scholarly journals Mixed problem for weakly hyperbolic equations of second order with degenerate first order boundary condition

1984 ◽  
Vol 60 (3) ◽  
pp. 97-100
Author(s):  
Masaru Taniguchi
Keyword(s):  
Boundary Condition ◽  
Mixed Problem ◽  
Hyperbolic Equations ◽  
Second Order ◽  
First Order

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

2015 ◽  
Vol 25 (11) ◽  
pp. 1530032 ◽  
Author(s):  
Liangliang Li ◽  
Yu Huang ◽  
Goong Chen ◽  
Tingwen Huang
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Hyperbolic Equations ◽  
Nonlinear Boundary Conditions ◽  
Second Order ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Order Linear ◽  
First Order

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

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