0 引言

1963年, 美国气象学家LORENZ^[1]提出了大气热对流过程的动力学模型, 即著名的Lorenz系统，并且发现了著名的蝴蝶效应.随后，众多学者对Lorenz系统的各种动力学行为展开了深入研究, 揭示了Lorenz系统复杂动力学行为的演化过程及混沌的产生机制^[2-14]，并将Lorenz系统应用于自然科学的各个领域.

著名学者STENFLO在研究大气的热对流运动时，给出了大气热对流运动的四维混沌模型^[15-16]：

$ \left\{ \begin{array}{l} \frac{{{\text{d}}x}}{{{\text{d}}t}} = \alpha \left( {y - x} \right) + \gamma w,\\ \frac{{{\text{d}}y}}{{{\text{d}}t}} = cx - xz - y,\\ \frac{{{\text{d}}z}}{{{\text{d}}t}} = xy - \beta z,\\ \frac{{{\text{d}}w}}{{{\text{d}}t}} = - x - \alpha w, \end{array} \right. $

(1)

其中α, β, γ, c为系统(1)的正参数，变量w用于描述气流的旋转, 参数γ(>0)是与w相对应的旋转数(rotation number), α(>0)表示普朗特数(Prandtl number), c(>0)表示瑞利数(Rayleigh number)，β(>0)为几何参数(geometric parameter).当c=26, β=0.7, γ=1.5, α=1时，系统(1)的混沌吸引子如图 1所示，在平面yoz上的吸引子如图 2所示.

1 主要结果及证明

系统(1)的一些动力学行为：模型的推导、混沌行为产生的机理等，文献[12]已有研究.下文主要研究系统(1)的最终界和全局吸引性.

引理1  定义集合

$ \begin{array}{*{20}{c}} {{\Gamma _1} = \left\{ {\left( {{x_1},{y_1},{z_1},{w_1}} \right)\left| {\frac{{x_1^2}}{{{a^2}}} + \frac{{y_1^2}}{{{b^2}}} + \frac{{{{\left( {{z_1} - c} \right)}^2}}}{{{c^2}}} + } \right.} \right.}\\ {\left. {\frac{{w_1^2}}{{{d^2}}} = 1,a > 0,b > 0,c > 0,d > 0} \right\},} \end{array} $

令

$ \begin{array}{*{20}{c}} {G\left( {{x_1},{y_1},{z_1},{w_1}} \right) = x_1^2 + y_1^2 + z_1^2 + w_1^2,}\\ {H\left( {{x_1},{y_1},{z_1},{w_1}} \right) = x_1^2 + y_1^2 + {{\left( {{z_1} - 2c} \right)}^2} + w_1^2,} \end{array} $

则有

$ \begin{array}{l} \mathop {\max G}\limits_{\left( {{x_1},{y_1},{z_1},{w_1}} \right) \in {\Gamma _1}} = \mathop {\max H}\limits_{\left( {{x_1},{y_1},{z_1},{w_1}} \right) \in {\Gamma _1}} = \\ \left\{ \begin{array}{l} \frac{{{a^4}}}{{{a^2} - {c^2}}},a \ge b,a \ge d,a \ge \sqrt 2 c,\\ \frac{{{b^4}}}{{{b^2} - {c^2}}},b > a,b \ge d,b \ge \sqrt 2 c,\\ \frac{{{d^4}}}{{{d^2} - {c^2}}},d > a,d > b,d \ge \sqrt 2 c,\\ 4{c^2},a < \sqrt 2 c,b < \sqrt 2 c,d < \sqrt 2 c. \end{array} \right. \end{array} $

证明  由多元函数求条件极值的拉格朗日乘数法即可证得.

