Regularity and complexity in dynamical systems

Chapter 2
Nonlinear Discrete Dynamical Systems

In this chapter, the basic concepts of nonlinear discrete systems will be presented. The
Local and global theory of stability and bifurcation for nonlinear discrete systems
will be discussed. The stability switching and bifurcation on specific eigenvectors
of the linearized system at fixed points under specific period will be presented.
The higher singularity and stability for nonlinear discrete systems on the specific
eigenvectors will be developed. A few special cases in the lower dimensional maps
will be presented for a better understanding of the generalized theory. The route to
chaos will be discussed briefly, and the intermittency phenomena relative to specific
bifurcations will be presented. The normalization group theory for 2-D discrete
systems will be presented via Duffing discrete systems.

2.1 Discrete Dynamical Systems

Definition 2.1 For α ⊆ R n and ⊆ R m with α ∈ Z, consider a vector function
fα : α × → α which is C r (r ≥ 1)-continuous, and there is a discrete (or
difference) equation in the form of

xk+1 = fα (xk , pα ) for xk , xk+1 ∈ α, k ∈ Z and pα ∈ . (2.1)

With an initial condition of xk = x0 , the solution of Eq. (2.1) is given by

xk = fα (fα (· · · (fα (x0 , pα ))))
k (2.2)
for xk ∈ α, k ∈ Z and p ∈ .

(i) The difference equation with the initial condition is called a discrete dynam-
ical system.
(ii) The vector function fα (xk , pα ) is called a discrete vector field on domain α .
(iii) The solution xk for each k ∈ Z is called a flow of discrete dynamical system.

A. C. J. Luo, Regularity and Complexity in Dynamical Systems, 63
DOI: 10.1007/978-1-4614-1524-4_2, © Springer Science+Business Media, LLC 2012

64 2 Nonlinear Discrete Dynamical Systems

Fig. 2.1 Maps and vector functions on each sub-domain for discrete dynamical system

(iv) The solution xk for all k ∈ Z on domain α is called the trajectory, phase
curve or orbit of discrete dynamical system, which is deﬁned as

= xk |xk+1 = fα (xk , pα ) for k ∈ Z and pα ∈ ⊆ ∪α α. (2.3)

(v) The discrete dynamical system is called a uniform discrete system if

xk+1 = fα (xk , pα ) = f(xk , p) for k ∈ Z and xk ∈ α (2.4)

Otherwise, this discrete dynamical system is called a non-uniform discrete
system.
Deﬁnition 2.2 For the discrete dynamical system in Eq. (2.1), the relation between
state xk and state xk+1 (k ∈ Z) is called a discrete map if

fα
Pα : xk −→ xk+1 and xk+1 = Pα xk (2.5)

with the following properties:

fα1 ,fα2 ,··· ,fαn
P(k,n) : xk − − − − → xk+n and xk+n = Pαn ◦ Pαn−1 ◦ · · · ◦ Pα1 xk
−−−− (2.6)

where

P(k;n) = Pαn ◦ Pαn−1 ◦ · · · ◦ Pα1 . (2.7)

If Pαn = Pαn−1 = · · · = Pα1 = Pα , then

(n)
P(α;n) ≡ Pα = Pα ◦ Pα ◦ · · · ◦ Pα (2.8)

with
(n) (n−1) (0)
Pα = Pα ◦ Pα and Pα = I. (2.9)

The total map with n-different submaps is shown in Fig. 2.1. The map Pαk with
the relation function fαk (αk ∈ Z) is given by Eq. (2.5). The total map P(k,n) is
given in Eq. (2.7). The domains αk (αk ∈ Z) can fully overlap each other or can be
completely separated without any intersection.

2.1 Discrete Dynamical Systems 65

Definition 2.3 For a vector function in fα ∈ R n , fα : R n → R n . The operator
norm of fα is defined by
n
||fα || = |fα(i) (xk , pα )|. (2.10)
i=1

For an n × n matrix fα (xk , pα ) = Aα xk and Aα = (aij )n×n , the corresponding norm
is defined by
n
||Aα || = |aij |. (2.11)
i,j=1

Definition 2.4 For α ⊆ R n and ⊆ R m with α ∈ Z, the vector function
fα (xk , pα ) with fα : α × →R n is differentiable at x ∈
k α if

∂fα (xk , pα ) fα (xk + xk , pα ) − fα (xk , pα )
= lim . (2.12)
∂xk (xk ,p) xk →0 xk

∂fα /∂xk is called the spatial derivative of fα (xk , pα ) at xk , and the derivative is
given by the Jacobian matrix

∂fα (xk , pα ) ∂fα(i)
= . (2.13)
∂xk ∂xk(j) n×n

Definition 2.5 For α ⊆ R n and ⊆ R m , consider a vector function f(xk , p)
with f : α × → R n, where x ∈
k α and p ∈ with k ∈ Z. The vector function
f(xk , p) satisfies the Lipschitz condition

||f(yk , p) − f(xk , p)|| ≤ L||yk − xk || (2.14)

with xk , yk ∈ α and L a constant. The constant L is called the Lipschitz constant.

2.2 Fixed Points and Stability

Definition 2.6 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p) in
Eq. (2.4).
∗
(i) A point xk ∈ α is called a fixed point or a period-1 solution of a discrete
∗
nonlinear system xk+1 = f(xk , p) under map Pk if for xk+1 = xk = xk
∗ ∗
xk = f(xk , p) (2.15)

The linearized system of the nonlinear discrete system xk+1 = f(xk , p) in
∗
Eq. (2.4) at the fixed point xk is given by


∗ ∗
yk+1 = DP(xk , p)yk = Df(xk , p)yk (2.16)

where
∗ ∗
yk = xk − xk and yk+1 = xk+1 − xk+1 . (2.17)

(ii) A set of points xj∗ ∈ αj ( αj ∈ Z) is called the fixed point set or period-1
point set of the total map P(k;n) with n-different submaps in nonlinear discrete
system of Eq. (2.2) if
∗ ∗
xk+j+1 = fαj (xk+j , pαj ) for j ∈ Z+ and j = mod(j, n) + 1;
∗ ∗ (2.18)
xk+ mod (j,n) = xk .

The linearized equation of the total map P(k;n) gives
∗ ∗
yk+j+1 = DPαj (xk+j , pαj )yk+j = Dfαj (xk+j , pαj )yk+j
∗
with yk+j+1 = xk+j+1 − xk+j+1 and yk+j = xk+j − xk+j ∗ (2.19)
for j ∈ Z+ and j = mod(j, n) + 1.

The resultant equation for the total map is
∗
yk+j+1 = DP(k,n) (xk , p)yk+j for j ∈ Z+ (2.20)

where
∗ 1 ∗
DP(k,n) (xk , p) = DPαj (xk+j−1 , p)
j=n
∗ ∗ ∗
= DPαn (xk+n−1 , pαn ) · · · · · DPα2 (xk+1 , pα2 ) · DPα1 (xk , pα1 )
∗ ∗ ∗
= Dfαn (xk+n−1 , pαn ) · · · · · Dfα2 (xk+1 , pα2 ) · Dfα1 (xk , pα1 ).
(2.21)
∗
The fixed point xk lies in the intersected set of two domains k and k+1 , as
∗
shown in Fig. 2.2. In the vicinity of the fixed point xk , the incremental relations in the
two domains k and k+1 are different. In other words, setting yk = xk − xk and ∗

yk+1 = xk+1 − xk+1 ∗ , the corresponding linearization is generated as in Eq. (2.16).

Similarly, the fixed point of the total map with n-different submaps requires the
intersection set of two domains k and k+n , which are a set of equations to obtain
the fixed points from Eq. (2.18). The other values of fixed points lie in different
domains, i.e., xj∗ ∈ j (j = k + 1, k + 2, · · · , k + n − 1), as shown in Fig. 2.3.
The corresponding linearized equations are given in Eq. (2.19). From Eq. (2.20),
the local characteristics of the total map can be discussed as a single map. Thus,
the dynamical characteristics for the fixed point of the single map will be discussed
comprehensively, and the fixed points for the resultant map are applicable. The results
can be extended to any period-m flows with P (m) .
Definition 2.7 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p)
∗
in Eq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinear

2.2 Fixed Points and Stability 67

Fig. 2.2 A fixed point
between domains k and
k+1 for a discrete
dynamical system

Fig. 2.3 Fixed points with n-maps for a discrete dynamical system

∗ ∗
system in the neighborhood of xk is yk+1 = Df(xk , p)yk (yl = xl − xk and ∗

l = k, k + 1) in Eq. (2.16). The matrix Df(xk ∗ , p) possesses n real eigenvalues
1
|λj | < 1 (j ∈ N1 ), n2 real eigenvalues |λj | > 1 (j ∈ N2 ), n3 real eigenvalues
λj = 1 (j ∈ N3 ), and n4 real eigenvalues λj = −1 (j ∈ N4 ). N = {1, 2, · · · , n} and
Ni = {l1 , l2 , · · · , lni } ∪ ∅ (i = 1, 2, 4) with lm ∈ N (m = 1, 2, · · · , ni ). Ni ⊆ N ∪ ∅,
∪3 Ni = N, ∩3 Ni = ∅ and i=1 ni = n. The corresponding eigenvectors for con-
i=1 i=1
3

traction, expansion, invariance and flip oscillation are {vji } (ji ∈ Ni ) (i = 1, 2, 3, 4),
respectively. The stable, unstable, invariant and flip subspaces of yk+1 = Df(xk , p)yk ∗

in Eq. (2.16) are linear subspace spanned by {vji } (ji ∈ Ni ) (i = 1, 2, 3, 4),
respectively, i.e.,
∗
(Df(xk , p) − λj I)vj = 0,
E s = span vj ;
|λj | < 1, j ∈ N1 ⊆ N ∪ ∅

