Imperfect bifurcation of systems with slowly varying parameters and application to Duffing's equation

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎＰｈｙｓｉｃｓ,Ｍｅｃｈａｎｉｃｓ,ＣｈｅｍｉｓｔｒｙａｎｄＬｉｆｅＳｃｉｅｎｃｅｅｔｃ.,ｔｈｅｒｅｅｘｉｓｔｓｔｈｅｆｏｌｌｏｗｉｎｇｂｉｆｕｒｃａｔｉｏｎｐｒｏｂｌｅｍｗｉｔｈａｐａｒａｍｅｔｅｒ ｘ=ｆ(ｘ ,λ)　　 (ｘ∈Ｒｎ,λ∈Ｒ) ,(1 )ｗｈｅｒｅλｉｓａｃｏｎｔｒｏｌｌｉｎｇｐａｒａｍｅｔｅｒｏｆｂｉｆｕｒｃａｔｉｏｎａｎｄｆ(ｘ ,λ) ∈Ｒｎｉｓａｎｎ＿ｄｉｍｅｎｓｉｏｎａｌｖｅｃｔｏｒｆｉｅｌｄ .Ｔｈｅｂｉｆｕｒｃａｔｉｏｎｐｒｏｂｌｅｍｗｉｔｈａｃｏｎｓｔａｎ…     

Quasi-equilibrium problems in noncompact generalized convex spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＸａｎｄＹｂｅｎｏｎｅｍｐｔｙｓｅｔｓａｎｄ2ＸｂｅｔｈｅｆａｍｉｌｙｏｆａｌｌｓｕｂｓｅｔｓｏｆＸ.ＬｅｔＴ:Ｘ→Ｙｂｅａｓｉｎｇｌｅ＿ｖａｌｕｅｄｍａｐｐｉｎｇ,Ａ:Ｘ→2Ｘｂｅａｓｅｔ＿ｖａｌｕｅｄｍａｐｐｉｎｇａｎｄｆ:Ｘ×Ｙ→Ｒ∪±∞ｂｅａｆｕｎｃｔｉｏｎ.Ｔｈｅｑｕａｓｉ＿ｅｑｕｉｌｉｂｒｉｕｍｐｒｏｂｌｅｍＱＥＰ(Ｔ,Ａ,ｆ)ｉｓｔｏｆｉｎｄ^ｘ∈Ｘｓｕｃｈｔｈａｔ^ｘ∈Ａ(^ｘ),ｆ(^ｘ,Ｔ^ｘ)≤ｆ(ｙ,Ｔ^ｘ),　　ｙ∈Ａ(^ｘ).(1)ＴｈｅＱＥＰ(Ｔ,Ａ,ｆ)ｗａｓｉｎｔｒｏｄ…     

[0,k_i]_1~m-FACTORIZATIONS ORTHOGONAL TO A SUBGRAPH

1　ＢａｓｉｃＤｅｆｉｎｉｔｉｏｎｓａｎｄＮｏｔａｔｉｏｎｓＷｅｏｎｌｙｄｅａｌｗｉｔｈｆｉｎｉｔｅｕｎｄｉｒｅｃｔｅｄｓｉｍｐｌｅｇｒａｐｈｓ.ＬｅｔＧｂｅａｇｒａｐｈｗｉｔｈｖｅｒｔｅｘｓｅｔＶ(Ｇ)ａｎｄｅｄｇｅｓｅｔＥ(Ｇ) .Ｆｏｒｅａｃｈｘ∈Ｖ(Ｇ) ,ｗｅｄｅｎｏｔｅｔｈｅｄｅｇｒｅｅｏｆｘｉｎＧｂｙｄＧ(ｘ) .Ｌｅｔｇ ,ｆｂｅｔｗｏｉｎｔｅｇｅｒ＿ｖａｌｕｅｄｆｕｎｃｔｉｏｎｓｄｅｆｉｎｅｄｏｎＶ(Ｇ)ｓｕｃｈｔｈａｔｇ(ｘ)≤ｆ(ｘ)ｆｏｒａｌｌｘ∈Ｖ(Ｇ) .Ａ (ｇ ,ｆ)＿ｆａｃｔｏｒｏｆＧｉｓａｓ…     

