  In this paper, we propose and analyse a novel class of exponential
collocation methods for solving conservative or dissipative systems based on
exponential integrators and collocation methods. It is shown that these novel
methods can be of arbitrarily high order and exactly or nearly preserve first
integrals or Lyapunov functions. We also consider order estimates of the new
methods. Furthermore, we explore and discuss the application of our methods in
important stiff gradient systems, and it turns out that our methods are
unconditionally energy-diminishing and strongly damped even for very stiff
gradient systems. Practical examples of the new methods are derived and the
efficiency and superiority are confirmed and demonstrated by three numerical
experiments including a nonlinear Schr\"{o}dinger equation. As a byproduct of
this paper, arbitrary-order trigonometric/RKN collocation methods are also
presented and analysed for second-order highly oscillatory/general systems. The
paper is accompanied by numerical results that demonstrate the great potential
of this work.