引理2  对任意的λ>0, m>0, α>0, β>0, γ>0, c>0, 令

$ \begin{array}{l} \Gamma = \left\{ {\left( {x,y,z,w} \right)\left| {\frac{{\lambda {x^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }}}} + } \right.} \right.\\ \;\;\;\;\;\;\frac{{m{y^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \frac{{m\beta {{\left( {z - \frac{{\lambda \alpha + mc}}{{2m}}} \right)}^2}}}{{\frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \\ \;\;\;\;\;\;\left. {\frac{{\lambda \alpha \gamma {w^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} = 1} \right\}, \end{array} $

$ \begin{array}{l} V\left( {x,y,z,w} \right) = \lambda {x^2} + m{y^2} + m{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)^2} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\lambda \gamma {w^2},\;\;\;\;\;\forall \lambda > 0,m > 0, \end{array} $

则有

$ \mathop {\max V}\limits_{\left( {x,y,z,w} \right) \in \Gamma } = \left\{ \begin{array}{l} \frac{{{\beta ^2}{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha \left( {\beta - \alpha } \right)}},\;\;\;\;\alpha \le 1,\beta \ge 2\alpha ,\\ \frac{{{\beta ^2}{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\left( {\beta - 1} \right)}},\;\;\;\;\alpha > 1,\beta \ge 2,\\ \frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{m},\;\;\;\;\;\beta < 2\alpha ,\beta < 2. \end{array} \right. $

证明  令

$ \sqrt \lambda x = {x_1},\sqrt m y = {y_1},\sqrt m z = {z_1},\sqrt {\lambda \gamma } w = {w_1}, $

$ \frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }} = a_1^2,\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}} = b_1^2,\frac{{\lambda \alpha + mc}}{{2\sqrt m }} = {c_1}, $

$ \frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }} = d_1^2, $

则有

$ \begin{array}{l} V\left( {x,y,z,w} \right) = \lambda {x^2} + m{y^2} + \\ \;\;\;\;\;\;\;\;\;m{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)^2} + \lambda \gamma {w^2} = \\ \;\;\;\;\;\;\;\;\;x_1^2 + y_1^2 + {\left( {{z_1} - 2{c_1}} \right)^2} + w_1^2, \end{array} $

$ \begin{array}{l} \Gamma = \left\{ {\left( {x,y,z,w} \right)\left| {\frac{{\lambda {x^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }}}} + \frac{{m{y^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + } \right.} \right.\\ \;\;\;\;\;\;\left. {\frac{{m\beta {{\left( {z - \frac{{\lambda \alpha + mc}}{{2m}}} \right)}^2}}}{{\frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \frac{{{w^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4\lambda m\alpha \gamma }}}} = 1} \right\} = \\ \;\;\;\;\;\;\left\{ {\left( {{x_1},{y_1},{z_1},{w_1}} \right)\left| {\frac{{x_1^2}}{{{a^2}}} + \frac{{y_1^2}}{{{b^2}}} + \frac{{{{\left( {{z_1} - c} \right)}^2}}}{{{c^2}}} + } \right.} \right.\\ \;\;\;\;\;\;\left. {\frac{{w_1^2}}{{{d^2}}} = 1,a \ne 0,b \ne 0,c \ne 0,d \ne 0} \right\}. \end{array} $

由引理1即可得到结论.

引理3  定义

$ \sum { = \left\{ {\left( {{y_1},{z_1}} \right)\left| {\frac{{y_1^2}}{{{b^2}}} + \frac{{{{\left( {{z_1} - c} \right)}^2}}}{{{c^2}}} = 1,b \ne 0,c \ne 0} \right.} \right\}} , $

令

$ \begin{array}{*{20}{c}} {G\left( {{y_1},{z_1}} \right) = y_1^2 + z_1^2,\;\;\;H\left( {{y_1},{z_1}} \right) = y_1^2 + }\\ {{{\left( {{z_1} - 2c} \right)}^2},\;\;\;\;\left( {{y_1},{z_1}} \right) \in \Sigma ,} \end{array} $

则有

$ \mathop {\max G}\limits_{\left( {{y_1},{z_1}} \right) \in \Sigma } = \mathop {\max H}\limits_{\left( {{y_1},{z_1}} \right) \in \Sigma } = \left\{ \begin{array}{l} \frac{{{b^4}}}{{{b^2} - {c^2}}},\;\;\;\;b \ge \sqrt 2 c,\\ 4{c^2},\;\;\;\;\;b < \sqrt 2 c. \end{array} \right. $

证明  由多元函数求条件极值的拉格朗日乘数法便可证得.