NONOSCILLATION RESULTS FOR A CLASS OF THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＦｉｎｄｉｎｇｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｆｏｒｎｏｎｏｓｃｉｌｌａｔｏｒｙｏｆｓｏｌｕｔｉｏｎｓｉｓａｐｒｏｂｌｅｍｏｆｇｅｎｅｒａｌｉｎｔｅｒｅｓｔｉｎｔｈｅｔｈｅｏｒｙｏｆｏｒｄｉｎａｒｙａｎｄｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｉｎｔｈｉｓｗｏｒｋｗｅｃｏｎｓｉｄｅｒｄｄｓ σ(ｓ) ｄｄｓ ｒ(ｓ) ｄｘｄｓ +ｑ(ｓ) ｄｘｄｓβ +ｐ(ｓ)ｘα =ｆ(ｓ) ,(1)ｗｈｅｒｅσ ,ｒ,ｑ,ｐａｎｄｆａｒｅｒｅａｌ＿ｖａｌｕｅｄｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉ…     

Quasi-Equilibrium Problems and Constrained Multiobjective Games in Generalized Convex Space

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＸａｎｄＹｂｅｎｏｎｅｍｐｔｙｓｅｔｓａｎｄ2 ＸｂｅｔｈｅｆａｍｉｌｙｏｆａｌｌｓｕｂｓｅｔｓｏｆＸ .ＬｅｔＴ :Ｘ→Ｙｂｅａｓｉｎｇｌｅ＿ｖａｌｕｅｄｍａｐｐｉｎｇ ,Ａ :Ｘ→ 2 Ｘｂｅａｓｅｔ＿ｖａｌｕｅｄｍａｐｐｉｎｇ ,ψ :Ｘ×Ｙ×Ｘ→Ｒ∪ ±∞ ａｎｄｆ:Ｘ×Ｙ→Ｒ∪ ±∞ ｂｅｔｗｏｆｕｎｃｔｉｏｎｓ.Ｔｈｅｇｅｎｅｒａｌｑｕａｓｉ＿ｅｑｕｉｌｉｂｒｉｕｍｐｒｏｂｌｅｍＧＱＥＰ (Ｔ ,Ａ ,ψ)ｉｓｔｏｆｉｎｄ ^ｘ∈Ｘｓｕｃｈｔｈａｔ^ｘ∈Ａ( ^ｘ) ,ψ( ^ｘ ,Ｔ^ｘ ,ｙ)≤ 0…     

A new approach for the computation of Hopf bifurcation points

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｐａｒａｍｅｔｅｒｄｅｐｅｎｄｅｎｔｎｏｎｌｉｎｅａｒｐｒｏｂｌｅｍｆ(ｘ,λ) =0 ,　ｆ:Ｘ×Ｒ →Ｘ ,( 1 )ｗｈｅｒｅＸ=Ｒｎ,λｉｓｒｅａｌｐａｒａｍｅｔｅｒ,ｆ∈Ｃｒ(ｒ≥ 2 ) .Ｔｈｅｏｒｉｇｉｎａｌｐｒｏｂｌｅｍｃｏｕｌｄｂｅａｓｙｓｔｅｍｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ ,ｂｕｔｈｅｒｅｗｅｗｉｌｌａｓｓｕｍｅｔｈａｔａｓｕｉｔａｂｌｅｄｉｓｃｒｅｔｉｚａｔｉｏｎｈａｓｂｅｅｎｍａｄｅａｎｄｔｈｅｐｒｏｂｌｅｍｈａｓｔｈ…     

PANSYSTEMS METHODOLOGY AND CONSTRUCTION OF MAGIC SQUARES

ＩｎｔｒｏｄｕｃｔｉｏｎＡｒｅｆｉｎｅｄｅｐｉｔｏｍｅｏｆｖａｒｉｏｕｓｐａｎｓｙｓｔｅｍｓｍｅｔｈｏｄｓｉｓｔｏｃｏｍｐｒｅｈｅｎｄａｎｄｏｐｅｒａｔｅａｂｏｕｔ 2 0ｐａｎｓｙｓｔｅｍｓｃａｔｅｇｏｒｉｅｓｗｈｉｃｈｃａｎｂｅｅｘｐａｎｄｅｄａｓｔｈｅｆｏｌｌｏｗｉｎｇｐｉｔｈｙｆｏｒｍｕｌａｓｅｘｐｒｅｓｓｉｏｎｓ:Ｔｈｅｐａｎｄｅｒｉｖａｔｉｖｅｓ (ｏｒｔｈｅｃｈａｎｇｅ＿ｃｈａｎｇｅｒｅｌａｔｉｏｎｓ)ｏｆｔｈｅｅｘｔｅｒｉｏｒ＿ｉｎｔｅｒｉｏｒ(ｏｒｔｈｅｆｕｎｃｔｉｏｎ＿ｓｔｒｕｃｔｕ…     