∗
(Df(xk , p) − λj I)vj = 0,
E u = span vj ;
|λj | > 1, j ∈ N2 ⊆ N ∪ ∅
(2.22)
∗
(Df(xk , p) − λj I)vj = 0,
E i = span vj ;
λj = 1, j ∈ N3 ⊆ N ∪ ∅

∗
(Df(xk , p) − λj I)vj = 0,
E f = span vj .
λj = −1, j ∈ N4 ⊆ N ∪ ∅


where

E s = Em ∪ Eos with
s

∗
(Df(xk , p) − λj I)vj = 0,
Em = span vj
s ;
0 < λj < 1, j ∈ N1 ⊆ N ∪ ∅
m (2.23)

∗
(Df(xk , p) − λj I)vj = 0,
Eos = span vj ;
−1 < λj < 0, j ∈ N1 ⊆ N ∪ ∅
o

E u = Em ∪ Eou with
u

∗
(Df(xk , p) − λj I)vj = 0,
Em = span vj
u ;
λj > 1, j ∈ N2 ⊆ N ∪ ∅
m (2.24)

∗
(Df(xk , p) − λj I)vj = 0,
Eos = span vj ;
−1 < λj , j ∈ N2 ⊆ N ∪ ∅
o

where subscripts “m” and “o” represent the monotonic and oscillatory evolutions.
Definition 2.8 Consider a 2n-dimensional, discrete, nonlinear dynamical system
∗
xk+1 = f(xk , p) in Eq. (2.4) with a fixed point xk . The linearized system of the
∗ ∗
discrete nonlinear system in the neighborhood of xk is yk+1 = Df(xk , p)yk (yl =
xl −xk∗ and l = k, k+1) in Eq. (2.16). The matrix Df(x∗ , p) has complex eigenvalues
k
αj ± iβj with eigenvectors uj ± ivj (j ∈ {1, 2, · · · , n}) and the base of vector is

B = u1 , v1 , · · · , uj , vj , uj+1 , · · · , un , vn . (2.25)
∗
The stable, unstable and center subspaces of yk+1 = Dfk (xk , p)yk in Eq. (2.16)
are linear subspaces spanned by {uji , vji } (ji ∈ Ni , i = 1, 2, 3), respectively.
N = {1, 2, · · ·, n} plus Ni = {l1 , l2 , · · · , lni } ∪ ∅ ⊆ N ∪ ∅ with lm ∈ N (m =
1, 2, · · · , ni ). ∪3 Ni = N with ∩3 Ni = ∅ and i=1 ni = n. The stable, unstable
i=1 i=1
3

and center subspaces of yk+1 = Df(xk ∗ , p)y in Eq. (2.16) are defined by
k
⎧ ⎫
⎪
⎨ rj = αj2 + βj2 < 1, ⎪
⎬
E s = span (uj , vj ) Df(xk , p) − (αj ± iβj )I (uj ± ivj ) = 0, ⎪ ;
∗
⎪
⎩ ⎭
j ∈ N1 ⊆ {1, 2, · · · , n} ∪ ∅
⎧ ⎫
⎪
⎨ rj = αj2 + βj2 > 1, ⎪
⎬
E u = span (uj , vj ) Df(xk , p) − (αj ± iβj )I (uj ± ivj ) = 0, ⎪ ;
∗
⎪
⎩ ⎭
j ∈ N2 ⊆ {1, 2, · · · , n} ∪ ∅

⎧ ⎫
⎪
⎨ rj = αj2 + βj2 = 1, ⎪
⎬
E c = span (u , v )
j j Df(xk∗ , p) − (α ± iβ )I (u ± iv ) = 0, . (2.26)
⎪
⎩ j j j j ⎪
⎭
j ∈ N3 ⊆ {1, 2, · · · , n} ∪ ∅

Definition 2.9 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p) in
∗
Eq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinear system
∗ ∗ ∗
in the neighborhood of xk is yk+1 = Df(xk , p)yk (yl = xl − xk and l = k, k + 1)
in Eq. (2.16). The fixed point or period-1point is hyperbolic if no eigenvalues of
∗
Df(xk , p) are on the unit circle (i.e., |λj | = 1 for j = 1, 2, · · · , n).
Theorem 2.1 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p) in
∗
Eq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinear system
∗ is y ∗ ∗
in the neighborhood of xk k+1 = Df(xk , p)yk (yl = xl − xk and l = k, k + 1) in
Eq. (2.16). The eigenspace of Df(xk ∗ , p) (i.e., E ⊆ R n ) in the linearized dynamical

system is expressed by direct sum of three subspaces

E = Es ⊕Eu ⊕Ec (2.27)

where E s , E u and E c are the stable, unstable and center subspaces, respectively.
Proof This proof is the same as the linear system in Appendix B.
∗
in Eq. (2.4) with a fixed point xk . Suppose there is a neighborhood of the equilibrium
∗ as U (x∗ ) ⊂
xk k k k , and in the neighborhood,

∗ ∗
||f(xk + yk , p) − Df(xk , p)yk ||
lim =0 (2.28)
||yk ||→0 ||yk ||

and
∗
yk+1 = Df(xk , p)yk . (2.29)

(i) A C r invariant manifold
∗ ∗ ∗
Sloc (xk , xk ) = {xk ∈ U (xk )| lim xk+j = xk and xk+j
j→+∞
(2.30)
∗
∈ U (xk ) with j ∈ Z+ }
∗
is called the local stable manifold of xk , and the corresponding global stable
manifold is defined as
∗ ∗
S (xk , xk ) = ∪j∈Z− f(Uloc (xk+j , xk+j ))
(2.31)
= ∪j∈Z− f (j) (Uloc (xk , xk )).
∗


∗
(ii) A C r invariant manifold Uloc (xk , xk )
∗ ∗ ∗
Uloc (xk , xk ) = {xk ∈ U (xk )| lim xk+j = xk and xk+j
j→−∞
(2.32)
∗
∈ U (xk ) with j ∈ Z− }

is called the unstable manifold of x∗ , and the corresponding global stable
manifold is defined as
∗ ∗
U (xk , xk ) = ∪j∈Z+ f(Uloc (xk+j , xk+j ))
(2.33)
= ∪j∈Z+ f (j) (Uloc (xk , xk )).
∗

(iii) A C r−1 invariant manifold Cloc (x, x∗ ) is called the center manifold of x∗ if
Cloc (x, x∗ ) possesses the same dimension of E c for x∗ ∈ S (x, x∗ ), and the
tangential space of Cloc (x, x∗ ) is identical to E c .
As in continuous dynamical systems, the stable and unstable manifolds are unique,
but the center manifold is not unique. If the nonlinear vector field f is C ∞ -continuous,
then a C r center manifold can be found for any r < ∞.
Theorem 2.2 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p)
∗
in Eq. (2.4) with a hyperbolic fixed point xk . The corresponding solution is xk+j =
f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the hyperbolic fixed
∗ ∗ ∗
point xk (i.e., Uk (xk ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk ). The
linearized system is yk+j+1 = Df(xk ∗ , p)y ∗ and l = k + j, k + j + 1)
k+j (yl = xl − xk
∗
in Uk (xk ). If the homeomorphism between the local invariant subspace E(xk ) ⊂ ∗

U (xk∗ ) and the eigenspace E of the linearized system exists with the condition in

Eq. (2.28), the local invariant subspace is decomposed by
∗ ∗ ∗
E(xk , xk ) = Sloc (xk , xk ) ⊕ Uloc (xk , xk ). (2.34)

(a) The local stable invariant manifold Sloc (x, x∗ ) possesses the following
properties:
∗ ∗ ∗
(i) for xk ∈ Sloc (xk , xk ), Sloc (xk , xk ) possesses the same dimension of E s and
the tangential space of Sloc (xk , xk ∗ ) is identical to E s ;
∗ ∗ ∗
(ii) for xk ∈ Sloc (xk , xk , xk+j ∈ Sloc (xk , xk ) and lim xk+j = xk for all j ∈ Z+ ;
j→∞
∗ ∗
(iii) for xk ∈ Sloc (xk , xk , ||xk+j − xk || ≥ δ for δ > 0 with j, j1 ∈ Z+ and j ≥
/
j1 ≥ 0.
∗
(b) The local unstable invariant manifold Uloc (xk , xk ) possesses the following prop-
erties:
∗ ∗ ∗
(i) for xk ∈ Uloc (xk , xk ), Uloc (xk , xk ) possesses the same dimension of E u and
the tangential space of Uloc (xk , xk ∗ ) is identical to E u ;
∗ ∗
(ii) for xk ∈ Uloc (xk , xk ), xk+j ∈ Uloc (xk , xk ) and lim xk+j = x∗ for all
j→−∞
j ∈ Z−


