Structure and Behavior of Dynamic Systems System Dynamics Structure and Behavior of Dynamic Systems Morteza Bazrafshan

Structure and Behavior of Dynamic Systems The behavior of a system arises from its structure. We intend to provides an overview of dynamics focusing on the relationship between structure and behavior. The basic modes of behavior in dynamic systems are identified along with the feedback structures generating them. رفتار یک سیستم از ساختار آن ناشی می شود. رفتار پایه در سیستم های دینامیکی بر اساس ساختار فیدبک ها مطالعه می شود.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Change takes many forms, and the variety of dynamics around us is astounding. In fact, most dynamics are instances of a fairly small number of distinct patterns of behavior.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR The most fundamental modes of behavior are exponential growth, goal seeking, and oscillation. Each of these is generated by a simple feedback structure: - Growth arises from positive feedback - Goal seeking arises from negative feedback - Oscillation arises from negative feedback with time delays in the loop. Other common modes of behavior, including S-shaped growth, S-shaped growth with overshoot and oscillation, and overshoot and collapse, arise from nonlinear interactions of the fundamental feedback structures. بیشتر شکل رفتارها که در این درس مورد بررسی قرار می گیرد رشد نمایی، جستجوی هدف و نوسان می باشد. هر یک از رفتارها بر اساس فیدبک ها ایجاد می شود.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Exponential Growth Exponential growth arises from positive (self-reinforcing) feedback. The larger the quantity, the greater its net increase, further augmenting the quantity and leading to ever-faster growth Pure exponential growth has the remarkable property that the doubling time is constant: the state of the system doubles in a fixed period of time, no matter how large. زمان رشد معمولا بر مبنای زمان مورد نیاز برای دو برابر شدن وضعیت سیستم مورد ارزیابی قرار می گیرد.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Exponential Growth Positive feedback need not always generate growth. It can also create self-reinforcing decline As when a drop in stock prices erodes investor confidence which leads to more selling, lower prices, and still lower confidence. What about linear growth? Linear growth is actually quite rare. Linear growth requires that there be no feedback from the state of the system to the net increase rate, because the net increase remains constant even as the state of the system changes. What appears to be linear growth is often actually exponential, but viewed over a time horizon too short to observe the acceleration. افزایش خطی به معنی عدم تاثیرپذیری از وضعیت سیستم میباشد. معمولا رشد خطی جزیی از یک رشد نمایی است که در بازه کوتاه زمانی مشاهده می شود.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Exponential Growth Average growth rate 3.45%/Year Doubling time 20 Years Average growth rate 1926-1995 3.5%/Year D.time 20 Years 1970-1995 6.8%/Year D.time 10 Years Billion $/Year Average growth rate 1900-1950 0.86%/Year D.time 80 Years 1950-2000 1.76%/Year D.time 40 Years Average growth rate 34%/Year D.time 2 Years Upper Bound Best Fit Exponantial

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR When a Rate Is Not a Rate In dynamic modeling, the term "rate" generally refers to the absolute rate of change in a quantity. The term "birth rate" here refers to the number of people born per time period. Often, however, the term "rate" is used as shorthand for the fractional rate of change of a variable. For example, the birth rate is often interpreted as the number of births per year per thousand people Similarly, we commonly speak of the unemployment rate. The word "rate" in these cases actually means "ratio": the unemployment rate is the ratio of the number of unemployed workers to the labor force. Select variable names that minimize the chance for confusion. Be sure to check the units of measure for your rates. The units of measure for rates of flow are units/time period; the units of measure for fractional rates of flow are units per unit per time period = 1/time periods. در هنگام استفاده از مفهوم نرخ باید توجه شود که این مفهوم ایجاد تغییر را نشان دهد. همانند نرخ تولد اما نرخ بیکاری مفهوم دینامیک ندارد.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Goal Seeking Negative loops seek balance, equilibrium, and stasis. Negative feedback loops act to bring the state of the system in line with a goal or desired state. When the relationship between the size of the gap and the corrective action is linear, the rate of adjustment is exactly proportional to the size of the gap and the resulting goal-seeking behavior is exponential decay. As the gap falls, so too does the adjustment rate. Just as exponential growth is characterized by its doubling time, pure exponential decay is characterized by its half life -the time it takes for half the remaining gap to be eliminated هرگاه رابطه بین تفاوت (gap ) فعالیت اصلاحی خطی باشد نرخ اصلاح به صورت نمایی خواهد شد. به همین دلیل با کم شدن gap زمان اصلاح بسیار طولانی می شود. همانند وضعیت افزایشی در حالت کاهشی زمان نصف شدن وضعیت سیستم دارای اهمیت می باشد.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Goal Seeking

