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Some Fundamental Concepts in the Solution of Differential Equations
One Approach to Solving a System of Ordinary Differential Equations
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altitude angle angular angular velocity approximation assume attractor axis ball bifurcation bifurcation diagram bounce calculation center of mass chaos chaotic comet components conﬁrm consider constant coordinates cricket ball curve deﬁned density depends diagram differential equations dynamical system Earth energy equilibrium Euler’s method ﬁeld Figure ﬁnal ﬁnd ﬁrst ﬁrst-order ﬁxed point ﬂight function gravitational horizontal illustrated in ﬁgure increase initial conditions initial speed integration investigate iteration Jupiter lift force limit cycle Lorenz equations mass Moon moving Newton’s method numerical values orbit oscillations parameters pendulum periodic phase-plane pitch plane plot Poincaré maps population predation radius rocket rotation satellite shown in ﬁgure Skylab solution solved spacecraft spin stable starting conditions step stepsize subroutine Suppose swing TeMax trajectory truncation error variables vary velocity vertical yo-yo zero