Predator Extinction arose from Chaos of the Prey: the Chaotic Behavior of a Homomorphic Two-Dimensional Logistic Map in the Form of Lotka-Volterra Equations

• Speaker: Wei Shan Lee, Pui Ching Middle School Macau
• Abstract: A two-dimensional homomorphic logistic map that preserves features of Lotka-Volterra Equations was proposed. In order to examine the Lotka-Volterra chaos, in addition to ordinary iteration plots of population, Lyapunov exponents either calculated directly from eigenvalues of Jacobian of the \$2\$D logistic mapping, or from time-series algorithms of both Rosenstein and Eckmann et al. were calculated, among which discrepancies were compared. Bifurcation diagrams may be divided into five categories depending on different topological shapes, among which flip bifurcation and Neimark-Sacker bifurcation were observed, the latter showing closed orbits around limit circles in the phase portrait and phase space diagram. Our model restored the 1- D logistic map of the prey at the absence of the predator, as well as the normal competing behavior between two species when the initial population of the two is equal. In spite of the possibility for two species going into chaos simultaneously, it is also possible that with the same inter-species parameters as normal but with predator population 10 times more than that of the prey, under certain growth rate the latter becomes chaotic, and the former dramatically reduces to zero, referring to total annihilation of the predator species. Interpreting humans as the predator and natural resources as the prey in the ecological system, the aforementioned conclusion may imply that not only excessive consumption of the natural resources, but its chaotic state triggered by overpopulation of humans may also backfire in a manner of total extinction on human species. Fortunately, a little chance may exist for survival of human race, as isolated fixed points in bifurcation diagram of the predator reveal. Joint work with Hou Fai Chan, Ka Ian Im, Kuan Ieong Chan, and U Hin Cheang, Pui Ching Middle School Macau.

Origination of Eskov Chaos as a Precursor of Failure

• Speaker: Mikhail Zimin, 2554620 Ontario LTD.
• Abstract: In some manuscripts of V. M. Eskov, statistical instability of samples for living systems is described. This effect was called as Eskov chaos. Further researches showed that this phenomenon may be a precursor of failure. For instance, such phenomenon is observed for precipitation rate. It results in Eskov chaos in avalanche risk. Also, it causes Eskov chaos in estimations of mud flow danger. So, taking into account this factor is important in forecasting slope processes connected with destruction of snow or soil. Other example is Eskov chaos in the biological forerunners of earthquakes. In particular, chaotic motion of rats was on display many times before earthquakes at North Caucasus. Therefore, it preceded failure of rock. Given examples show that analysis of statistical instability of samples may be useful for estimating risk of failure of various objects. References 1. Eskov, V. V., Phenomenon of statistical instability of the third type systems ' complexity / V. V. Eskov, T. V. Gavrilenko, Y. V. Vokhmina // Technical physics. ' 2017. ' Vol. 62(11). ' P. 1611 ' 1616. Joint work with Olga Kumukova, High-Mountain Geophysical Institute, Nalchik, Russian Federation, and Svetlana Zimina, 2554620 Ontario LTD.

Distributed Position and Velocity Delay Effects in a Van der Pol System with Time-periodic Feedback

• Speaker: Roy Choudhury, University of Central Florida
• Abstract: The effects of a distributed delay on a parametrically forced Van der Pol limit cycle oscillator are considered. Delays in self-excited systems, modeling time lags due to a variety of factors, have been discussed earlier in the control and modification of limit cycle and quasiperiodic system responses. Those earlier studies are extended here to include the effects of periodically amplitude modulated delays in both the position and velocity. A normal form or “slow flow” is employed to search for various bifurcations and transitions between regimes of different dynamics, including amplitude death and quasi-periodicity. The existence of quasiperiodic solutions then motivates the derivation of a second slow flow. A detailed comparison of results and predictions from the second slow flow to numerical solutions is made. The second slow flow is then employed to approximate the amplitudes of the quasi-periodic solutions, yielding close agreement with the numerical results. Finally, the effect of varying the delay parameter is briefly considered, and the results and conclusions are summarized. Joint work with Ryan Roopnarain, Rollins College.

Estimation of Time Remaining to Dangerous State by Changing Average Risk in the Method of Ordered Risk Minimization 

• Speaker: Svetlana Zimina, Analytic, 2554620 Ontario LTD.
• Abstract: Despite considerable progress in dangerous states' forecast, computing time interval to their formation remains very complicated problem. Therefore, any technique helping to solve it presents some features of interest. In utilizing the method of ordered risk minimization, average risk is consequentially calculated for dependences beginning from the simplest. The one under which average risk is minimum is considered as providing optimum approximation of initial data. If, when using Chebyshev polynomials, optimum degree is equal to zero, situation may be considered as stable. However, due to the passage of time difference between average risk for n = 0 (n ' degree of polynomial approximant) and n > 0 may decrease. This suggests that optimum degree (nopt) will become more than zero after a time, and situation will be unstable and may be treated as a catastrophe. This time can be estimated with the help of found approximations. Calculations are performed for analysis of the somatic temperature, modeling of stage ominous calm in rabies, and radiation sickness, evaluating degrees of expressiveness of biological precursors of earthquakes and mudflows. Knowing time to dangerous state, it is possible to take corresponding measures and avoid risk of loss of values and fatalities. Joint work with Olga Kumukova, High-Mountain Geophysical Institute, Mikhail Zimin, 2554620 Ontario LTD., and Oksana Kulikova, the Siberian State Automobile and Highway University.