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Oscillation Oscillation is the third fundamental mode of behavior observed in dynamic systems. Like goal-seeking behavior, oscillations are caused by negative feedback loops. In an oscillatory system, the state of the system constantly overshoots its goal or equilibrium state, reverses, then undershoots, and so on. The overshooting arises from the presence of significant time delays in the negative loop. The time delays cause corrective actions to continue even after the state of the system reaches its goal, forcing the system to adjust too much, and triggering a new correction in the opposite direction افزایش ناگهانی به دلیل تاخیرات زمانی در لوپ‌های منفی می باشد.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Oscillation

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Oscillation Oscillations are among the most common modes of behavior in dynamic systems. There are many types of oscillation, including: Damped oscillations Limit cycles Chaos Oscillations can arise if there is a significant delay in any part of the negative loop. - There may be delays in perceiving the state of the system caused by the measurement and reporting system. - There may be delays in initiating corrective actions after the discrepancy is perceived due to the time required to reach a decision. - There may be delays between the initiation of a corrective action and its effect on the state of the system.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Oscillation It takes time for a company to measure and report inventory levels, time for management to meet and decide how much to produce, and more time while raw materials procurement, the labor force, and other needed resources respond to the new production schedule. Sufficiently long delays at anyone of these points could cause inventory to oscillate.

FUNDAMENTAL MODES OF DYNAMIC BEHAVIOR Oscillation

INTERACTIONS OF THE FUNDAMENTAL MODES S-shaped Growth No real quantity can grow forever: eventually one or more constraints halt the growth. A commonly observed mode of behavior in dynamic systems is S-shaped growth-growth is exponential at first, but then gradually slows until the state of the system reaches an equilibrium level. The shape of the curve resembles a stretched-out "S" To understand the structure underlying S-shaped growth it is helpful to use the ecological concept of carrying capacity. The carrying capacity of any habitat is the number of organisms of a particular type it can support and is determined by the resources available in the environment and the resource requirements of the population. As a population approaches its carrying capacity, resources per capita diminish thereby reducing the fractional net increase rate until there are just enough resources per capita to balance births and deaths

INTERACTIONS OF THE FUNDAMENTAL MODES

INTERACTIONS OF THE FUNDAMENTAL MODES S-shaped Growth A system generates S-shaped growth only if two critical conditions are met: First, the negative loops must not include any significant time delays Second, the carrying capacity must be fixed.

INTERACTIONS OF THE FUNDAMENTAL MODES S-Shaped Growth with Overshoot S-shaped growth requires the negative feedbacks that constrain growth to act swiftly as the carrying capacity is approached. Often, however, there are significant time delays in these negative loops. Time delays in the negative loops lead to the possibility that the state of the system will overshoot and oscillate around the carrying capacity

INTERACTIONS OF THE FUNDAMENTAL MODES

INTERACTIONS OF THE FUNDAMENTAL MODES Overshoot and Collapse The second critical assumption underlying S-shaped growth is that the carrying capacity is fixed. Often, however, the ability of the environment to support a growing population is eroded or consumed by the population itself. For example, the population of deer in a forest can grow so large that they overbrowse the vegetation, leading to starvation and a precipitous decline in the population.

INTERACTIONS OF THE FUNDAMENTAL MODES About $600 M Loss