定理1  对任意的λ>0, m>0, α>0, β>0, γ>0, c>0，

$ \begin{array}{l} {\Omega _{\lambda ,m}} = \left\{ {\left( {x,y,z,w} \right)\left| {\lambda {x^2} + m{y^2} + m{{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)}^2} + } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\lambda \gamma {w^2} \le {R^2},\forall \lambda > 0,\forall m > 0} \right\} \end{array} $

(2)

为系统(1)正半轨线的一个最终界和正向不变集.其中，

$ {R^2} = \left\{ \begin{array}{l} \frac{{{\beta ^2}{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha \left( {\beta - \alpha } \right)}},\;\;\;\alpha \le 1,\beta \ge 2\alpha ,\\ \frac{{{\beta ^2}{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\left( {\beta - 1} \right)}},\;\;\;\alpha > 1,\beta \ge 2,\\ \frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{m},\;\;\;\;\beta < 2\alpha ,\beta < 2. \end{array} \right. $

证明  定义广义李雅普诺夫函数

$ \begin{array}{l} V\left( {x,y,z,w} \right) = \lambda {x^2} + m{y^2} + \\ \;\;\;\;\;m{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)^2} + \lambda \gamma {w^2},\;\;\forall \lambda > 0,m > 0, \end{array} $

对上述函数求导有

$ \begin{array}{l} \frac{{{\text{d}}V\left( {x,y,z,w} \right)}}{{{\text{d}}t}}\left| {_{\left( 1 \right)}} \right. = 2\lambda x\frac{{{\text{d}}x}}{{{\text{d}}t}} + 2my\frac{{{\text{d}}y}}{{{\text{d}}t}} + \\ \;\;\;\;\;\;2m\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)\frac{{{\text{d}}z}}{{{\text{d}}t}} + 2\lambda \gamma w\frac{{{\text{d}}w}}{{{\text{d}}t}} = \\ \;\;\;\;\;\;2\lambda x\left( {\alpha y - \alpha x + \gamma w} \right) + 2my\left( {cx - xz - y} \right) + \\ \;\;\;\;\;\;2m\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)\left( {xy - \beta z} \right) + \\ \;\;\;\;\;\;2\lambda w\left( { - x - \alpha w} \right) = - 2\lambda \alpha {x^2} - 2m{y^2} - \\ \;\;\;\;\;\;2m\beta {z^2} + 2\beta \left( {\lambda \alpha + mc} \right)z - 2\lambda \gamma \alpha {w^2}, \end{array} $

令$ \frac{\text{d}V}{\text{d}t}=0 $, 可得到椭球面Γ₀：

$ \begin{array}{l} {\Gamma _0} = \left\{ {\left( {x,y,z,w} \right)\left| {\frac{{\lambda {x^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }}}} + \frac{{m{y^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + } \right.} \right.\\ \left. {\;\;\;\;\;\;\;\frac{{m{{\left( {z - \frac{{\lambda \alpha + mc}}{{2m}}} \right)}^2}}}{{\frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \frac{{{w^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4\lambda m\alpha \gamma }}}} = 1} \right\}, \end{array} $

在Γ₀外面有$ \frac{\text{d}V}{\text{d}t}<0 $, 而在Γ₀内部有$ \frac{\text{d}V}{\text{d}t}>0 $.从而函数V(x, y, z, w)只能在四维椭球面Γ₀上取得最大值.因V(x, y, z, w)为连续函数，且Γ₀为有界闭集，所以函数V(x, y, z, w)能够在四维椭球面Γ₀上取得最大值.