∗
(iii) for xk ∈ Uloc (x, x∗ ), ||xk+j −xk || ≥ δ for δ > 0 with j1 , j ∈ Z− and j ≤ j1 ≤ 0.
/
Proof See Hirtecki (1971).
∗
Eq. (2.4) with a fixed point xk . The corresponding solution is xk+j = f(xk+j−1 , p)
∗
with j ∈ Z. Suppose there is a neighborhood of the fixed point xk (i.e., Uk (xk ) ⊂ ∗
r (r ≥ 1)−continuous in U (x∗ ). The linearized system is
α ), and f(xk , p) is C k k
∗ ∗ ∗
yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) in Uk (xk ). If the homeomorphism
between the local invariant subspace E(xk ∗ ) ⊂ U (x∗ ) and the eigenspace E of the
k
linearized system exists with the condition in Eq. (2.28), in addition to the local stable
∗
and unstable invariant manifolds, there is a C r−1 center manifold Cloc (xk , xk ). The
center manifold possesses the same dimension of E c for x∗ ∈ C (x , x∗ ), and the
loc k k
tangential space of Cloc (x, x∗ ) is identical to E c . Thus, the local invariant subspace
is decomposed by
∗ ∗ ∗ ∗
E(xk , xk ) = Sloc (xk , xk ) ⊕ Uloc (xk , xk ) ⊕ Cloc (xk , xk ). (2.35)

Proof See Guckenhiemer and Holmes (1990).
in Eq. (2.4) on domain α ⊆ R n . Suppose there is a metric space ( α , ρ), then the
map P under the vector function f(xk , p) is called a contraction map if
(1) (2) (1) (2) (1) (2)
ρ(xk+1 , xk+1 ) = ρ(f(xk , p), f(xk , p)) ≤ λρ(xk , xk ) (2.36)

(1) (2) (1) (2) (1) (2)
for λ ∈ (0, 1) and xk , xk ∈ α with ρ(xk , xk ) = ||xk − xk ||.
Eq. (2.4) on domain α ⊆ R n . Suppose there is a metric space ( α , ρ), if the map
P under the vector function f(xk , p) is a contraction map, then there is a unique fixed
∗
point xk which is globally stable.
Proof Consider
(1) (2) (1) (2) (1) (2)
ρ(xk+j+1 , xk+j+1 ) = ρ(f(xk+j , p), f(xk+j , p)) ≤ λρ(xk+j , xk+j )
(1) (2) (1) (2)
= λρ(f(xk+j−1 , p), f(xk+j−1 , p)) ≤ λ2 ρ(xk+j−1 , xk+j−1 )
.
.
.
(1) (2) (1) (2)
= λj−1 ρ(f(xk+1 , p), f(xk+1 , p)) ≤ λj ρ(xk , xk )

As j → ∞ and 0 < λ < 1, thus, we have
(1) (2) (1) (2)
lim ρ(xk+j+1 , xk+j+1 ) = lim λj ρ(xk , xk ) = 0
j→∞ j→∞


(2) (2)
∗
If xk+j+1 = xk = xk , in domain α ∈ R n , we have

(1) (2) ∗ (1)
lim ρ(xk+j+1 , xk+j+1 ) = lim ||xk+j+1 − xk || = 0
j→∞ j→∞

∗ ∗
Consider two fixed points xk1 and xk2 . The above equation gives
∗ ∗ ∗ ∗
||xk1 − xk2 || = lim ||xk1 − xk+j+1 + xk+j+1 − xk2 ||
j→∞
∗ ∗
≤ lim ||xk1 − xk+j+1 || + lim ||xk+j+1 − xk2 || = 0
j→∞ j→∞

Therefore, the fixed point is unique and globally stable. This theorem is proved.
∗
in Eq. (2.4) with a fixed point xk . The corresponding solution is given by xk+j =
∗
f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point xk (i.e.,
Uk (xk∗) ⊂ r (r ≥ 1)-continuous in U (x∗ ). The linearized
∗ ∗ ∗
system is yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) in Uk (xk ). Consider a
real eigenvalue λi of matrix Df(xk ∗ , p) (i ∈ N = {1, 2, · · · , n}) and there is a
(i) (i)
corresponding eigenvector vi . On the invariant eigenvector vk = vi , consider yk =
(i) (i) (i) (i) (i) (i)
ck vi and yk+1 = ck+1 vi = λi ck vi , thus, ck+1 = λi ck .
(i)
(i) xk on the direction vi is stable if

(i) (i)
lim |ck | = lim |(λi )k | × |c0 | = 0 for |λi | < 1. (2.37)
k→∞ k→∞

(i)
(ii) xk on the direction vi is stable if

(i) (i)
lim |ck | = lim |(λi )k | × |c0 | = ∞ for |λi | > 1. (2.38)
k→∞ k→∞

(i)
(iii) xk on the direction vi is invariant if

(i) (i) (i)
lim c = lim (λi )k c0 = c0 for λi = 1. (2.39)
k→∞ k k→∞

(i)
(iv) xk on the direction vi is flipped if

(i) (i) (i)
⎫
lim ck = lim (λi )2k × c0 = c0 ⎬
2k→∞
(i)
2k→∞
(i) (i) for λi = −1. (2.40)
lim ck = lim (λi )2k+1 × c0 = −c0 ⎭
2k+1→∞ 2k+1→∞

(i)
(v) xk on the direction vi is degenerate if

(i) (i)
ck = (λi )k c0 = 0 for λi = 0. (2.41)


∗
∗
∗) ⊂
Uk (xk r (r ≥ 1)-continuous in U (x∗ ). Consider a pair of
α ), and f(xk , p) is C
∗
k k
√
complex eigenvalues αi ± iβi of matrix Df(xk , p) (i ∈ N = {1, 2, · · · , n}, i = −1)
and there is a corresponding eigenvector ui ± ivi . On the invariant plane of
(i) (i) (i) (i) (i)
(uk , vk ) = (ui , vi ), consider xk = xk+ + xk− with

(i) (i) (i) (i) (i) (i)
xk = ck ui + dk vi , xk+1 = ck+1 ui + dk+1 vi . (2.42)
(i) (i) (i)
Thus, ck = (ck , dk )T with
(i) (i) (i)
ck+1 = Ei ck = ri Ri ck (2.43)
where
αi βi cos θi sin θi
Ei = and Ri = ,
−βi αi − sin θi cos θi
(2.44)
ri = αi2 + βi2 , cos θi = αi /ri and sin θi = βi /ri ;

and
k
k αi βi cos kθi sin kθi
Ei = and Rik = . (2.45)
−βi αi − sin kθi cos kθi

(i)
(i) xk on the plane of (ui , vi ) is spirally stable if
(i) (i)
lim ||ck || = lim rik ||Rik || × ||c0 || = 0 for ri = |λi | < 1. (2.46)
k→∞ k→∞
(i)
(ii) xk on the plane of (ui , vi ) is spirally unstable if
(i) (i)
lim ||ck || = lim rik ||Rik || × ||c0 || = ∞ for ri = |λi | > 1. (2.47)
k→∞ k→∞
(i)
(iii) xk on the plane of (ui , vi ) is on the invariant circles if,

(i) (i) (i)
||ck || = rik ||Rik || × ||c0 || = ||c0 || for ri = |λi | = 1. (2.48)
(i)
(iv) xk on the plane of (ui , vi ) is degenerate in the direction of ui if βi = 0.
∗
∗
∗) ⊂
Uk (xk r (r ≥ 1)-continuous in U (x∗ ) with Eq. (2.28).
∗ ∗
The linearized system is yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) in Uk (xk ). ∗

The matrix Df(xk ∗ , p) possesses n eigenvalues λ (k = 1, 2, · · · n).
k


(i) ∗
The fixed point xk is called a hyperbolic point if |λi | = 1 (i = 1, 2, · · · , n).
(ii) ∗
The fixed point xk is called a sink if |λi | < 1 (i = 1, 2, · · · , n).
(iii) ∗
The fixed point xk is called a source if |λi | > 1(i ∈ {1, 2, · · · n}).
(iv) ∗
The fixed point xk is called a center if |λi | = 1 (i = 1, 2, · · · n) with distinct
eigenvalues.
∗
∗
∗) ⊂
∗ ∗

The matrix Df(xk ∗ , p) possesses n eigenvalues λ (i = 1, 2, · · · n).
i
∗
(i) The fixed point xk is called a stable node if |λi | < 1(i = 1, 2, · · · , n).
∗
(ii) The fixed point xk is called an unstable node if |λi | > 1 (i = 1, 2, · · · , n).
(iii) The fixed point xk ∗ is called an (l : l )-saddle if at least one |λ | > 1(i ∈ L ⊂
1 2 i 1
{1, 2, · · · n}) and the other |λj | < 1 (j ∈ L2 ⊂ {1, 2, · · · n}) with L1 ∪ L2 =
{1, 2, · · · , n} and L1 ∩ L2 = ∅.
∗
(iv) The fixed point xk is called an lth-order degenerate case if λi = 0 (i ∈ L ⊆
{1, 2, · · · n}).
∗
∗
∗) ⊂
∗ ∗

