• The Pure and Applied Mathematics
• /
• v.15 no.3
• /
• pp.329-342
• /
• 2008

There have been many experimental and observational evidences which indicate the predator response to prey density needs not always monotone increasing as in the classical predator-prey models in population dynamics. Holling type functional response depicts situations in which sufficiently large number of the prey species increases their ability to defend or disguise themselves from the predator. In this paper we investigated the stability and instability property for a Holling type predator-prey system of a generalized form. Hopf type bifurcation properties of the non-diffusive system and the diffusion effects on instability and bifurcation values are studied.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.25 no.3
• /
• pp.433-441
• /
• 2001

Natural convection of a fluid with intermediate Prand시 number of Pr=0.2 in a horizontal annulus is considered, and the bifurcation phenomena and chaotic flows are numerically investigated. The unsteady two-dimensional streamfunction-vorticity equation is solved with finite difference method. The steady downward flow with two counter-rotating eddies bifurcates to a simple periodic flow with a fundamental frequency. And afterwards, second Hopf bifurcation occurs, and a quasi-periodic flow with two incommensurable frequencies appears. However, a new time-periodic flow is established after experiencing quasi-periodic states. As Rayleigh number is increased further, the chaotic flow regime is reached after a sequence of successive Hopf bifurcation to quasi-periodic and chaotic flow regimes. A scenario similar to the Ruelle-Takens-Newhouse scenario of the onset of chaos is observed.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.655-666
• /
• 2017

In this paper, we study the pattern formation to a general Degn-Harrison reaction model. We show Turing instability happens by analyzing the stability of the unique positive equilibrium with respect to the PDE model and the corresponding ODE model, which indicate the existence of the non-constant steady state solutions. We also show the existence periodic solutions of the PDE model and the ODE model by using Hopf bifurcation theory. Numerical simulations are presented to verify and illustrate the theoretical results.

• Journal of applied mathematics & informatics
• /
• v.28 no.5_6
• /
• pp.1379-1393
• /
• 2010

In this paper, a stage-structured predator prey model (stage structure on prey) with two discrete time delays has been discussed. The two discrete time delays occur due to maturation delay and gestation delay. Linear stability analysis for both non-delay as well as with delays reveals that certain thresholds have to be maintained for coexistence. Numerical simulation shows that the system exhibits Hopf bifurcation, resulting in a stable limit cycle.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.14 no.7
• /
• pp.561-568
• /
• 2004

The nonlinear dynamic characteristic of a straight tube conveying fluid with constraints and an attached mass on the tube is examined in this study An experimental apparatus with an elastomer tube conveying water which has an attached mass and constraints is made and comparisons are made between the theoretical results from the non-linear equation of motion of piping system and the experimental results. The comparisons show that the tube is destabilized as the magnitude of the attached mass increases, and stabilized as the position of the attached mass closes to the fixed end. In case of a small end-mass, the system shows complicated and different types of solutions. For a constant end-mass. the system undergoes a series of bifurcations after the first Hopf bifurcation, as the flow velocity increases. which causes chaotic motions of the tube eventually.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.34 no.9
• /
• pp.859-866
• /
• 2010

Nonlinear dynamics of pulsating instability-diffusional-thermal instability with Lewis numbers sufficiently higher than unity-in counterflow diffusion flames, is numerically investigated by imposing a Damkohler number perturbation. The flame evolution exhibits three types of nonlinear behaviors, namely, decaying pulsating behavior, diverging behavior (which leads to extinction), and stable limit-cycle behavior. The stable limit-cycle behavior is observed in counterflow diffusion flames, but not in diffusion flames with a stagnant mixing layer. The critical value of the perturbed Damkohler number, which indicates the region where the three different flame behaviors can be observed, is obtained. A stable simple limit cycle, in which two supercritical Hopf bifurcations exist, is found in a narrow range of Damkohler numbers. As the flame temperature is increased, the stable simple limit cycle disappears and an unstable limit cycle corresponding to subcritical Hopf bifurcation appears. The period-doubling bifurcation is found to occur in a certain range of Damkohler numbers and temperatures, which leads to extend the lower boundary of supercritical Hopf bifurcation.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.20 no.9
• /
• pp.2965-2972
• /
• 1996

Transitional flow in an obstructed channel is investigated using numerical simulation. Two-dimensional thin obstacles are mounted symmetrically in the vertical direction and periodically in the streamwise direction. Flow separation occurs at the tip of the sharp obstacles. Depending on the Reynolds number, the flow undergoes Hopf bifurcation as the primary instability leading to a two-dimensional unsteady periodic solution. At higher Reynolds numbers, the unsteady solution exhibits a symmetry-breaking bifurcation which results in an unsteady asymmetric solution. The results are compared with experiments currently available, and show a good agreement.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.20 no.9
• /
• pp.2973-2980
• /
• 1996

Secondary instability in an obstructed channel is investigated using direct numerical simulation. Flow geometry under consideration is a plane channel with two-dimensional thin obstacles mounted symmetrically in the vertical direction and periodically in the streamwise direction. Flow separation occurs at the tip of the sharp obstacles. As a basic flow, we consider an unsteady periodic solution which results from Hopf bifurcation. Depending on the Reynolds number, the basic flow becomes unstable to three-dimensional disturbances, which results in a chaotic flow. Numerical results obtained are consistent with experimental findings currently available.

• The Pure and Applied Mathematics
• /
• v.4 no.1
• /
• pp.61-69
• /
• 1997

In this paper, we study the dynamics of a two-parameter unfolding system $\chi$' = y, y' = $\beta$y＋$\alpha$f($\chi\alpha\pm\chiy$＋yg($\chi$), where f($\chi$,$\alpha$) is a second order polynomial in $\chi$ and g($\chi$) is strictly nonlinear in $\chi$. We show that the higher order term yg($\chi$) in the system does not change qulitative structure of the Hopf bifurcations near the fixed points for small $\alpha$ and $\beta$ if the nontrivial fixed point approaches to the origin as $\alpha$ approaches zero.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.631-638
• /
• 1996

The parabolic free boundary problem with Puschino dynamics is given by (see in [3]) $$ (1) { \upsilon_t = D\upsilon_{xx} - (c_1 + b)\upsilon + c_1 H(x - s(t)) for (x,t) \in \Omega^- \cup \Omega^+, { \upsilon_x(0,t) = 0 = \upsilon_x(1,t) for t > 0, { \upsilon(x,0) = \upsilon_0(x) for 0 \leq x \leq 1, { \tau\frac{dt}{ds} = C)\upsilon(s(t),t)) for t > 0, { s(0) = s_0, 0 < s_0 < 1, $$ where $\upsilon(x,t)$ and $\upsilon_x(x,t)$ are assumed continuous in $\Omega = (0,1) \times (0, \infty)$.