On Differential Equations (1950-1974)

"Part of the charm in solving a differential equation is in the feeling that we are getting something for nothing. So little information appears to go into the solution that there is a sense of surprise over the extensive results that are derived." (George R Stibitz & Jules A Larrivee, "Mathematics and Computers", 1957)

"Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"We completely agree that description by differential equations is not only a clumsy but, in principle, inadequate way to deal with many problems of organization." (Ludwig von Bertalanffy, "General System Theory", 1968)

"My definition of global analysis is simply the study of differential equations, both ordinary and partial, on manifolds and vector space bundles. Thus one might consider global analysis as differential equations from a global, or topological point of view." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"To abstract the qualitative features of a differential equation on M, the concept of a phase portrait become important. Usually the phase portrait means the picture of the solution curves of the differential equation. [...] Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological  equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"A fairly general procedure for mathematical study of a physical system with explication of the space of states of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate For the state space should be a differentiable manifold; and in case the has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with physical is the notion of how a state progresses in time. The corresponding object is a dynamical system or a first order ordinary differential equation on the manifold of states." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"[A] system is represented by a mathematical model which may take many forms, such as algebraic equations, finite state machines, difference equations, ordinary differential equations, partial differential equations, and functional equations. The system model may be uncertain, as the mathematical model may not be known completely." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"Generally speaking, when a mathematician introduces a new term, he ordinarily pays little attention to whether it has a contrasting term to go with it. For instance, there is a class of ordinary differential equations but there are no 'extraordinary' differential equations. Actually, ordinary differential equations are equations in one independent variable, whereas differential equations involving many independent variables are termed partial differential equations and not extraordinary differential equations. [...] Of course, when introducing new terms, one should be guided by something more than vanity. The objects and phenomena under consideration must be important enough and the class of phenomena or objects should merit having a new term to describe them." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"It is well known that an initial value problem for a nonlinear ordinary differential equation may very well fail to have a solution for all time; the solution may blow up after a finite time. The same is true for quasi-linear hyperbolic partial differential equations: solutions may break down after a finite time when their first derivatives blow up." (Peter D Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves", 1974)

Previous Post <<||>> Next Post