140
Z. Zheng
shown in (4.11). In other words, the slow mode dominates the evolution of the system
and of course, the fast mode can be reduced in terms of adiabatic elimination, leaving
only the equation of the slow mode. Therefore, the slow mode will determine the
dynamical tendency of the system (7) in the vicinity of the critical point and can be
identified as the order parameter. The emergence of order parameters in a complex
system is the central point of the slaving principle, which was proposed by Hermann
Haken [21].
The above discussion exhibits a typical competition of two modes in a twodimensional dynamical system. The insightful thought embedded in the slaving
principle can be naturally extended to complex systems with a large number of
competing modes. Suppose an n-dimensional dynamical system ˙
x = F(α, x), where
α is the controlling parameter. The equations of motion can be written in the following
canonical form near the critical point x = 0:
˙
x = A(α)x + B(x,α),
(4.14)
where x is an n-dimensional state vector x = (x 1 , x 2 , . . . , x n )
T , A = A(α) is an n×n
Jacobian matrix, and B(x, α) is an n-dimensional nonlinear function vector of x. The
eigenvalues of the matrix A are {λ i (α)}, which are aligned as the descending order
according to their real parts, i.e.
Reλ 1 ≥ Reλ 2 ≥ . . . ≥ Reλ n .
Assume that x = 0 is a stable solution of (4.14) in a certain parameter regime of
α, i.e. the real parts of all eigenvalues are negative,
Reλ i (α) < 0, i = 1, . . . , n.
(4.15)
By modulating the parameter α to a critical point, say, α > α c , when Re λ 1
changes from negative to positive, i.e.
0 < Reλ 1 << 1,
(4.16)
and other eigenvalues {Reλ 2 , Reλ 3 , . . . , Reλ n } remain negative. In this case the solution x = 0 becomes unstable. By introducing the linear transformation matrixT of
the Jacobian A as
˜
A = T
−1 AT
(4.17)
so that the new matrix is diagonalized as.
Z. Zheng
shown in (4.11). In other words, the slow mode dominates the evolution of the system
and of course, the fast mode can be reduced in terms of adiabatic elimination, leaving
only the equation of the slow mode. Therefore, the slow mode will determine the
dynamical tendency of the system (7) in the vicinity of the critical point and can be
identified as the order parameter. The emergence of order parameters in a complex
system is the central point of the slaving principle, which was proposed by Hermann
Haken [21].
The above discussion exhibits a typical competition of two modes in a twodimensional dynamical system. The insightful thought embedded in the slaving
principle can be naturally extended to complex systems with a large number of
competing modes. Suppose an n-dimensional dynamical system ˙
x = F(α, x), where
α is the controlling parameter. The equations of motion can be written in the following
canonical form near the critical point x = 0:
˙
x = A(α)x + B(x,α),
(4.14)
where x is an n-dimensional state vector x = (x 1 , x 2 , . . . , x n )
T , A = A(α) is an n×n
Jacobian matrix, and B(x, α) is an n-dimensional nonlinear function vector of x. The
eigenvalues of the matrix A are {λ i (α)}, which are aligned as the descending order
according to their real parts, i.e.
Reλ 1 ≥ Reλ 2 ≥ . . . ≥ Reλ n .
Assume that x = 0 is a stable solution of (4.14) in a certain parameter regime of
α, i.e. the real parts of all eigenvalues are negative,
Reλ i (α) < 0, i = 1, . . . , n.
(4.15)
By modulating the parameter α to a critical point, say, α > α c , when Re λ 1
changes from negative to positive, i.e.
0 < Reλ 1 << 1,
(4.16)
and other eigenvalues {Reλ 2 , Reλ 3 , . . . , Reλ n } remain negative. In this case the solution x = 0 becomes unstable. By introducing the linear transformation matrixT of
the Jacobian A as
˜
A = T
−1 AT
(4.17)
so that the new matrix is diagonalized as.