The theory in [9] shows that every

dynamical system of order one can be prolonged to an Euler-Lagrange

dynamical system of order two whose trajectories are geodesies of a Lagrangian defined by the velocity vector field (Lagrange structure of order one).

The time derivative of V along the trajectory of the error

dynamical system (2) is as follows

that is, the cone of nonnegative vectors, and consider the

dynamical systemSection 2 provides the necessary background information on the invariance and conserved quantities of

dynamical system and especially the Noether's theorem.

Now we will formulate some theorems describing the chaotic and stable behaviour of the above

dynamical system.

Truth be told, the derivation of a

dynamical system from the payoff matrix of the underlying game is tricky business.

17] The

dynamical system is said to converge to the solution set [K.

Despite this sensitivity, however, chaotic trajectories in a given

dynamical system still generally end up on an attractor that has a particular geometry -- albeit one that can look extremely convoluted and complicated.

This volume contains 15 selected papers discussing connections between fractal geometry and

dynamical system in pure mathematics and connections between these two and other fields of mathematics.

The term "chaos" refers to situations in which the behavior of a

dynamical system -- whether a pendulum or a collection of gravitationally interacting bodies -- depends sensitively on initial conditions.

Recurring Themes and Applications: Important themes and applications, such as

dynamical system formulation, phase portraits, linearization, stability of equilibrium solutions, vibrating systems, and frequency response are revisited and reexamined in different applications and mathematical settings.

To predict the future course of a

dynamical system such as a swinging pendulum, physicists have traditionally counted on solving a suitable equation to find a formula describing the system's position or state at any time.