International Symposium in Commemoration of the 65th Anniversary of
the Foundation of Kim Il Sung University (Mathematics)
20－21. Sep. Juche100(2011) Pyongyang DPR Korea
Investigation on Bifurcation of Map Satisfying Condition of
Periodic Doubling Bifurcation
Dr. Vice-prof. Kim Sang Mun
College of Mathematics, Kim Il Sung University
Abstract In this paper, we consider sufficient conditions for an invariant double circle to occur
in a 1 parameter discrete dynamical systems on R×S.
Key words discrete dynamical system, doubling bifurcation, double circle
We consider the sufficient conditions for invariant double circle to occur in a discrete
dynamical system.
In the previous papers, were investigated sufficient conditions for periodic doubling bifurcation
to occur, in 1 parameter-1 dimension systems and sufficient conditions for Naimark-Sacar
bifurcation to occur, in 1 parameter-2 dimension systems.
Then, was investigated a sufficient condition for periodic doubling bifurcation to occur in
the case that higher order derivative is equal to zero, in 1 parameter-1 dimension systems.
And were studied sufficient conditions for Hopf- bifurcation to occur in the case that partial derivative
of higher order is equal to zero and the normal form of Hopf- bifurcation. In this paper we consider
conditios for invariant double circle to occur for the discrete dynamical system on R×S.
Theorem 1 If λ (3 < λ < λ0 ) irrational number and Fλ (r , [θ ]) = ( f λ (r ), H ([θ ])) then there
exists λ0 (> 3) such that the
Let
f λ (r ) = λr (1 − r ), r > 0, λ > 0,
H ([θ ]) = [θ + α ], [θ ] ∈ S 1 , here α is an irrational
number; Fλ (r , [θ ]) = ( f λ (r ), H ([θ ])) (where [ ] is a symbol of equivalent class of R / Z ).
Then there exists λ0 (> 3) such that for each
λ (3 < λ < λ0 ) there exist the invariant circles Γλ1 , Γλ2 with respect to Fλ2 and the
following are true:
① Γλ1 I Γλ2 = φ
② Fλ (Γλ1 ) = Γλ2 ,
Fλ (Γλ2 ) = Γλ1
③ Fλ (Γλ1 U Γλ2 ) = Γλ1 U Γλ2
Proof Since f λ (r ) satisfies the condition of periodic doubling bifurcation, there exists some
λ0 (> 3) such that for each λ (3 < λ < λ0 ) , there exist
f λ ( r1 ) = r2 , f λ ( r2 ) = r1 , r1 ≠ r2 .[1]
Let define Γλ1 and Γλ2 as follows:
Γλ1 : = {r1 (λ )} × S 1 , Γλ2 : = {r2 (λ )} × S 1
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r1 (λ ), r2 (λ ) > 0
satisfying
Investigation on Bifurcation of Map Satisfying Condition of Periodic Doubling Bifurcation
Let write r1 (λ ), r2 (λ ) as r1 , r2 and suppose (r1 , [0]) ∈ Γλ1 , then
Fλ1 (r1 , [0]) = (r2 , [α ]) ∈ Γλ2 ⎫
⎪
Fλ2 (r1 , [0]) = (r1 , [2α ]) ∈ Γλ1 ⎪
⎪
Fλ3 (r1 , [0]) = (r2 , [3α ]) ∈ Γλ2 ⎬
⎪
Fλ4 (r1 , [0]) = (r1 , [4α ]) ∈ Γλ1 ⎪
⎪⎭
M
If k is a nonnegative integer, then
Fλ2 k (r1 , [0]) = (r1 , [2kα ]) ∈ Γλ1 , Fλ2k +1 (r1 , [0]) = (r2 , [(2k + 1)α ]) ∈ Γλ2 .
First prove that {Fλ2k (r1 , [0])} is dense in Γλ1 .
The set {Fλ2 k (r1 , [0])} consists of different points. Therefore, since Γλ1 is compact, this
sequence has an accumulation point in Γλ1 . And {r1} × S 1 is a complete metric space with distance.
d ((r1 , [ x]), (r1 , [ y ])) = inf{| x − y + k | ; k ∈ Z }
Since {Fλ2 k (r1 , [0])} has an accumulation point in Γλ1
∀ε > 0, ∃k1 , k 2 (k1 > k 2 ) ∈ Z + ; d ( Fλ2k1 (r1 , [0]), Fλ2k2 (r1 , [0])) = inf{| (2k1 − 2k 2 )α + k | ; k ∈ Z} < ε .
If p : =| 2k1 − 2k 2 | then p is an even number.
