Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 60
... Consider a spring - mass system with a nonlinear restoring force satisfying m d2x dt 2 - -kx αx3 , where a > 0. Which positions are equilibrium positions ? Are they stable ? 18.5 . Consider a system which satisfies m d2x dt 2 = -kx + ...
... Consider a spring - mass system with a nonlinear restoring force satisfying m d2x dt 2 - -kx αx3 , where a > 0. Which positions are equilibrium positions ? Are they stable ? 18.5 . Consider a system which satisfies m d2x dt 2 = -kx + ...
Page 184
... Consider equation 42.1 . If 0 < α < 4 : ( a ) Show that the population may go beyond the equilibrium ( at most once ) . ( b ) Give an example of an " initial " population that reaches equilibrium only after going beyond the equilibrium ...
... Consider equation 42.1 . If 0 < α < 4 : ( a ) Show that the population may go beyond the equilibrium ( at most once ) . ( b ) Give an example of an " initial " population that reaches equilibrium only after going beyond the equilibrium ...
Page 254
... Consider the three - species ecosystem dF dt dR dt = F ( a + bR - k1G ) - R ( c dF - k2G ) dG dt = G ( k + k3F − k1R ) , assuming all coefficients are positive constants . ( a ) If G = 0 , show that this represents a predator - prey ...
... Consider the three - species ecosystem dF dt dR dt = F ( a + bR - k1G ) - R ( c dF - k2G ) dG dt = G ( k + k3F − k1R ) , assuming all coefficients are positive constants . ( a ) If G = 0 , show that this represents a predator - prey ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
NEWTONS LAW AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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analyze approximation Assume c₁ c₂ calculated conservation of cars Consider corresponding d2x dt2 de/dt density wave velocity depends derived determine dq/dp dx dt dx/dt equilibrium population equilibrium position equilibrium solution example exercise exponential Figure flow-density force formula friction function growth rate highway increases initial conditions initial density initial traffic density initial value problem integral intersect isoclines logistic equation mass mathematical model maximum method of characteristics motion moving Newton's nonlinear pendulum number of cars observer occurs ordinary differential equation oscillation P/Pmax P₁ partial differential equation period phase plane Pmax potential energy problem qualitative region result sharks shock velocity Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow traffic light trajectories Umax Umaxt unstable equilibrium variables velocity-density x₁ yields zero др