Dynamic systems are everywhere, from bouncing springs to complex ecosystems. They're the backbone of engineering, control systems, and even nature itself. Understanding these systems helps us make sense of the world around us.

This intro to dynamic systems shows how math models can describe everything from electrical circuits to robot movements. It's a powerful tool that connects different fields, helping us solve real-world problems in innovative ways.

Dynamic systems in engineering

Mechanical systems

• Involve the study of forces, motion, and energy
• Represented by mathematical models such as Newton's laws of motion and Lagrangian mechanics
• Common examples include:
  • Spring-mass-damper systems
  • Pendulums
  • Multi-body systems (robotics, vehicles)

Electrical systems

• Involve the study of electric circuits, electromagnetic fields, and signal processing
• Represented by mathematical models such as Kirchhoff's laws and Maxwell's equations
• Common examples include:
  • RLC circuits (resistor, inductor, capacitor)
  • Power systems (generators, transformers, transmission lines)
  • Communication networks (wireless, optical, satellite)

Fluid systems

• Involve the study of fluid mechanics, thermodynamics, and heat transfer
• Represented by mathematical models such as the Navier-Stokes equations and the laws of thermodynamics
• Common examples include:
  • Hydraulic systems (pumps, valves, actuators)
  • Aerodynamic systems (aircraft, wind turbines)
  • Heat exchangers (radiators, condensers, evaporators)

Dynamic systems in control

Control systems

• Used to regulate the behavior of dynamic systems by applying feedback and feedforward control strategies
• Feedback control measures the output of a system, compares it to a desired reference value, and adjusts the input to minimize the error
• Feedforward control predicts the future behavior of a system based on a model and applies an appropriate input to achieve the desired output
• Common applications include manufacturing, transportation, aerospace, and biomedical engineering

Robotics

• Involves the design and control of autonomous or semi-autonomous machines that interact with their environment and perform tasks
• Dynamic systems in robotics include:
  • Kinematics and dynamics of robot manipulators
  • Control of mobile robots (wheeled, legged, aerial)
  • Coordination of multi-robot systems (swarms, teams)

Dynamic systems in nature

Biological systems

• Involve the study of living organisms and their interactions with each other and their environment
• Represented by mathematical models such as population dynamics and ecological networks
• Dynamic systems in biology include:
  • Regulation of gene expression (transcription, translation)
  • Dynamics of metabolic networks (enzyme kinetics, flux balance analysis)
  • Spread of infectious diseases (SIR models, contact networks)

Environmental systems

• Involve the study of natural processes such as climate, hydrology, and geomorphology
• Represented by mathematical models such as climate models and hydrological models
• Dynamic systems in environmental science include:
  • Global carbon cycle (sources, sinks, feedbacks)
  • Water cycle (precipitation, evaporation, runoff)
  • Dynamics of natural hazards (earthquakes, hurricanes, landslides)

Interdisciplinary nature of dynamic systems

A framework for studying complex systems

• Dynamic systems theory is a framework for studying complex systems that evolve over time
• Applied to a wide range of fields beyond engineering, including physics, chemistry, biology, economics, and social sciences
• Requires a multidisciplinary approach that integrates knowledge from different fields, such as mathematics, computer science, and domain-specific expertise

Enabling cross-fertilization of ideas and techniques

• Provides a common language and set of tools for analyzing and designing complex systems across different domains
• Led to the development of new fields such as:
  • Systems biology (application of dynamic systems concepts to biological systems)
  • Econophysics (application of physics concepts to economic systems)
  • Network science (study of complex networks in various domains)

Developing a broader perspective on problem-solving and innovation

• Appreciating the interdisciplinary nature of dynamic systems theory helps students develop a broader perspective on problem-solving and innovation
• Prepares them for careers in a wide range of industries and academia
• Encourages the integration of knowledge from different fields to tackle complex real-world problems