The matrix Df(xk ∗ , p) possesses n-pairs of complex eigenvalues λ (i = 1, 2, · · ·, n).
i
∗
(i) The fixed point xk is called a spiral sink if |λi | < 1 (i = 1, 2, · · ·, n) and
Im λj = 0 (j ∈ {1, 2, · · ·, n}).
∗
(ii) fixed point xk is called a spiral source if |λi | > 1 (i = 1, 2, · · ·, n) with Im λj =
0 (j ∈ {1, 2, · · ·, n}).
∗
(iii) fixed point xk is called a center if |λi | = 1 with distinct Im λi = 0 (i ∈
{1, 2, · · ·, n}).
As in Appendix B, the refined classification of the linearized, discrete, nonlinear
system at fixed points should be discussed. The generalized stability and bifurcation
of flows in linearized, nonlinear dynamical systems in Eq. (2.4) will be discussed as
follows.
∗
f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point xk (i.e., ∗

Uk (xk∗) ⊂ r (r ≥ 1)-continuous in U (x∗ ) with Eq. (2.28). The
∗ ∗
linearized system is yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) in Uk (xk ). The ∗

matrix Df(xk ∗ , p) possesses n eigenvalues λ (i = 1, 2, · · ·n). Set N = {1, 2, · · ·, n},
i
Np = {l1 , l2 , · · ·, lnp } ∪ ∅ with lqp ∈ N (qp = 1, 2, · · ·, np , p = 1, 2, · · ·, 7) and


p=1 np +2 p=5 np = n. ∪p=1 Np = N and ∩p=1 Np = ∅.Np = ∅ if nqp = 0.Nα = Nα
4 7 7 7 m

∪ Nα o (α = 1, 2) and N m ∩ N o = ∅ with nm + no = n , where superscripts “m” and
α α α α α
∗
“o” represent monotonic and oscillatory evolutions. The matrix Df(xk , p) possesses
n1 -stable, n2 -unstable, n3 -invariant and n4 -flip real eigenvectors plus n5 -stable, n6 -
unstable and n7 -center pairs of complex eigenvectors. Without repeated complex
eigenvalues of |λi | = 1(i ∈ N3 ∪ N4 ∪ N7 ), an iterative response of xk+1 = f(xk , p) is
an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| n5 : n6 : n7 ) flow in the neighborhood of the
m o m o

fixed point xk ∗ . With repeated complex eigenvalues of |λ | = 1 (i ∈ N ∪ N ∪ N ),
i 3 4 7
an iterative response of xk+1 = f(xk , p) is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]
m o m o

|n5 : n6 : [n7 , l; κ7 ]) flow in the neighborhood of the fixed point xk , where κp ∈ ∗

{∅, mp } (p = 3, 4, 7). The meanings of notations in the aforementioned structures
are defined as follows:
(i) [n1 , n1 ] represents that there are n1 -sinks with n1 -monotonic convergence
m o m

and n1 o -oscillatory convergence among n -directions of v (i ∈ N ) if |λ | < 1
1 i 1 i
(k ∈ N1 and 1 ≤ n1 ≤ n) with distinct or repeated eigenvalues.
(ii) [n2 , n2 ] represents that there are n2 -sources with n2 -monotonic diver-
m o m

gence and n2 o -oscillatory divergence among n -directions of v (i ∈ N ) if
2 i 2
|λi | > 1 (k ∈ N2 and 1 ≤ n2 ≤ n) with distinct or repeated eigenvalues.
(iii) n3 = 1 represents an invariant center on 1-direction of vi (i ∈ N3 ) if λk =
1 (i ∈ N3 and n3 = 1).
(iv) n4 = 1 represents a flip center on 1-direction of vi (i ∈ N4 ) if λi = −1 (i ∈
N4 and n4 = 1).
(v) n5 represents n5 -spiral sinks on n5 -pairs of (ui , vi ) (i ∈ N5 ) if |λi | < 1 and
Im λi = 0 (i ∈ N5 and 1 ≤ n5 ≤ n) with distinct or repeated eigenvalues.
(vi) n6 represents n6 -spiral sources on n6 -directions of (ui , vi ) (i ∈ N6 ) if |λi | > 1
and Im λi = 0 (i ∈ N6 and 1 ≤ n6 ≤ n) with distinct or repeated eigenvalues.
(vii) n7 represents n7 -invariant centers on n7 -pairs of (ui , vi ) (i ∈ N7 ) if |λi | = 1
and Im λi = 0 (i ∈ N7 and 1 ≤ n7 ≤ n) with distinct eigenvalues.
(viii) ∅ represents none if nj = 0 (j ∈ {1, 2, · · ·, 7}).
(ix) [n3 ; κ3 ] represents (n3 − κ3 ) invariant centers on (n3 − κ3 ) directions of
vi3 (i3 ∈ N3 ) and κ3 -sources in κ3 -directions of vj3 (j3 ∈ N3 and j3 = i3 )
if λi = 1 (i ∈ N3 and n3 ≤ n) with the (κ3 + 1)th-order nilpotent matrix
N33 +1 = 0 (0 < κ3 ≤ n3 − 1).
κ

(x) [n3 ; ∅] represents n3 invariant centers on n3 -directions of vi (i ∈ N3 ) if
λi = 1 (i ∈ N3 and 1 < n3 ≤ n) with a nilpotent matrix N3 = 0.
(xi) [n4 ; κ4 ] represents (n4 − κ4 ) flip oscillatory centers on (n4 − κ4 ) directions
of vi4 (i4 ∈ N4 ) and κ4 -sources in κ4 -directions of vj4 (j4 ∈ N4 and j4 = i4 )
if λi = −1 (i ∈ N4 and n4 ≤ n) with the (κ4 + 1)th-order nilpotent matrix
N44 +1 = 0 (0 < κ4 ≤ n4 − 1).
κ

(xii) [n4 ; ∅] represents n4 flip oscillatory centers on n4 -directions of vi (i ∈ N3 )
if λi = −1 (i ∈ N4 and 1 < n4 ≤ n) with a nilpotent matrix N4 = 0.
(xiii) [n7 , l; κ7 ] represents (n7 − κ7 ) invariant centers on (n7 − κ7 ) pairs of
(ui7 , vi7 )(i7 ∈ N7 ) and κ7 sources on κ7 pairs of (uj7 , vj7 ) (j7 ∈ N7 and
j7 = i7 ) if |λi | = 1 and Im λi = 0 (i ∈ N7 and n7 ≤ n) for (l + 1)


pairs of repeated eigenvalues with the (κ7 + 1)th-order nilpotent matrix
N77 +1 = 0 (0 < κ7 ≤ l).
κ

(xiv) [n7 , l; ∅] represents n7 -invariant centers on n7 -pairs of (ui , vi )(i ∈ N6 ) if
|λi | = 1 and Im λi = 0 (i ∈ N7 and 1 ≤ n7 ≤ n) for (l + 1) pairs of repeated
eigenvalues with a nilpotent matrix N7 = 0.
∗
f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point xk (i.e., ∗

Uk (xk∗) ⊂ r (r ≥ 1)-continuous in U (x∗ ) with Eq. (2.28). The
∗
linearized system is yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) in Uk (xk ). The ∗ ∗

matrix Df(xk ∗ , p) possesses n eigenvalues λ (i = 1, 2, · · ·n). Set N = {1, 2, · · ·, n},
i
Np = {l1 , l2 , · · ·, lnp } ∪ ∅ with lqp ∈ N (qp = 1, 2, · · ·, np , p = 1, 2, · · ·, 7) and
p=1 np + 2 p=5 np = n. ∪p=1 Np = N and ∩p=1 Np = ∅.Np = ∅ if nqp = 0. Nα =
4 7 7 7

Nα m ∪ N o (α = 1, 2) and N m ∩ N o = ∅ with nm + no = n , where super-
α α α α α α
scripts “m” and “o” represent monotonic and oscillatory evolutions. The matrix
∗
Df(xk , p) possesses n1 -stable, n2 -unstable, n3 -invariant, and n4 -flip real eigenvec-
tors plus n5 -stable, n6 -unstable and n7 -center pairs of complex eigenvectors. With-
out repeated complex eigenvalues of |λk | = 1(k ∈ N3 ∪ N4 ∪ N7 ), an iterative
response of xk+1 = f(xk , p) is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| n5 : n6 : n7 )
m o m o
∗
flow in the neighborhood of the fixed point xk . With repeated complex eigenval-
ues of |λi | = 1 (i ∈ N3 ∪ N4 ∪ N7 ), an iterative response of xk+1 = f(xk , p) is an
([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) flow in the neighborhood
m o m o
∗
of the fixed point xk , where κp ∈ {∅, mp }(p = 3, 4, 7).
I. Non-degenerate cases
∗
(i) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : ∅) hyperbolic
m o m o

point.
∗
(ii) The fixed point xk is an ([n1 , n1 ] : [∅, ∅] : ∅ : ∅|n5 : ∅ : ∅)-sink.
m o

(iii) The fixed point xk∗ is an ([∅, ∅] : [nm , no ] : ∅ : ∅|∅ : n : ∅)-source.
2 2 6
∗
(iv) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : ∅|∅ : ∅ : n/2)-circular
center.
∗
(v) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : ∅|∅ : ∅ : [n/2, l; ∅])-
circular center.
∗
(vi) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : ∅|∅ : ∅ : [n/2, l; m])-point.
∗
(vii) The fixed point xk is an ([n1 , n1 ] : [∅, ∅] : ∅ : ∅|n5 : ∅ : n7 )-point.
m o

(viii) The fixed point xk ∗ is an ([∅, ∅] : [nm , no ] : ∅ : ∅|∅ : n : n )-point.
2 2 6 7
∗
(ix) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : n7 )-point.
m o m o

II. Simple special cases
∗
(i) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n; ∅] : ∅|∅ : ∅ : ∅)-invariant
center (or static center).
∗
(ii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n; m] : ∅|∅ : ∅ : ∅)-point.
∗
(iii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : [n; ∅]|∅ : ∅ : ∅)-flip center.
∗
(iv) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : [n; m]|∅ : ∅ : ∅)-point.


