With any of normal modes you may tie finite energy, sometimes momentum etc. So during evolution, for linear system, such modes do not couple each other, and system in one of this state leaves in it forever. Every linear physical system has its spectrum of normal modes, and if coupled with some external random source of energy white noise , its evolution runs through such states from the lowest possible energy to the greatest.
It depends on initial conditions and boundary values and restrictions but for finite systems and linear equations Fourier Transform gives you transformation from linear differential equation to matrix one which is nearly always soluble and has clear theory and meaning whilst Laplace Transform from DE to algebraic one with all advantages and disadvantages of it. Laplace transform gives you solution in terms of decaying exponents so it is quite useful in relaxation processes, but it has no physical interpretation, usually no invariants are connected to any "vectors" of such representation, there is no discrete version of such transform with physical meaning.
It is used in various engineering problems such that electrical circuits, queue theory etc. Definitely it would be easier to advice you what method of solution to use if you would describe what is the process you are trying to describe. References: try to Google such words: energy spectrum, normal modes, eigenstates, eigenvectors in context of linear differential equations - solving DE by means of integral transforms in practical way is usually described in books on Mathematical Methods in Physics, and is connected to response functions, distribution theory, Hilbert and Banach functional spaces etc.
It is very very broad area. What is more, if you asking in specific context for example in context of stochastic processes, or quantum mechanics , then probably you are looking for some certain interpretation of such transforms and not for formal theory. This differences sometimes tricky because mount of mathematical books focus on existence theorems etc.
A key quantity is here the abstract metric of the linear space and its binary product. The inherent difficulty of uncovering irreducible degeneracies, i. The need for a consistent evaluation of the conjugate operator representations has been investigated and analyzed in some detail. It is quite surprising to realize the consequences offered by the change from a positive to a non-positive definite metric.
Not only becomes relativity, self-references and in general telicity, the latter referring to processes owing their goal-directedness to the influence of an evolved program Mayr, , conceivable, but the formulation unfolds a syntax that organizes communication simpliciter, i. The description entails an extension to open system dynamics providing a self-referential amplification underpinning the signature of life as well as the evolution of consciousness via long-range correlative information, ODLCI.
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