Bifurcation and stability of periodic solutions of Duffing equations

Nonlinearity ◽  
2008 ◽  
Vol 21 (11) ◽  
pp. 2485-2503 ◽  
Author(s):  
Hongbin Chen ◽  
Yi Li
Keyword(s):  
Periodic Solutions ◽  
Duffing Equations ◽  
Bifurcation And Stability

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

Bifurcation and stability of periodic solutions of differential equations with state-dependent delays

2003 ◽  
Vol 14 (1) ◽  
pp. 3-14 ◽  
Author(s):  
D. SCHLEY
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Perturbation Methods ◽  
Population Models ◽  
Delay Function ◽  
State Dependent ◽  
The Stability ◽  
Bifurcation And Stability

We consider periodic solutions which bifurcate from equilibria in simple population models which incorporate a state-dependent time delay of the discrete kind. The delay is a function of the current size of the population. Solutions near equilibria are constructed using perturbation methods to determine the sub/supercriticality of the bifurcation and hence their stability. The stability of the bifurcating solutions depends on the qualitative form of the delay function. This is in contrast to the stability of an equilibrium, which is determined purely by the actual value of this function at the equilibrium.

scholarly journals Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tiantian Ma ◽  
Zaihong Wang
Keyword(s):  
Periodic Solutions ◽  
Autonomous System ◽  
Positive Periodic Solutions ◽  
Duffing Equations ◽  
Time Map ◽  
Twist Theorem ◽  
The Given ◽  
Infinitely Many Periodic Solutions

We study the periodic solutions of Duffing equations with singularitiesx′′+g(x)=p(t). By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided thatgsatisfies the singular condition and the time map related to autonomous systemx′′+g(x)=0tends to zero.

Coincidence degree and periodic solutions of Duffing equations

1998 ◽  
Vol 34 (3) ◽  
pp. 443-460 ◽  
Author(s):  
Ma Shiwang ◽  
Wang Zhicheng ◽  
Yu Jianshe
Keyword(s):  
Periodic Solutions ◽  
Coincidence Degree ◽  
Duffing Equations

Positive periodic solutions to the forced non-autonomous Duffing equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiří Šremr
Keyword(s):  
Periodic Solutions ◽  
Positive Solutions ◽  
Periodic Problem ◽  
Duffing Equation ◽  
Positive Periodic Solutions ◽  
Duffing Equations ◽  
Existence And Multiplicity ◽  
Constant Coefficients ◽  
Multiplicity Of Positive Solutions

Abstract We study the existence and multiplicity of positive solutions to the periodic problem for a forced non-autonomous Duffing equation u ′′ = p ⁢ ( t ) ⁢ u - h ⁢ ( t ) ⁢ | u | λ ⁢ sgn ⁡ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-h(t)\lvert u\rvert^{\lambda}\operatorname{sgn}u+f(t);\quad u(0)=u(\omega),\ u^{\prime}(0)=u^{\prime}(\omega), where p , h , f ∈ L ⁢ ( [ 0 , ω ] ) p,h,f\in L([0,\omega]) and λ > 1 \lambda>1 . The obtained results are compared with the results known for the equations with constant coefficients.

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