Dynamic Systems Theory

FIELDS OF STUDY: Thermodynamics; Classical Mechanics; Harmonics

ABSTRACT: Dynamic systems theory examines the properties of systems that change over time, as opposed to static systems that do not undergo time-related change. Systems that change incrementally are treated with difference equations. Systems that undergo continuous change are treated with differential calculus.

PRINCIPAL TERMS

• dynamic systems: systems that are subject to change.
• entropy: the degree of randomness of the components within a system.
• mixing: being able to transform a dynamic system with multiple phase spaces in its initial state in more than one way over time to reach a target state that has completely overlapping phase spaces.
• periodicity: the extent to which a property repeats over time; regular recurrence.
• phase space: a space in which all of a dynamic system’s possible states are represented.
• thermodynamics: the study of the transfer of heat energy into other forms of energy, and vice versa.

Dynamic Systems and Static Systems

Any physical system can be described either as a static system or a dynamic system. The components of a dynamic system change, while those of a static system do not. A golf ball on a tee is a static system because it undergoes no change. When the ball is struck by a golf club, it becomes part of a dynamic system that includes the golf club, the golfer, and Earth through the force of gravity and the atmosphere. These components act to change the state of the system over time. Mechanical systems are dynamic systems while they are performing functions. When they are idle, they are static. Chemical systems are dynamic systems at all times because atoms and molecules in the system are in a constant state of motion. Motion is defined as a change in location over time.

Atomic and molecular motion reflect changes in the heat energy of the system, as described by thermodynamics. A central feature of thermodynamics and of systems in general is the entropy of the system. Entropy is the extent to which the system is in a state of disorder or randomness. A crystalline solid, for example, has low entropy compared to the same material in the liquid state. In the solid form, atoms and molecules are locked into a rigid, ordered array, though they still move due to vibrations and rotations of bonds. As the temperature of the solid rises, the atoms and molecules move more energetically. At the material’s melting point, these motions are energetic enough to overcome the forces that maintain the orderly array of molecules. The atoms and molecules can then move more freely instead of remaining locked in place. Because the orderliness of the system has decreased, its entropy has increased.

Entropy can be seen in a simple cup of coffee when sugar is added to it. As the coffee’s water molecules interact with the sugar’s sucrose molecules, the temperature of the sucrose increases to that of the water, and the sugar dissolves. Thus, the sugar becomes an integrated component of the overall system.

Mixing and Phase

The cup of coffee is a dynamic system that undergoes mixing as the coffee combines with milk and sugar. In its initial state, the system has multiple phase spaces. A phase space is a space that shows all of the possible states of a system. For the cup of coffee system, the sugar, coffee, and milk each have a separate phase space. The cup of coffee system can be disturbed in several ways. It can be stirred. It can be swirled. It can be shaken. Yet no matter which way the cup of coffee is disturbed, over time, the contents combine to reach the same target state, a drinkable cup of coffee. In this target state, the phase spaces of the coffee, sugar, and milk completely overlap one another.

Systematic Change

Dynamic systems can undergo change in different ways. The mathematics used in dynamic systems theory differs accordingly. Systems that change incrementally are described using difference equations that use specific value differences in their solution. Systems that change continuously require the use of differential equations that use derivative and integral calculus in their solution. Some systems that undergo continuous change do so in a cyclic manner. They may change their value between two limiting values or over a regular period of time. This periodicity can be described using differential calculus based on sine and cosine functions.

The broad study of particle dynamics examines those systematic changes that affect motion and energy, and is described using both Newtonian and relativistic mechanics. Newtonian mechanics describes the actions of the planets and other bodies that make up the solar system. Although Newtonian mechanics can also be used to describe the motion and energy of atoms and molecules, these phenomena are more precisely described by quantum mechanics and other disciplines. On the fundamental subatomic level, quantum field theories alone are adequate to describe the motion and energy of the relevant particles.

Working with Dynamic Systems

The primary feature of any dynamic system is the way in which the motion and energy of its components change over time. Complex machines, while carrying out their defined functions, are dynamic systems having strict boundaries. A robotic welding machine, for example, cannot carry out functions beyond those defined within the computer algorithm that controls the robot’s movements. There are many applications, however, that do not have such boundaries. Chemical engineering uses processes that depend on mixing for their effectiveness and the facilitation of their control. An explosion, such as occurs in an internal combustion engine, is a dynamic system. The flow of water through a turbine to generate electricity is also a dynamic system. Understanding such systems enables both their control and their improvement, as well as the development of new systems and applications.

Biological systems are perhaps the ultimate examples of dynamic systems. In the human body, for example, a multitude of chemical reactions involving many different molecules take place every second. The natural world itself is an adaptive dynamic system that has undergone a continual process of dynamic change over billions of years. One example of adaptive dynamic change is the coevolution of different species due to some beneficial relationship. Bees and flowers are an example of coevolution in which both species have evolved features that benefit the other species. Bees have evolved in ways that better utilize the nectar of flowers to maintain the survival of bees, while flowers have evolved nectar production and structural features that better enable visiting bees to pollinate the flowers that they visit to obtain nectar. The mathematics of biological dynamic systems are accordingly complex, but they are also essential for understanding the ecologies of those systems.

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