γ 22 〈u; v〉∂Ω þ a3 ðγ 11 þ γ 22 Þ〈∇u; ∇v〉 þa3 γ 11 1−β 1−β þγ 11 a1 α‖u‖2 þ γ 22 a2 α‖v‖2 −a3 ðγ 11 α þ γ 22 αÞ〈u; v〉:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dE ¼ b11 ‖U n ‖2 7 a12 a21 þ μ1 〈U n ; V n 〉 dt μ1
which implies dE n ≤−d E dt ð85Þ
n
with d ¼ 2ð1−εÞinf ðjb11 j; jb22 jÞ.
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I. Torcicollo / International Journal of Non-Linear Mechanics 57 (2013) 31–38
Proof of Theorem 4. In the presence of non-linear terms one obtains dV ¼ AIð‖U n ‖2 þ ‖V n ‖2 Þ þ Φ1 þ Φ2 : dt
ð92Þ
In the case (74) it follows that 1þε dV 2 ≤−cV þ CV 1þε dt k1 and hence " # dU 2 1þε ε ≤− c− CU U: dt k1 Then c−η k 1 V0 o C 2
!1þε ;
ηoc
implies V ≤V 0 e−ηt : In the case (76) one obtains dE n ≤−d E þ ð2Þ1þε CE1þε dt n
with d ¼ 2ð1−εÞinf ðjb11 j; jb22 jÞÞ and hence Eε0 o
n
d −η Cð2Þ1þε
;
ηod
n
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