scholarly journals Bifurcation diagram of stationary solutions of the 2D Kuramoto-Sivashinsky equation in periodic domains.

2021 ◽  
Vol 1730 (1) ◽  
pp. 012077
Author(s):  
Nikolay M. Evstigneev ◽  
Oleg I. Ryabkov
Keyword(s):  
Bifurcation Diagram ◽  
Stationary Solutions ◽  
Periodic Domains ◽  
Sivashinsky Equation

Pretzel Orbits in Simulations of Epitaxial Crystal Growth

1998 ◽  
Vol 08 (07) ◽  
pp. 1629-1639 ◽  
Author(s):  
Christoph Menke
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Stationary Solutions ◽  
Heteroclinic Orbits ◽  
Third Order ◽  
Unstable Manifolds ◽  
Epitaxial Crystal Growth ◽  
Sivashinsky Equation ◽  
Stability Results

A third order autonomous ordinary differential equation is studied that describes stationary solutions of a nonlinear partial differential equation. The PDE models the growth of an epitaxial film on misoriented crystal substrates and is similar to the Kuramoto–Sivashinsky equation, but contains an additional nonlinear term. The equilibria, the periodic solutions, and the heteroclinic orbits of the ODE are analyzed, and stability results are given. Parameter regions are identified where the equilibria and the periodic solutions are unstable, but other bounded solutions exist. Their phase portrait is a double focus ("pretzel") that connects the stable and the unstable manifolds of the equilibria.

scholarly journals Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

2020 ◽  
Vol 117 (38) ◽  
pp. 23227-23234
Author(s):  
Yong-Cong Chen ◽  
Chunxiao Shi ◽  
J. M. Kosterlitz ◽  
Xiaomei Zhu ◽  
Ping Ao
Keyword(s):  
Stochastic Dynamics ◽  
Nonlinear Partial Differential Equations ◽  
Saddle Points ◽  
Stationary Solutions ◽  
Stochastic Simulations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Topological Network ◽  
Sivashinsky Equation ◽  
General Method

We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen.37, L25–L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto–Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.

scholarly journals Complexity and Chimera States in a Network of Fractional-Order Laser Systems

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 341
Author(s):  
Shaobo He ◽  
Hayder Natiq ◽  
Santo Banerjee ◽  
Kehui Sun
Keyword(s):  
Numerical Simulation ◽  
Numerical Solution ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Dynamical Model ◽  
Chimera States ◽  
Laser Systems ◽  
Complexity Algorithm ◽  
Derivative Order

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network.

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