BIFURCATIONS OF SUBHARMONIC SOLUTIONS IN PERIODIC PERTURBATION OF A HYPERBOLIC LIMIT CYCLE

Ｃｏｎｓｉｄｅｒｐｌａｎａｒｐｅｒｉｏｄｉｃｐｅｒｔｕｒｂｅｄｓｙｓｔｅｍ ｘ=ｆ(ｘ) +εｇ(ｔ,ｘ,ε ,δ) ,　　ｘ∈Ｒ2 ,( 1 )ｗｈｅｒｅε∈Ｒ ,δ∈Ｄ ＲｎｗｉｔｈＤｃｏｍｐａｃｔ,ａｎｄｆ,ｇａｒｅＣ3ｆｕｎｃｔｉｏｎｓａｎｄｇｉｓＴ＿ｐｅｒｉｏｄｉｃｉｎｔ,Ｔ >0 .Ｌｅｔｓｙｓｔｅｍ ( 1 )ｈａｓａｌｉｍｉｔｃｙｃｌｅＬ0 :ｘ =ｕ(ｔ) ,0 ≤ｔ≤Ｔ0 ｆｏｒε =0ｗｉｔｈＴ0 ｔｈｅｐｅｒｉｏｄｏｆＬ0 .ＳｕｐｐｏｓｅＴ0 /Ｔｉｓｒａｔｉｏｎａｌ,ｔｈａｔｉｓＴ0 /Ｔ=ｍ/ｋ,　　 (ｍ ,ｋ) =1 . ( 2 )Ａｓｗｅｋｎｏ…     

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｄｙｎａｍｉｃｓｙｓｔｅｍｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ : ｘ=ｆ(ｘ) ,( 1 )ｗｈｅｒｅｘａｎｄｆａｒｅｎ＿ｄｉｍｅｎｓｉｏｎａｌｒｅａｌｖｅｃｔｏｒｓ,ａｎｄｔｈｅｄｏｔｉｎｄｉｃａｔｅｓｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｉｍｅ ,ｔ.Ｓｕｐｐｏｓｅｔｈａｔｘ=ｕ(ｔ)ｉｓａｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆＥｑ .( 1 )ｗｉｔｈｐｅｒｉｏｄｉｃ,ｓａｙ ,Ｔ .ＫｒｙｌｏｖａｎｄＢ…     

REMARKS ON THE PAPER“THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION -DIFFUSION EQUATIONS”

ＴｈｅｆｏｌｌｏｗｉｎｇＬｅｍｍａｓｔａｔｅｄ (ｉｎ :ＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ (ＥｎｇｌｉｓｈＥｄｉｔｉｏｎ) ,2 0 0 1 ,2 2 ( 1 1 ) :1 3 52 -1 3 56)ｗａｓａｐｐｌｉｅｄｔｏｄｅｓｃｒｉｂｅｔｈｅａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｓｏｆｓｏｌｕｔｉｏｎｓｏｆａｃｌａｓｓｏｆｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎ .Ｏｎａｃｃｏｕｎｔｏｆｔｈｅｓｐｅｃｉａｌｐｒｏｐｅｒｔｙｏｆｔｈｅｓｕｐｅｒｓｏｌｕｔｉｏｎ(ｉ.ｅ .,ａｎｏｎｚｅｒｏｓｏｌｕｔｉｏｎ…     

GLOBAL ATTRACTIVITY FOR A CLASS OF NONLINEARDELAY DIFFERENCE EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｇｌｏｂａｌａｔｔｒａｃｔｉｖｉｔｙｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｘｎ+ａｎｘｎ+ｆ(ｎ ,∑0ｓ=-ｋ ｑｓ,ｎｘｓ+ｎ) =0　 (ｎ=0 ,1 ,2 ,… ) . ( 1 )ＴｈｅｉｎｉｔｉａｌｖａｌｕｅｏｆＥｑ .( 1 )ｉｓｘｓ =φｓ　 (ｓ=-ｋ ,-ｋ+ 1 ,… ,-1 ,0 ) . ( 2 )Ｉｎｔｈｅｓｅｑｕｅｌ,ｗｅａｌｗａｙｓｓｕｐｐｏｓｅｔｈａｔＤ1) ａｎ ｉｓａｎｏｎｎｅｇａｔｉｖｅｒｅａｌｓｅｑｕｅｎｃｅ ,0 ≤ａｎ <1ａｎｄｋｉｓａｐｏｓｉｔｉｖｅ…     

Extended self-similar scaling law of multi-scale eddy structure in wall turbulence