∗
(v) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [n4 ; κ4 ]|∅ : ∅ : ∅)-
point.
∗
(vi) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [1; ∅] : [n4 ; κ4 ]|∅ : ∅ : ∅)-point.
∗
(vii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [1; ∅]|∅ : ∅ : ∅)-point.
∗
(viii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [∅; ∅]|∅ : ∅ : n7 )-
point.
∗
(ix) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [1; ∅] : [∅; ∅]|∅ : ∅ : n7 )-point.
∗ is an ([∅, ∅] : [∅, ∅] : [n ; κ ] : [∅; ∅]|∅ : ∅ : [n , l; κ ])-
(x) The fixed point xk 3 3 7 7
point.
∗
(xi) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [∅; ∅] : [n4 ; κ4 ]|∅ : ∅ : n7 )-
point.
∗
(xii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [∅; ∅] : [n4 ; κ4 ]|∅ : ∅ : [n7 , l; κ7 ])-
point.
∗
(xiii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [n4 ; κ4 ]|∅ : ∅ : n7 )-
point.
∗
(xiv) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [n4 ; κ4 ]|∅ : ∅ : [n7 , l; κ7 ])-
point.
III. Complex special cases
∗
(i) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [1; ∅] : [∅; ∅]|n5 : n6 : n7 )-
m o m o

point.
∗
(ii) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [1; ∅] : [∅; ∅]|n5 : n6 :
m o m o

[n7 , l; κ7 ])-point.
∗
(iii) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [∅; ∅] : [1; ∅]|n5 : n6 : n7 )-
m o m o

point.
∗
(iv) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [∅; ∅] : [1; ∅]|n5 : n6 :
m o m o

[n7 , l; κ7 ])-point.
∗
(v) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : n7 )-
m o m o

point.
∗
(vi) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 :
m o m o

[n7 , l; κ7 ])-point.
∗
(vii) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : n7 )-
m o m o

point.
∗
(viii) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 :
m o m o

[n7 , l; κ7 ])-point.
∗
f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point
∗ ∗
xk (i.e., Uk (xk ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk ) with ∗

Eq. (2.28). The linearized system is yk+j+1 = Df(xk ∗ , p)y ∗
k+j (yk+j = xk+j − xk )
in Uk (xk ∗ ). The matrix Df(x∗ , p) possesses n eigenvalues λ (i = 1, 2, · · ·n). Set N =
k i
{1, 2, · · ·, n}, Np = {l1 , l2 , · · ·, lnp }∪∅ with lqp ∈ N(qp = 1, 2, · · ·, np , p = 1, 2, 3,4)
and p=1 np = n. ∪4 Np = N and ∩4 Np = ∅. Np = ∅ if nqp = 0. Nα = Nα ∪
4
p=1 p=1
m


Nα (α = 1, 2) and Nα ∩ Nα = ∅ with nα + nα = nα where superscripts “m”
o m o m o

and “o” represent monotonic and oscillatory evolutions. The matrix Df(xk , p) pos- ∗

sesses n1 -stable, n2 -unstable, n3 -invariant and n4 -flip real eigenvectors. An iterative
response of xk+1 = f(xk , p) is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| flow in
m o m o

the neighborhood of the fixed point xk p ∗ . κ ∈ {∅, m } (p = 3, 4).
p

∗
(i) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅| saddle.
m o m o
∗ is an ([nm , no ] : [∅, ∅] : ∅ : ∅|-sink.
(ii) The fixed point xk 1 1
∗
(iii) The fixed point xk is an ([∅, ∅] : [n2 , n2 ] : ∅ : ∅|-source.
m o

II. Simple special cases
∗
(i) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n; ∅] : ∅|-invariant center (or
static center).
∗
(ii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n; m] : ∅|-point.
∗
(iii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : [n; ∅]|-flip center.
∗
(iv) The fixed point xk is an ([∅, ∅] : [∅, ∅] : ∅ : [n; m]|-point.
∗
(v) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [n4 ; κ4 ]|-point.
∗
(vi) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [1; ∅] : [n4 ; κ4 ]|-point.
∗
(vii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [1; ∅]|-point.
∗
(viii) The fixed point xk is an ([∅, ∅] : [∅, ∅] : [n3 ; κ3 ] : [∅; ∅]|-point.
∗
(ix) T The fixed point xk is an ([∅, ∅] : [∅, ∅] : [∅; ∅] : [n4 ; κ4 ]|-point.
III. Complex special cases
∗
(i) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [1; ∅] : [∅; ∅]|-point.
m o m o
∗ is an ([nm , no ] : [nm , no ] : [∅; ∅] : [1; ∅]|-point.
(ii) The fixed point xk 1 1 2 2
∗
(iii) The fixed point xk is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|-point.
m o m o

∗
∈ R 2n in Eq. (2.4) with a fixed point xk . The corresponding solution is given by
xk+j = f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point
∗ ∗
xk (i.e., Uk (xk ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk ) with ∗

in Uk (xk ∗ ). The matrix Df(x∗ , p) possesses 2n eigenvalues λ (i = 1, 2, · · ·, n). Set
k i
N = {1, 2, · · ·, n}, Np = {l1 , l2 , · · ·, lnp } ∪ ∅ with lqp ∈ N (qp = 1, 2, · · ·, np , p =
5, 6, 7) and p=5 np = n. ∪7 Np = N and ∩7 Np = ∅. if nqp = 0. The matrix
7
p=5 p=5
Df(xk ∗ , p) possesses n -stable, n -unstable and n -center pairs of complex eigenvec-
5 6 7
tors. Without repeated complex eigenvalues of |λk | = 1(k ∈ N7 ), an iterative response
of xk+1 = f(xk , p) is an |n5 : n6 : n7 ) flow. With repeated complex eigenvalues of
|λk | = 1 (k ∈ N7 ), an iterative response of xk+1 = f(xk , p) is an |n5 : n6 : [n7 , l; κ7 ])
∗
flow in the neighborhood of the fixed point xk , where κp ∈ {∅, mp }(p = 7).
∗
(i) The fixed point xk is an |n5 : n6 : ∅) spiral hyperbolic point.
∗
(ii) The fixed point xk is an |n : ∅ : ∅) spiral sink.


(iii) ∗
The fixed point xk is an |∅ : n : ∅) spiral source.
(iv) ∗
The fixed point xk is an |∅ : ∅ : n)-circular center.
(v) ∗
The fixed point xk is an |n5 : ∅ : n7 )-point.
(vi) ∗
The fixed point xk is an |∅ : n6 : n7 )-point.
(vii) ∗
The fixed point xk is an |n5 : n6 : n7 )-point.
II. Special cases
(i) ∗
The fixed point xk is an |∅ : ∅ : [n, l; ∅])-circular center.
(ii) ∗
The fixed point xk is an |∅ : ∅ : [n, l; m])-point.
(iii) ∗
The fixed point xk is an |n5 : ∅ : [n7 , l; κ7 ])-point.
(iv) ∗
The fixed point xk is an |∅ : n6 : [n7 , l; κ7 ])-point.
(v) ∗
The fixed point xk is an |n5 : n6 : [n7 , l; κ7 ])-point.

2.3 Bifurcation and Stability Switching

To understand the qualitative changes of dynamical behaviors of discrete systems
with parameters in the neighborhood of fixed points, the bifurcation theory for fixed
points of nonlinear dynamical system in Eq. (2.4) will be investigated.
∗
∗ ∗

in Uk (xk ∗ ). The matrix Df(x∗ , p) possesses n eigenvalues λ (i = 1, 2, · · ·n). Set
k i
N = {1, 2, · · ·, n}, Np = {l1 , l2 , · · ·, lnp } ∪ ∅ with lqp ∈ N (qp = 1, 2, · · ·, np ,
p = 1, 2, · · ·, 7) and p=1 np + 2 p=5 np = n. ∪7 Np = N and ∩7 Np = ∅.
4 7
p=5 p=1
Np = ∅ if nqp = 0. Nα = Nα m ∪ N o (α = 1, 2) and N m ∩ N o = ∅ with nm +no = n ,
α α α α α α
where superscripts “m” and “o” represent monotonic and oscillatory evolutions.
∗
The matrix Df(xk , p) possesses n1 -stable, n2 -unstable, n3 -invariant and n4 -flip real
eigenvectors plus n5 -stable, n6 -unstable and n7 -center pairs of complex eigenvectors.
Without repeated complex eigenvalues of |λk | = 1(k ∈ N3 ∪ N4 ∪ N7 ), an iterative
response of xk+1 = f(xk , p) is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| n5 : n6 : n7 )
m o m o
∗
flow in the neighborhood of the fixed point xk . With repeated complex eigenval-
ues of |λk | = 1 (k ∈ N3 ∪ N4 ∪ N7 ), an iterative response of xk+1 = f(xk , p) is an
([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) flow in the neighborhood
m o m o
∗
of the fixed point xk , where κp ∈ {∅, mp }(p = 3, 4, 7).
I. Simple switching and bifurcation
m o m o ∗
(i) An ([n1 , n1 ] : [n2 , n2 ] : 1 : ∅|n5 : n6 : ∅) state of the fixed point (xk0 , p0 ) is
a switching of the ([n1 1m , no + 1] : [nm , no ] : ∅ : ∅|n : n : ∅) spiral saddle
2 2 5 6
and ([n1 , n1 ] : [n2 + 1, n2 ] : ∅ : ∅|n5 : n6 : ∅) spiral saddle for the fixed
m o m o
∗
point (xk , p).