Therefore d ( Fλp (r1 , [0]), (r1 , [0])) = d ( Fλ2 k1 (r1 , [0]), Fλ2 k 2 (r1 , [0])) < ε .
If k ′ ∈ N then d ( Fλk ′p (r1 , [0]), Fλ( k ′−1) p (r1 , [0])) = d ( Fλp (r1 , [0]), (r1 , [0])) < ε .
Therefore {r1} × S 1 is divided into parts whose length is smaller than ε by the sequence
Fλp (r1 , [0]), Fλ2 p (r1 , [0]), L .
Therefore {Fλ2 k (r1 , [0])} is dense in Γλ1 . That is
{Fλ2 k (r1 , [0])} = Γλ1 .
(1)
Similarly,
{Fλ2 k +1 (r1 , [0])} = Γλ2 .
(2)
From (1), (2) and the property of invariant sets, Γλ1 = Fλ2 (Γλ1 ), Γλ2 = Fλ2 (Γλ2 ) and from the
definition Γλ1 I Γλ2 = φ .Thus the result ① is proved.
Let prove ②. If p ∈ Fλ (Γλ1 ) then there exists some u ∈ Γλ1 such that p = Fλ (u ) .
And there exists {(r1 , [2k1α ])} ⊂ Γλ1 such that u = lim (r1 , [2k1α ]) .
k1 →∞
1
ϕ : R → R / Z , ϕ ( x) : = [ x ] , ϕ : R → S , ϕ ( x) : = e
2πix
then π is a homeomorphism and
ϕ ( x) = π o ϕ ( x) , therefore ϕ is continuous.
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,
, π : R / Z → S 1 , π [ x] = e 2πix , x ∈ R
International Symposium(Mathematics)
Since Fλ (r , [θ ]) = ( f λ (r ), H ([θ ])) = (λr (1 − r ), ϕ (θ + α )) , Fλ is continuous with respect to
(r , [θ ]) . Thus, p = Fλ (u ) = lim (r2 , [(2k1 + 1)α ]) and since Γλ2 is compact, p = Fλ (u ) ∈ Γλ2 .
k n →∞
Therefore
Fλ (Γλ1 ) ⊂ Γλ2
(3)
2
If v ∈ Γλ then there exist the sequence {r2 , [(2k n + 1)α ]} ⊂ Γλ such that
2
v = lim (r2 , [(2k1 + 1)α ]) .
k n →∞
Since Fλ is continuous, v = Fλ ( lim (r1 , [2k nα ])) and by the compactness of Γλ1 .
k n →∞
If
u = lim (r1 , [2k nα ]) then v = Fλ (u ), u ∈ Γλ1 . Thus
k n →∞
Γλ2 ⊂ Fλ (Γλ1 ) .
1
2
2
(4)
1
By (3), (4), Fλ (Γλ ) = Γλ and Fλ (Γλ ) = Γλ .
The proof of ③ is omitted.
Theorem 2 Assume that f : R 2 → R satisfies the condition of periodic doubling bifurcation
in a neighborhood of f (λ , ⋅ ) and g : R 2 ∋ (λ , r ) a g (λ , r ) ∈ R is continuous with respect to
r and for 2-periodic points r1 , r2 of f (λ , ⋅ ) , g (λ , r1 ), g (λ , r2 ) are irrational numbers, and
Θ λ : R × S 1 → S 1 , Θ λ (r , [θ ]) : = [θ ] + [ g (λ , r )] .
If Fλ (r , [θ ]) : = ( f (λ , r ), Θ λ (r , [θ ]))
λ1 ∈ R
then there exists
1
such that for each
λ ∈ (λ0 , λ1 ) (or λ ∈ (λ1 , λ0 ) ) there exist invariant circle Γλ , Γλ with respect to Fλ2 such that
① Γλ1 I Γλ2 = φ , Fλ2 (Γλ1 ) = Γλ2 ,
2
Fλ2 (Γλ2 ) = Γλ2
② Fλ (Γλ1 ) = Γλ2 , Fλ (Γλ2 ) = Γλ1
③ Fλ (Γλ1 U Γλ2 ) = Γλ1 U Γλ2
Reference
[1] 김일성종합대학학보(자연과학), 52, 11, 25, 주체95(2006).
[2] 김일성종합대학학보(자연과학), 52, 12, 2, 주체95(2006).
[3] 김일성종합대학학보(자연과학), 54, 10, 34, 주체97(2008).
[4] G. Iooss; Bifurcation of Maps and Application, North–Holland Publishing Company, 12, 1979.
[5] F. Balibrea et al.; Nonlinear Analysis, 52, 405, 2003.
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