90
3 Computer Simulation of Dynamic Systems
The main property of a dynamic system is that, knowing its state at a certain
moment in time, it is possible to find a state at any subsequent moment in time. For
this, it is enough to apply the law of evolution to the initial state.
The mathematical model of a dynamic system is considered given if the parameters
(coordinates) of the system are entered that determine its state unambiguously and
the law of state evolution in time is indicated.
Depending on the degree to which various factors are considered, different
mathematical models can be associated with the same dynamic system.
Denote the coordinates of the system by x = (x 1 , x 2 , . . . , x n ). These coordinates
are called state variables or phase variables. The time coordinate is denoted by t. The
law of evolution of a dynamic system in time in general is a system of algebraic–
differential equations
F(x, μ, t) = 0
In this equation, the only independent variable is time t, and the state variables
x depend on time, i.e., x = x(t), μ is the vector of parameters that are assumed
to be independent of time t, F = (F 1 , F 2 , . . . , F n ) are some given functions that,
generally speaking, can be nonlinear and include time derivatives of variables state
x.
Naturally, other types of equations can also be used, for example, partial differential equations. Then, some of the state variables can become independent
variables.
If we consider the quantities x 1 , x 2 , . . . , x n as the coordinates of the point x in the
n-dimensional space
n , we get a clear geometric representation of the state of the
dynamical system in the form of this point. This point is called the image or phase
point, and the state space is called the phase space of the system. The change in the
state of the system in time corresponds to the movement of the phase point along a
line called the phase trajectory.
In this tutorial, we restrict ourselves to considering dynamical systems that can
be specified by a mathematical model described by a system of ordinary differential
equations of the first order:
dx
dt
= F(x, μ, t)
where t is time, x = x(t) are state variables (phase variables), μ is a vector of timeindependent parameters, F = (F, F 2 , . . . , F n ) are some given functions that can be
interpreted as phase velocity points in the space
n .
As a rule, even complex engineering problems allow the use of mathematical
models of this kind.
3 Computer Simulation of Dynamic Systems
The main property of a dynamic system is that, knowing its state at a certain
moment in time, it is possible to find a state at any subsequent moment in time. For
this, it is enough to apply the law of evolution to the initial state.
The mathematical model of a dynamic system is considered given if the parameters
(coordinates) of the system are entered that determine its state unambiguously and
the law of state evolution in time is indicated.
Depending on the degree to which various factors are considered, different
mathematical models can be associated with the same dynamic system.
Denote the coordinates of the system by x = (x 1 , x 2 , . . . , x n ). These coordinates
are called state variables or phase variables. The time coordinate is denoted by t. The
law of evolution of a dynamic system in time in general is a system of algebraic–
differential equations
F(x, μ, t) = 0
In this equation, the only independent variable is time t, and the state variables
x depend on time, i.e., x = x(t), μ is the vector of parameters that are assumed
to be independent of time t, F = (F 1 , F 2 , . . . , F n ) are some given functions that,
generally speaking, can be nonlinear and include time derivatives of variables state
x.
Naturally, other types of equations can also be used, for example, partial differential equations. Then, some of the state variables can become independent
variables.
If we consider the quantities x 1 , x 2 , . . . , x n as the coordinates of the point x in the
n-dimensional space
n , we get a clear geometric representation of the state of the
dynamical system in the form of this point. This point is called the image or phase
point, and the state space is called the phase space of the system. The change in the
state of the system in time corresponds to the movement of the phase point along a
line called the phase trajectory.
In this tutorial, we restrict ourselves to considering dynamical systems that can
be specified by a mathematical model described by a system of ordinary differential
equations of the first order:
dx
dt
= F(x, μ, t)
where t is time, x = x(t) are state variables (phase variables), μ is a vector of timeindependent parameters, F = (F, F 2 , . . . , F n ) are some given functions that can be
interpreted as phase velocity points in the space
n .
As a rule, even complex engineering problems allow the use of mathematical
models of this kind.