(ii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : 1|n5 : n6 : ∅) state of the fixed point (xk0 , p0 ) is
m o m o ∗

a switching of the ([n1 1 m , no + 1] : [nm , no ] : ∅ : ∅| n : n : ∅) spiral saddle
2 2 5 6
and ([n1 , n1 ] : [n2 , n2 + 1] : ∅ : ∅|n5 : n6 : ∅) spiral saddle for the fixed
m o m o
∗
point (xk , p).
(iii) An ([n1 1m , no ] : [∅, ∅] : 1 : ∅|n : ∅ : ∅) state of the fixed point (x∗ , p ) is a
5 k0 0
stable saddle-node bifurcation of the ([n1 +1, n1 ] : [∅, ∅] : ∅ : ∅|n5 : ∅ : ∅)
m o

spiral sink and ([n1 , n1 ] : [1, ∅] : ∅ : ∅|n5 : ∅ : ∅) spiral saddle for the fixed
m o

point (xk ∗ , p).

(iv) An ([n1 , n1 ] : [∅, ∅] : ∅ : 1|n5 : ∅ : ∅) state of the fixed point (xk0 , p0 ) is a
m o ∗

stable period-doubling bifurcation of the ([n1 1 m , no + 1] : [∅, ∅] : ∅ : ∅|n :
5
∅ : ∅) sink and ([n1 , n1 ] : [∅, 1] : ∅ : ∅|n5 : ∅ : ∅) spiral saddle for the
m o
∗
fixed point (xk , p).
(v) An ([∅, ∅] : [n2 , n2 ] : 1 : ∅|∅ : n6 : ∅) state of the fixed point (xk0 , p0 ) is
m o ∗

an unstable saddle-node bifurcation of the ([∅, ∅] : [n2 m + 1, no ] : ∅ : ∅|∅ :
2
n6 : ∅) spiral source and ([1, ∅] : [n2 , n2 ] : ∅ : ∅|∅ : n6 : ∅) spiral saddle
m o
∗
for the fixed point (xk , p).
(vi) An ([∅, ∅] : [n2 2 m , no ] : ∅ : 1|∅ : n : ∅) state of the fixed point (x∗ , p )
6 k0 0
is an unstable period-doubling bifurcation of the ([∅, ∅] : [n2 , n2 + 1] : ∅ : m o

∅|∅ : n6 : ∅) spiral source and ([∅, 1] : [n2 , n2 ] : ∅ : ∅|∅ : n6 : ∅) spiral
m o

saddle for the fixed point (xk ∗ , p).

(vii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : 1) state of the fixed point (xk0 , p0 ) is
m o m o ∗

a switching of the ([n1 1 m , no ] : [nm , no ] : ∅ : ∅|n + 1 : n : ∅) spiral saddle
2 2 5 6
and ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 + 1 : ∅) saddle for the fixed point
m o m o
∗
(xk , p).
(viii) An ([n1 , n1 ] : [∅, ∅] : ∅ : ∅|n5 : ∅ : 1) state of the fixed point (xk0 , p0 ) is a
m o ∗

Neimark bifurcation of the ([n1 1 m , no ] : [∅, ∅] : ∅ : ∅|n + 1 : ∅ : ∅) spiral
5
sink and ([n1 , n1 ] : [∅, ∅] : ∅ : ∅|n5 : 1 : ∅) spiral saddle for the fixed point
m o
∗
(xk , p).
(ix) An ([∅, ∅] : [n2 , n2 ] : ∅ : ∅|∅ : n6 : 1) state of the fixed point (xk0 , p0 ) is an
m o ∗

unstable Neimark bifurcation of the ([∅, ∅] : [n2 2 m , no ] : ∅ : ∅|∅ : n + 1 : ∅)
6
spiral source and ([∅, ∅] : [n2 , n2 ] : ∅ : ∅|1 : n6 : ∅) spiral saddle for the
m o
∗
(x) An ([n1 1 m , no ] : [nm , no ] : 1 : ∅|n : n : n ) state of the fixed point (x∗ , p )
2 2 5 6 7 k0 0
is a switching of the ([n1 + 1, n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : n7 ) state and
m o m o

([n1 , n1 ] : [n2 + 1, n2 ] : ∅ : ∅|n5 : n6 : n7 ) state for the fixed point (xk , p).
m o m o ∗

(xi) An ([n1 1 m , no ] : [nm , no ] : ∅ : 1|n : n : n ) state of the fixed point (x∗ , p )
2 2 5 6 7 k0 0
is a switching of the ([n1 , n1 + 1] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : n7 ) state and
m o m o

([n1 , n1 ] : [n2 , n2 + 1] : ∅ : ∅|n5 : n6 : n7 ) state for the fixed point (xk , p).
m o m o ∗

(xii) An ([n1 1 m , no ] : [nm , no ] : 1 : ∅|n : n : [n , l; ∅]) state of the fixed point
2 2 5 6 7
∗
(xk0 , p0 ) is a switching of the ([n1 + 1, n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 :
m o m o

[n7 , l; κ7 ]) state and ([n1 1m , no ] : [nm + 1, no ] : ∅ : ∅|n : n : [n , l; κ ]) state
2 2 5 6 7 7
∗
(xiii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : 1|n5 : n6 : [n7 , l; κ7 ]) state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm , no + 1] : [nm , no ] : ∅ : ∅|n : n :
(xk0 0 1 1 2 2 5 6

2.3 Bifurcation and Stability Switching 81

[n7 , l; κ7 ]) state and ([n1 , n1 ] : [n2 , n2 + 1] : ∅ : ∅|n5 : n6 : [n7 , l; κ7 ])) state
m o m o

for the fixed point (xk ∗ , p).

(xiv) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 : n7 + 1) state of the fixed point
m o m o
∗ , p ) is a switching of its ([nm , no ] : [nm , no ] : ∅ : ∅|n + 1 : n : n )
(xk0 0 1 1 2 2 5 6 7
state and ([n1 , n1 ] : [n2 , n2 ] : ∅ : ∅|n5 : n6 + 1 : n7 ) state for the fixed point
m o m o
∗
(xk , p).
(xv) An ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : n7 + 1) state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm , no ] : [nm , no ] : [n ; κ ] : [n ; κ ]| n + 1 : n
(xk0 0 1 1 2 2 3 3 4 4 5 6
: n7 ) state and ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 + 1 : n7 ) state for
m o m o
∗
the fixed point (xk , p).
II. Complex switching
(i) An ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : ∅|n5 : n6 : n7 ) state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm + n , no ] : [nm , no ] : ∅ : ∅|n : n : n )
(xk0 0 1 3 1 2 2 5 6 7
state and ([n1 , n1 ] : [n2 + n3 , n2 ] : ∅ : ∅|n5 : n6 : n7 ) state for the fixed point
m o m o
∗
(xk , p).
(ii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : [n4 ; κ4 ]|n5 : n6 : n7 ) state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm , no + n ] : [nm , no ] : ∅ : ∅|n : n : n )
(xk0 0 1 1 4 2 2 5 6 7
state and ([n1 , n1 ] : [n2 , n2 + n4 ] : ∅ : ∅|n5 : n6 : n7 ) state for the fixed point
m o m o
∗
(xk , p).
(iii) An ([n1 , n1 ] : [n2 , n2 ] : [n3 + k3 ; κ3 ] : ∅|n5 : n6 : n7 ) state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm + k , no ] : [nm , no ] : [n ; κ ] : ∅|n : n :
(xk0 0 1 3 1 2 2 3 3 5 6
n7 ) state and ([n1 , n1 ] : [n2 + k3 , n2 ] : [n3 ; κ3 ] : ∅|n5 : n6 : n7 ) state for the
m o m o
∗
(iv) An ([n1 1 m , no ] : [nm , no ] : ∅ : [n + k ; κ ]|n : n : n ) state of the fixed point
2 2 4 4 4 5 6 7
∗
(xk0 , p0 ) is a switching of the ([n1 , n1 + k4 ] : [n2 , n2 ] : ∅ : [n4 , κ4 ]|n5 : n6 :
m o m o

n7 ) state and ([n1 , n1 ] : [n2 , n2 + k4 ] : ∅ : [n4 ; κ4 ]|n5 : n6 : n7 ) state for the
m o m o

fixed point (xk ∗ , p).