下面计算该最大值$ \underset{\left( x, y,z, w \right)\in {{\Gamma }_{0}}}{\mathop{\max }\limits}\, V\left( x, y,z, w \right)={{R}^{2}} $, 等价于求下面的条件极值问题：

$ \left\{ \begin{array}{l} \max \left\{ {\lambda {x^2} + m{y^2} + m{{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)}^2} + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\lambda \gamma {w^2},\left( {x,y,z,w} \right) \in {\Gamma _0}} \right\},\\ {\Gamma _0} = \left\{ {\left( {x,y,z,w} \right)\left| {\frac{{\lambda {x^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m\alpha }}}} + } \right.} \right.\\ \;\;\;\;\;\;\;\frac{{m{y^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \frac{{m{{\left( {z - \frac{{\lambda \alpha + mc}}{{2m}}} \right)}^2}}}{{\frac{{{{\left( {\lambda \alpha + mc} \right)}^2}}}{{4m}}}} + \\ \;\;\;\;\;\;\;\left. {\frac{{{w^2}}}{{\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{4\lambda m\alpha \gamma }}}} = 1} \right\}. \end{array} \right. $

由引理2便可得到结论.

容易证明式(2)为系统(1)正半轨线的一个最终界和正向不变集.

注1  (ⅰ) 令λ=1, m=1, 则

$ {\Omega _{1,1}} = \left\{ {\left( {x,y,z,w} \right)\left| {{x^2} + {y^2} + {{\left( {z - \alpha - c} \right)}^2} + \gamma {w^2} \le {l^2}} \right.} \right\} $

是系统(1)正半轨线的一个最终界和不变集，其中，

$ {l^2} = \left\{ \begin{array}{l} \frac{{{\beta ^2}{{\left( {\alpha + c} \right)}^2}}}{{4\alpha \left( {\beta - \alpha } \right)}},\;\;\;\;\alpha \le 1,\beta \ge 2\alpha ,\\ \frac{{{\beta ^2}{{\left( {\alpha + c} \right)}^2}}}{{4\left( {\beta - 1} \right)}},\;\;\;\;\alpha > 1,\beta \ge 2,\\ {\left( {\alpha + c} \right)^2},\;\;\;\;\beta < 2,\beta < 2\alpha . \end{array} \right. $

当c=26, β=0.7, γ=1.5, α=1时，则有

$ {\Omega _{1,1}} = \left\{ {\left( {x,y,z,w} \right)\left| {{x^2} + {y^2} + {{\left( {z - 27} \right)}^2} + \frac{3}{2}{w^2} \le {{27}^2}} \right.} \right\}. $

Ω_{1, 1}在xoyz空间中的投影如图 3所示.

(ⅱ) 基于集合交集的思想，对∀λ>0, ∀m>0, 则$ \bigcap\limits_{\lambda >0, m>0}{{{\Omega }_{\lambda, m}}} $为系统(1)的最终界和正向不变集.

定理2  令(x(t), y(t), z(t), w(t))为系统(1)的任意一个解.则对任意的α>0, β>0, γ>0, c>0，

$ \Phi = \left\{ {\left( {y,z} \right)\left| {{y^2} + {{\left( {z - c} \right)}^2} \le {l^2}} \right.} \right\} $

(3)

为系统(1)的y(t), z(t)的一个最终界.其中，

$ {l^2} = \left\{ \begin{array}{l} \frac{{{\beta ^2}{c^2}}}{{4\left( {\beta - 1} \right)}},\;\;\;\beta \ge 2,\\ {c^2},\;\;\;\beta < 2. \end{array} \right. $

证明  定义广义李雅普诺夫函数

$ {V_1}\left( {y,z} \right) = {y^2} + {\left( {z - c} \right)^2}, $

则有

$ \left\{ {\frac{{{\text{d}}{V_1}\left( {y,z} \right)}}{{{\text{d}}t}}\left| {_{\left( 1 \right)}} \right. = - 2{y^2} - 2\beta {z^2} + 2\beta cz} \right\}, $