ＩｎｔｒｏｄｕｃｔｉｏｎＭｕｃｈｗｏｒｋｈａｓｂｅｅｎｄｅｖｏｔｅｄｉｎｔｈｅｌａｓｔｆｅｗｄｅｃａｄｅｓｔｏｔｈｅｍｅａｓｕｒｅｍｅｎｔａｎｄｍｏｄｅｌｉｎｇｏｆｔｈｅｓｃａｌｉｎｇｌａｗｏｆｓｔｒｕｃｔｕｒｅｆｕｎｃｔｉｏｎｏｆｔｕｒｂｕｌｅｎｔｆｌｏｗｓ.Ｔｈｅｓｏ＿ｃａｌｌｅｄ“ｖｅｌｏｃｉｔｙｓｔｒｕｃｔｕｒｅｆｕｎｃｔｉｏｎｏｆｏｒｄｅｒｎ”ｆｏｒｔｕｒｂｕｌｅｎｔｆｌｏｗｓｉｓｄｅｆｉｎｅｄａｓ〈ΔＶ(ｒ) ｎ〉 ,ｗｈｅｒｅΔＶ(ｒ) =Ｖ(ｘ ｒ) -Ｖ(ｘ)ｉｓｔｈｅｖｅｌｏｃｉｔｙｃｏｍｐ…     

ON SOME PROPERTIES OF LIENARD SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒａｇｅｎｅｒａｌｉｚｅｄＬｉｅｎａｒｄｓｙｓｔｅｍｓｏｆｔｈｅｆｏｒｍ ｘ=ｈ(ｙ) -Ｆ(ｘ) , ｙ=-ｇ(ｘ) ,( 1 )ｗｈｅｒｅｔｈｅｆｕｎｃｔｉｏｎｓＦ ,ｇａｎｄｈａｒｅＬｉｐｓｃｈｉｔｚｃｏｎｔｉｎｕｏｕｓａｎｄｓａｔｉｓｆｙＦ( 0 ) =ｈ( 0 ) =0 ,ｘｇ(ｘ) >0 ,　　ｘ≠ 0 . ( 2 )Ｔｈｅｒｅｈａｖｅｂｅｅｎｍａｎｙｓｔｕｄｉｅｓｆｏｒｌｏｃａｌａｎｄｇｌｏｂａｌｐｒｏｐｅｒｔｉｅｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｓｙｓｔｅｍ ( 1 ) ,ｓｅｅ[1 -9] .ＳｅｔＣ+=(ｘ,ｙ)｜ｈ(ｙ…     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅰ)

ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｏｔｉｃｉｓｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｎｔｈｅｄｅｆｉｎｉｔｅｓｙｓｔｅｍ .Ｔｈｅｄｅｆｉｎｉｔｅｎｅｓｓｉｓｃａｕｓｅｄｂｙｔｈｅｓｙｓｔｅｍｉｎｔｅｒｎａｌｓｂｕｔｎｏｔｂｙｔｈｅｅｘｔｅｒｎａｌｄｉｓｔｕｒｂａｎｃｅ[1- 4],ｗｈｉｌｅｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｓｔｈｅｉｒｒｅｇｕｌａｒａｎｄｄｉｆｆｉｃｕｌｔ＿ｔｏ＿ｂｅ＿ｐｒｅｄｉｃｔｅｄｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍ[5 ,6 ].Ｗｈａｔｉｓｍｏｓｔａｔｔｒａｃｔｉｖｅ…     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅱ)

ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｏｔｉｃｉｓｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｎｔｈｅｄｅｆｉｎｉｔｅｓｙｓｔｅｍ .Ｔｈｅｄｅｆｉｎｉｔｅｎｅｓｓｉｓｃａｕｓｅｄｂｙｔｈｅｓｙｓｔｅｍｉｎｔｅｒｎａｌｓｂｕｔｎｏｔｂｙｔｈｅｅｘｔｅｒｎａｌｄｉｓｔｕｒｂａｎｃｅ,ｗｈｉｌｅｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｓｔｈｅｉｒｒｅｇｕｌａｒａｎｄｄｉｆｆｉｃｕｌｔ＿ｔｏ＿ｂｅ＿ｐｒｅｄｉｃｔｅｄｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍ .Ｗｈａｔｉｓｍｏｓｔａｔｔｒａｃｔｉｖｅｏｆｔｈｅｃ…     

Precise integration method for LQG optimal measurement feedback control problem

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Ishikawa Iterative Process in Uniformly Smooth Banach Spaces

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A complete expression of the asymptotic solution of differential equation with three-turning points

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