(v) An ([n1 , n1 ] : [n2 , n2 ] : [n3 + k3 ; κ3 ] : [n4 + k4 ; κ4 ]|n5 : n6 : n7 ) state of
m o m o

the fixed point (xk0 0 ∗ , p ) is a switching of the ([nm + k , no + k ] : [nm , no ] :
1 3 1 4 2 2
[n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : n7 ) state and ([n1 , n1 ] : [n2 + k3 , n2 + k4 ] :
m o m o

[n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : n7 ) state for the fixed point (xk , p). ∗

(vi) An ([n1 1 m , no ] : [nm , no ] : [n + k ; κ ] : ∅|n : n : [n , l; κ ]) state of the
2 2 3 3 3 5 6 7 7
∗
fixed point (xk0 , p0 ) is a switching of the ([n1 + k3 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] :
m o m o

∅|n5 : n6 : [n7 , l; κ7 ]) state and ([n1 , n1 ] : [n2 + k3 , n2 ] : [n3 , κ3 ] : ∅|n5 :
m o m o

n6 : [n7 , l; κ7 ]) state for the fixed point (xk ∗ , p).

(vii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : [n4 +k4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) state of the fixed
m o m o
∗ , p ) is a switching of the ([nm , no + k ] : [nm , no ] : ∅ : [n ; κ ]|n :
point (xk0 0 1 1 4 2 2 4 4 5
n6 : [n7 , l; κ7 ]) state and ([n1 , n1 ] : [n2 , n2 + k4 ] : ∅ : [n4 ; κ4 ]|n5 : n6 :
m o m o
∗
[n7 , l; κ7 ]) state for the fixed point (xk , p).
(viii) An ([n1 1 m , no ] : [nm , no ] : ∅ : [n +k ; κ ]|n : n : [n , l; κ ]) state of the fixed
2 2 4 4 4 5 6 7 7
∗
point (xk0 , p0 ) is a switching of the ([n1 , n1 + k4 ] : [n2 , n2 ] : ∅ : [n4 ; κ4 ]|n5 :
m o m o

n6 : [n7 , l; κ7 ]) state and ([n1 , n1 ] : [n2 , n2 + k4 ] : ∅ : [n4 ; κ4 ]|n5 : n6 :
m o m o

[n7 , l; κ7 ]) state for the fixed point (xk ∗ , p).


(ix) An ([n1 , n1 ] : [n2 , n2 ] : [n3 + k3 ; κ3 ] : [n4 + k4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) state
m o m o
∗
of the fixed point (xk0 , p0 ) is a switching of the ([n1 + k3 , n1 + k4 ] : [n2 , n2 ] :
m o m o

[n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) state and ([n1 1
m , no ] : [nm + k , no + k ] :
2 3 2 4
∗
[n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : [n7 , l; κ7 ]) state for the fixed point (xk , p).
(x) An ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]|n5 : n6 : [n7 + k7 , l; κ7 ]) state of
m o m o
∗
the fixed point (xk0 , p0 ) is a switching of the ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] :
m o m o

[n4 ; κ4 ]|n5 + k7 : n6 : [n7 , l; κ7 ]) state and ([n1 1 m , no ] : no + k ] : [n ; κ ] :
2 4 3 3
[n4 ; κ4 ]|n5 : n6 + k7 : [n7 , l; κ7 ]) state for the fixed point (xk , p). ∗

∗
∗ ∗
∗
Eq. (2.28). The linearized system is yk+j+1 = Df(xk , p)yk+j (yk+j = xk+j − xk ) ∗
∗ ∗
in Uk (xk ). The matrix Df(xk , p) possesses n eigenvalues λi (i = 1, 2, · · ·n). Set
p = 1, 2, 3, 4) and p=1 np = n. ∪4 Np = N and ∩4 Np = ∅.Np = ∅ if nqp = 0.
4
p=1 p=1
Nα = Nα m ∪ N o (α = 1, 2) and N m ∩ N o = ∅ with nm + no = n where superscripts
α α α α α α
“m” and “o” represent monotonic and oscillatory evolutions. The matrix Df(xk , p) ∗

possesses n1 -stable, n2 -unstable, n3 -invariant and n4 -flip real eigenvectors. An itera-
tive response of xk+1 = f(xk , p) is an ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| flow in
m o m o

the neighborhood of the fixed point xk p ∗ . κ ∈ {∅, m } (p = 3, 4).
p

I. Simple critical cases
m o m o ∗
(i) An ([n1 , n1 ] : [n2 , n2 ] : 1 : ∅| state of the fixed point (xk0 , p0 ) is a switching
of the ([n1 m + 1, no ] : [nm , no ] : ∅ : ∅| saddle and ([nm , no ] : [nm + 1, no ] :
1 2 2 1 1 2 2
∅ : ∅| saddle for the fixed point (xk , p). ∗
m o m o ∗
(ii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : 1| state of the fixed point (xk0 , p0 ) is a switching
m , no + 1] : [nm , no ] : ∅ : ∅| saddle and ([nm , no ] : [nm , no + 1] :
of the ([n1 1 2 2 1 1 2 2
∅ : ∅| saddle for the fixed point (xk , p). ∗
∗
(iii) An ([n1 , n1 ] : [∅, ∅] : 1 : ∅| state of the fixed point (xk0 , p0 ) is a stable
m o

saddle-node bifurcation of the ([n1 m + 1, no ] : [∅, ∅] : ∅ : ∅| sink and
1
∗
([n1 , n1 ] : [1, ∅] : ∅ : ∅| saddle for the fixed point (xk , p).
m o

(iv) An ([n1 1m , no ] : [∅, ∅] : ∅ : 1| state of the fixed point (x∗ , p ) is a stable
k0 0
period-doubling bifurcation of the ([n1 , n1 + 1] : [∅, ∅] : ∅ : ∅| sink and
m o
∗
([n1 , n1 ] : [∅, 1] : ∅ : ∅| saddle for the fixed point (xk , p).
m o

(v) An ([∅, ∅] : [n2 2 m , no ] : 1 : ∅| state of the fixed point (x∗ , p ) is an unstable
k0 0
saddle-node bifurcation of the ([∅, ∅] : [n2 + 1, n2 ] : ∅ : ∅| source and
m o
∗
([1, ∅] : [n2 , n2 ] : ∅ : ∅| saddle for the fixed point (xk , p).
m o

(vi) An ([∅, ∅] : [n2 2 m , no ] : ∅ : 1| state of the fixed point (x∗ , p ) is an unstable
k0 0
period-doubling bifurcation of the ([∅, ∅] : [n2 , n2 + 1] : ∅ : ∅| source and
m o
∗
([∅, 1] : [n2 , n2 ] : ∅ : ∅| saddle for the fixed point (xk , p).
m o

(vii) An ([n1 1m , no ] : [nm , no ] : 1 : ∅| state of the fixed point (x∗ , p ) is a switching
2 2 k0 0
of the ([n1 + 1, n1 ] : [n2 , n2 ] : ∅ : ∅| saddle and ([n1 , n1 ] : [n2 + 1, n2 ] :
m o m o m o m o

∅ : ∅| saddle for the fixed point (xk ∗ , p).


m o m o ∗
(viii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : 1| state of the fixed point (xk0 , p0 ) is a switching
m , no + 1] : [nm , no ] : ∅ : ∅| saddle and ([nm , no ] : [nm , no + 1] :
of the ([n1 1 2 2 1 1 2 2
∗
∅ : ∅| saddle for the fixed point (xk , p).
m o m o ∗
(i) An ([n1 , n1 ] : [n2 , n2 ] : [n3 ; κ3 ] : ∅| state of the fixed point (xk0 , p0 ) is a
switching of the ([n1 m + n , no ] : [nm , no ] : ∅ : ∅| saddle and ([nm , no ] :
3 1 2 2 1 1
m o ∗
[n2 + n3 , n2 ] : ∅ : ∅| saddle for the fixed point (xk , p).
m o m o ∗
(ii) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : [n4 ; κ4 ]| state of the fixed point (xk0 , p0 ) is a
switching of the ([n1 1 m , no + n ] : [nm , no ] : ∅ : ∅| saddle and ([nm , no ] :
4 2 2 1 1
[n2 , n2 + n4 ] : ∅ : ∅| saddle for the fixed point (xk , p).
m o ∗
m o m o ∗
(iii) An ([n1 , n1 ] : [n2 , n2 ] : [n3 + k3 ; κ3 ] : ∅| state of the fixed point (xk0 , p0 ) is
a switching of the ([n1 m + k , no ] : [nm , no ] : [n ; κ ] : ∅| state and ([nm , no ] :
3 1 2 2 3 3 1 1
m o ∗
[n2 + k3 , n2 ] : [n3 ; κ3 ] : ∅| state for the fixed point (xk , p).
m o m o ∗
(iv) An ([n1 , n1 ] : [n2 , n2 ] : ∅ : [n4 + k4 ; κ4 ]| state of the fixed point (xk0 , p0 ) is
a switching of the ([n1 1m , no + k ] : [nm , no ] : ∅ : [n ; κ ]| state and ([nm , no ] :
4 2 2 4 4 1 1
∗
[n2 , n2 + k4 ] : ∅ : [n4 ; κ4 ]| state for the fixed point (xk , p).
m o