令$ \frac{\text{d}{{V}_{1}}}{\text{d}t}=0$, 得到有界闭集Γ₂：

$ {\Gamma _2} = \left\{ {\left( {y,z} \right)\left| {\frac{{{y^2}}}{{\frac{{\beta {c^2}}}{4}}} + \frac{{{{\left( {z - \frac{c}{2}} \right)}^2}}}{{\frac{{{c^2}}}{4}}} = 1,\;\;\;\beta \ne 0,c \ne 0} \right.} \right\}, $

在Γ₂外有$ \frac{\text{d}{{V}_{1}}}{\text{d}t}<0 $，而在Γ₂内部，有$ \frac{\text{d}{{V}_{1}}}{\text{d}t}>0 $.从而函数V₁(y, z)只能在四维椭球面Γ₂上取得最大值.因V(x, y, z, w)为连续函数，且Γ₂为有界闭集，所以函数V₁(y, z)能够在椭球面Γ₂上取得最大值.

下面计算该最大值$ \underset{(y, z)\in {{\Gamma }_{2}}}{\mathop{\max }\limits}\, ~{{V}_{1}}(y, z)={{l}^{2}} $.等价于求条件极值问题：

$ \left\{ \begin{array}{l} \mathop {\max }\limits_{\left( {y,z} \right) \in {\Gamma _2}} {V_1}\left( {y,z} \right) = \max \left\{ {{y^2} + {{\left( {z - c} \right)}^2},\left( {y,z} \right) \in {\Gamma _2}} \right\},\\ {\text{s}}{\text{.t}}{\text{.}}\;\;\;\;{\Gamma _2} = \left\{ {\left( {y,z} \right)\left| {\frac{{{y^2}}}{{\frac{{\beta {c^2}}}{4}}} + \frac{{{{\left( {z - \frac{c}{2}} \right)}^2}}}{{\frac{{{c^2}}}{4}}} = 1,\;\;\;\beta \ne 0,c \ne 0} \right.} \right\}, \end{array} \right. $

令y=y₁, z=z₁, $ \frac{c}{2}={{c}_{1}} $, $ \frac{\beta {{c}^{2}}}{4}=b_{1}^{2} $, 由引理3即可得到结论.

注2  当c=26, β=0.7, γ=1.5, α=1时，有

$ \mathit{\Phi} =\left\{ \left( y,z \right)|{{y}^{2}}+{{\left( z-26 \right)}^{2}}\le {{26}^{2}} \right\}. $

由Φ可以得到系统(1)正半轨线在yoz平面上的最终界估计，见图 4.

由从吸引集外的轨线进入吸引集的速率估计，有

定理3  令X(t)=(x(t), y(t), z(t), w(t))为系统(1)的任意一个解.则对任意α>0, β>0, γ>0, c>0,

$ \begin{array}{l} \Omega = \left\{ {\left( {x,y,z,w} \right)\left| {\lambda {x^2} + m{y^2} + } \right.} \right.\\ \;\;\;\;\;\;m{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)^2} + \lambda \gamma {w^2} \le L,\\ \;\;\;\;\;\;\left. {\forall \lambda > 0,\forall m > 0} \right\} \end{array} $

(4)

为系统(1)的一个全局指数吸引集.其中，

$ L = \frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{{m\theta }},\;\;\;\theta = \min\left\{ {\alpha ,\beta ,1} \right\}. $

证明  定义广义李雅普诺夫函数

$ \begin{array}{*{20}{c}} {V\left( {x,y,z,w} \right) = \lambda {x^2} + m{y^2} + m{{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)}^2} + }\\ {\lambda \gamma {w^2},\;\;\;\;\forall \lambda > 0, m > 0,} \end{array} $