(v) An ([n1 , n1 ] : [n2 , n2 ] : [n3 + k3 ; κ3 ] : [n4 + k4 ; κ4 ]| state of the fixed point
m o m o
∗ , p ) is a switching of the ([nm +k , no +k ] : [nm , no ] : [n ; κ ] : [n ; κ ]|
(xk0 0 1 3 1 4 2 2 3 3 4 4
state and ([n1 , n1 ] : [n2 + k3 , n2 + k4 ] : [n3 ; κ3 ] : [n4 ; κ4 ]| state for the fixed
m o m o
∗
point (xk , p).
∗
∈ R 2n in Eq. (2.4) with a fixed point xk . The corresponding solution is given
by xk+j = f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed
∗ ∗ ∗
point xk (i.e., Uk (xk ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk ) with
k+j (yk+j = xk+j − xk ) in
Uk (xk∗ ). The matrix Df(x∗ , p) possesses 2n eigenvalues λ (i = 1, 2, · · ·, n). Set
k i
j = 5, 6, 7) and p=5 np = n. ∪7 Np = N and ∩7 Np = ∅. Np = ∅ if nqp = 0.
7
p=5 p=5
The matrix Df(xk ∗ , p) possesses n -stable, n -unstable and n -center pairs of com-
5 6 7
plex eigenvectors. Without repeated complex eigenvalues of |λi | = 1(i ∈ N7 ), an
iterative response of xk+1 = f(xk , p) is an |n5 : n6 : n7 ) flow in the neighborhood
∗
of the fixed point xk . With repeated complex eigenvalues of |λi | = 1 (i ∈ N7 ),
an iterative response of xk+1 = f(xk , p) is an |n5 : n6 : [n7 , l; κ7 ]) flow in the
∗
neighborhood of the fixed point xk , (κ7 ∈ {∅, m7 }).
I. Simple switching and bifurcation
∗
(i) An |n5 : n6 : 1) state of the fixed point (xk0 , p0 ) is a switching of the |n5 + 1 :
∗
n6 : ∅) spiral saddle and |n5 : n6 + 1 : ∅) saddle for the fixed point (xk , p).
∗ , p ) is a stable Neimark bifurcation
(ii) An |n5 : ∅ : 1) state of the fixed point (xk0 0
of the |n5 + 1 : ∅ : ∅) spiral sink and |n5 : 1 : ∅) spiral saddle for the fixed
∗
point (xk , p).


∗
(iii) An |∅ : n6 : 1) state of the fixed point (xk0 , p0 ) is an unstable Neimark
bifurcation of the |∅ : n6 + 1 : ∅) spiral source and |1 : n6 : ∅) spiral saddle
∗
∗
(iv) An |n5 : n6 : n7 + 1) state of the fixed point (xk0 , p0 ) is a switching of the
∗
|n5 + 1 : n6 : n7 ) state and |n5 : n6 + 1 : n7 ) state for the fixed point (xk , p).
(v) An |∅ : n6 : n7 + 1) state of the fixed point (xk0 0 ∗ , p ) is a switching of the
∗
|1 : n6 : n7 ) state and |n5 : n6 + 1 : n7 ) state for the fixed point (xk , p).
(vi) An |n5 : ∅ : n7 + 1) state of the fixed point (xk0 0 ∗ , p ) is a switching of the
∗
|n5 + 1 : ∅ : n7 ) state and |n5 : 1 : n7 ) state for the fixed point (xk , p).
∗
(i) An |n5 : n6 : [n7 , l; κ7 ]) state of the fixed point (xk0 , p0 ) is a switching of the
|n5 + n7 : n6 : ∅) spiral saddle and |n5 : n6 + n7 : ∅) spiral saddle for the
∗
∗
(ii) An |n5 : n6 : [n7 + k7 , l; κ7 ]) state of the fixed point (xk0 , p0 ) is a switching
of the |n5 + k7 : n6 : [n7 , l; κ7 ]) state and |n5 : n6 + k7 : [n7 , l; κ7 ]) state for
∗
∗
(iii) An |n5 : n6 : [n7 +k5 −k6 , l2 ; κ7 ]) state of the fixed point (xk0 , p0 ) is a switching
of the |n5 + k5 : n6 : [n7 , l1 ; κ7 ]) state and |n5 : n6 + k6 : [n7 , l3 ; κ7 ]) state for
∗

2.3.1 Stability and Switching

To extend the idea of Definitions 2.11 and 2.12, a new function will be defined to
determine the stability and the stability state switching.
∗
∈ R n in Eq. (2.4) with a fixed point xk . The corresponding solution is given by
xk+j = f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed
∗ ∗ ∗
in Uk (xk ∗ ) and there are n linearly independent vectors v (i = 1, 2, · · ·, n). For a
i
∗ (i) (i) (i) (i)
perturbation of fixed point yk = xk − xk , let yk = ck vi and yk+1 = ck+1 vi ,

(i) ∗
sk = viT · yk = viT · (xk − xk ) (2.49)

(i) (i)
where sk = ck ||vi ||2 . Define the following functions

∗
Gi (xk , p) = viT · [f(xk , p) − xk ] (2.50)


and
(1) (i)
G (i) (x, p) = viT · Dc(i) f(xk (sk ), p)
sk k
(i) (i)
= viT · Dxk f(xk (sk ), p)∂c(i) xk ∂sk ck (2.51)
k
(i)
= viT · Dx f(xk (sk ), p)vi ||vi ||−2

(m) (m) (i)
G (i) (x, p) = viT · D (i) f(xk (sk ), p)
sk sk
(m−1) (i)
(2.52)
= viT · Ds(i) (D (i) f(xk (sk ), p))
k sk

(i) (m) (m−1)
where Ds(i) (·) = ∂(·)/∂sk and D (i) (·) = Ds(i) (D (i) (·)).
k sk k sk

∗
xk+j = f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed
∗ ∗ ∗
in Uk (xk ∗ ) and there are n linearly independent vectors v (i = 1, 2, · · ·, n). For a
i
∗ (i) (i) (i) (i)
perturbation of fixed point yk = xk − xk , let yk = ck vi and yk+1 = ck+1 vi .
∗
(i) xk+j (j ∈ Z) at fixed point xk on the direction vi is stable if

∗ ∗
|viT · (xk+1 − xk )| < |viT · (xk − xk )| (2.53)
∗ ∗
for xk ∈ U (xk ) ⊂ α . The fixed point xk is called the sink (or stable node) on
the direction vi .
∗
(ii) xk+j (j ∈ Z) at fixed point xk on the direction vi is unstable if

∗ ∗
|viT · (xk+1 − xk )| > |viT · (xk − xk )| (2.54)
∗ ∗
for xk ∈ U (xk ) ⊂ α . The fixed point xk is called the source (or unstable
node) on the direction vi .
∗
(iii) xk+j (j ∈ Z) at fixed point xk on the direction vi is invariant if

∗ ∗
viT · (xk+1 − xk ) = viT · (xk − xk ) (2.55)
∗
for xk ∈ U (xk ) ⊂ ∗
α. The fixed point xk is called to be degenerate on the
direction vi .


(i) ∗
(iv) xk+j (j ∈ Z) at fixed point xk on the direction vi is symmetrically flipped if

∗ ∗
viT · (xk+1 − xk ) = −viT · (xk − xk ) (2.56)
∗
for xk ∈ U (xk ) ⊂ ∗
α. The fixed point xk is called to be degenerate on the
direction vi .
The stability of fixed points for a specific eigenvector is presented in Fig. 2.4.
The solid curve is viT · xk+1 = viT · f(xk , p). The circular symbol is the fixed point.
The shaded regions are stable. The horizontal solid line is for a degenerate case. The
vertical solid line is for a line with infinite slope. The monotonically stable node (sink)
is presented in Fig. 2.4a. The dashed and dotted lines are for viT · xk = viT · xk+1 and
viT · xk+1 = −viT · xk , respectively. The iterative responses approach the fixed point.
However, the monotonically unstable (source) is presented in Fig. 2.4b. The iterative
responses go away from the fixed point. Similarly, the oscillatory stable node (sink)
is presented in Fig. 2.4c. The dashed and dotted lines are for viT ·xk+1 = −viT ·xk and
viT · xk = viT · xk+1 , respectively. The oscillatory unstable node (source) is presented
in Fig. 2.4d.
Theorem 2.5 Consider a discrete, nonlinear dynamical system xk+1 = f(xk , p)
∗
xk+j = f(xk+j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed point
∗ ∗ ∗
xk (i.e., Uk (xk ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk ) with
in Uk (xk∗ ) and there are n linearly independent vectors v (i = 1, 2, · · ·, n). For a
i
∗ (i) (i) (i) (i)
perturbation of fixed point yk = xk − xk , let yk = ck vi and yk+1 = ck+1 vi .
∗
(i) xk+j (j ∈ Z) at fixed point xk on the direction vi is stable if and only if

G (1) (xk , p) = λi ∈ (−1, 1)
(i)
∗
(2.57)
sk

∗
for xk ∈ U (xk ) ⊂ α .
∗
(ii) xk+j (j ∈ Z) at fixed point xk on the direction vi is unstable if and only if

(1) ∗
G (i) (xk , p) = λi ∈ (1, ∞) and (−∞, −1) (2.58)
sk

∗
∗
(iii) xk+j (j ∈ Z) at fixed point xk on the direction vi is invariant if and only if

(1) ∗ (mi ) ∗
G(i) (xk , p) = λi = 1 and G (i) (xk , p) = 0 for mi = 2, 3, · · · (2.59)
sk sk

∗
(i) ∗
(iv) xk+j (j ∈ Z) at fixed point xk on the direction vi is symmetrically flipped if and
only if

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