当V(X(t))>L, V(X(t₀))>L时, 有

$ \begin{array}{l} \left\{ {\frac{{{\text{d}}V}}{{{\text{d}}t}}\left| {_{\left( 1 \right)}} \right. = 2\lambda x\frac{{{\text{d}}x}}{{{\text{d}}t}} + 2my\frac{{{\text{d}}y}}{{{\text{d}}t}} + } \right.\\ \;\;\;2m\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)\frac{{{\text{d}}z}}{{{\text{d}}t}} + 2\lambda \gamma w\frac{{{\text{d}}w}}{{{\text{d}}t}} = \\ \;\;\;2\lambda x\left( {\alpha y - \alpha x + \gamma w} \right) + 2my\left( {cx - xz - y} \right) + \\ \;\;\;2m\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)\left( {xy - \beta z} \right) + 2\lambda \gamma w\left( { - x - \alpha w} \right) = \\ \;\;\; - 2\lambda \alpha {x^2} - 2m{y^2} - 2m\beta {z^2} + 2\beta \left( {\lambda \alpha + mc} \right)z - \\ \;\;\;2\lambda \alpha \gamma {w^2} = - \lambda \alpha {x^2} - m{y^2} - m\beta {z^2} + \\ \;\;\;2\beta \left( {\lambda \alpha + mc} \right)z - \gamma \lambda \alpha {w^2} - \lambda \alpha {x^2} - m{y^2} - m\beta {z^2} - \\ \;\;\;\lambda \alpha \gamma {w^2} \le - \lambda \alpha {x^2} - m{y^2} - m\beta {\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)^2} - \\ \;\;\; - \lambda \gamma \alpha {w^2} + \frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{m} \le - \theta V\left( {x,y,z,w} \right) + \\ \;\;\;\left. {\frac{{\beta {{\left( {\lambda \alpha + mc} \right)}^2}}}{m} = - \theta \left( {V\left( {x,y,z,w} \right) - L} \right) < 0} \right\}, \end{array} $

当V(X(t))>L, V(X(t₀))>L时, 有

$ \begin{array}{*{20}{c}} {V\left( {X\left( t \right)} \right) \le V\left( {X\left( {{t_0}} \right)} \right){{\text{e}}^{ - \theta \left( {t - {t_0}} \right)}} + \int_{{t_0}}^t {\theta L{{\text{e}}^{ - \theta \left( {t - \tau } \right)}}{\text{d}}\tau } = }\\ {V\left( {X\left( {{t_0}} \right)} \right){{\text{e}}^{ - \theta \left( {t - {t_0}} \right)}} + L\left( {1 - {{\text{e}}^{ - \theta \left( {t - {t_0}} \right)}}} \right),} \end{array} $

从而有

$ \left[ {V\left( {X\left( t \right)} \right) - L} \right] \le \left[ {V\left( {{X_0}} \right) - L} \right]{{\text{e}}^{ - \theta \left( {t - {t_0}} \right)}}. $

令t→+∞, 对上述不等式两边取上极限，有

$ \overline {\mathop {\lim }\limits_{t \to + \infty } } V\left( {X\left( t \right)} \right) \le L. $

从而有

$ \begin{array}{l} \Omega = \left\{ {\left( {x,y,z,w} \right)\left| {\lambda {x^2} + m{y^2} + m{{\left( {z - \frac{{\lambda \alpha + mc}}{m}} \right)}^2} + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\lambda \gamma {w^2} \le L,\forall \lambda > 0,\forall m > 0} \right\} \end{array} $

为系统(1)的一个全局指数吸引集.

证毕！

注3  取λ=1, m=1, 则

$ \Omega = \left\{ {\left( {x,y,z,w} \right)\left| {{x^2} + {y^2} + {{\left( {z - \alpha - c} \right)}^2} + \gamma {w^2} \le \delta } \right.} \right\} $

为系统(1)的一个全局指数吸引集，其中，

$ \delta = \frac{{\beta {{\left( {\alpha + c} \right)}^2}}}{\theta },\;\;\;\theta = \min\left\{ {\alpha ,\beta ,1} \right\}. $

2 结论

研究了一类大气混沌模型的全局吸引性和最终界.本研究方法亦适用于其他混沌系统；研究结果对于该混沌系统的混沌控制及其应用有一定的参考价值.