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Summary on attractors
Topic: Attractors
by Aleks, 2018 Cohort
Note: This entry was created in 2018, when the task was to “summarise a key reading”, and so may not represent a good example to model current primer entries on.
Background#
Nearly every nontrivial real-world system is a nonlinear dynamical system. Better understanding the behaviour of these systems therefore has far-reaching applications and can help us to tackle complexity.
System: A set of interacting components that form a whole.
Nonlinear: The whole system is not just the sum of its parts.
Dynamical: The system changes over time based on its current state.
Attractors#
An attractor is a state toward which a system tends to evolve, for a wide range of initial system conditions. Attractors can have many types: fixed point(s), a finite number of points, limit cycles, or strange attractors. These correspond to different dynamical behaviours of the system.
A pendulum is an example of a fixed-point attractor. Regardless of where the pendulum is released, it will always end up in the centre bottom position over time due to gravity. Note that it is possible to keep a pendulum away from its attractor, it just requires a constant effort to continually push the pendulum. This is an example of an ineffective, yet easily implementable leverage point in the system. You could also remove the effects of gravity on the pendulum, stopping it from ever settling. This is an effective solution, yet very difficult to implement. This example illustrates that working within system boundaries is subject to their dynamical behaviour, whilst changing system parameters can fundamentally alter dynamical behaviour.
Predator and Prey populations within a closed environment demonstrate limit cycle behaviour in their attractor. Figure 1 shows predator and prey populations over time. As the prey population increases, predators have more food and their population grows. However, as the predator population increases, they can no longer be sustained by the prey population and their population falls. This cycle repeats indefinitely.
Predator and prey populations as functions of one another. Here we can clearly see a limit cycle as the attractor of this system. Over time the populations will continue to follow this trajectory.
A strange attractor is one that describes a chaotic system where the state changes deterministically but in unpredictable patterns, making them impossible to analyse. Because of this relatable examples of strange attractors are hard to find, as systems with them tend to look completely random.
Changing certain system parameters can fundamentally alter the behaviour of a system and change its attractor from something well-defined (fixed point, finite points, limit cycle) to a strange attractor. Therefore, extreme care should be taken in altering systems to avoid chaos.
Applications#
Complex problems inevitably involve nonlinear dynamical systems. For example, extreme inequality is a major disruptive force in society. However, it is naturally occurring as described by the Pareto principle, where 20% of the population hold 80% of the wealth. This principle is applicable over many other natural phenomena. This distribution of production can be considered an attractor of these systems. Its not clear how to alter these attractors, and caution should be taken to avoid changing parameters to the point of inducing chaos.
Living standards were greatly improved due to technological advances in the industrial revolution. However, over time as these advancements have continued it has become unclear whether they are completely beneficial. Technology can be thought of as a parameter that has fundamentally changed the behaviour of many systems, the climate for example, leading to chaos and uncertainty.
Overall, we should be aware of attractors within our systems, and be cautious in implementing changes to them, or risk descending into chaos.
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Disclaimer#
This content has been contributed by a student as part of a learning activity.
If there are inaccuracies, or opportunities for significant improvement on this topic, feedback is welcome on how to improve the resource.
You can improve articles on this topic as a student in "Unravelling Complexity", or by including the amendments in an email to: Chris.Browne@anu.edu.au
Note: This entry was created in 2018, when the task was to “summarise a key reading”, and so may not represent a good example to model current primer entries on.
Background#
Nearly every nontrivial real-world system is a nonlinear dynamical system. Better understanding the behaviour of these systems therefore has far-reaching applications and can help us to tackle complexity.
System: A set of interacting components that form a whole.
Nonlinear: The whole system is not just the sum of its parts.
Dynamical: The system changes over time based on its current state.
Attractors#
An attractor is a state toward which a system tends to evolve, for a wide range of initial system conditions. Attractors can have many types: fixed point(s), a finite number of points, limit cycles, or strange attractors. These correspond to different dynamical behaviours of the system.
A pendulum is an example of a fixed-point attractor. Regardless of where the pendulum is released, it will always end up in the centre bottom position over time due to gravity. Note that it is possible to keep a pendulum away from its attractor, it just requires a constant effort to continually push the pendulum. This is an example of an ineffective, yet easily implementable leverage point in the system. You could also remove the effects of gravity on the pendulum, stopping it from ever settling. This is an effective solution, yet very difficult to implement. This example illustrates that working within system boundaries is subject to their dynamical behaviour, whilst changing system parameters can fundamentally alter dynamical behaviour.
Predator and Prey populations within a closed environment demonstrate limit cycle behaviour in their attractor. Figure 1 shows predator and prey populations over time. As the prey population increases, predators have more food and their population grows. However, as the predator population increases, they can no longer be sustained by the prey population and their population falls. This cycle repeats indefinitely.
Predator and prey populations as functions of one another. Here we can clearly see a limit cycle as the attractor of this system. Over time the populations will continue to follow this trajectory.
A strange attractor is one that describes a chaotic system where the state changes deterministically but in unpredictable patterns, making them impossible to analyse. Because of this relatable examples of strange attractors are hard to find, as systems with them tend to look completely random.
Changing certain system parameters can fundamentally alter the behaviour of a system and change its attractor from something well-defined (fixed point, finite points, limit cycle) to a strange attractor. Therefore, extreme care should be taken in altering systems to avoid chaos.
Applications#
Complex problems inevitably involve nonlinear dynamical systems. For example, extreme inequality is a major disruptive force in society. However, it is naturally occurring as described by the Pareto principle, where 20% of the population hold 80% of the wealth. This principle is applicable over many other natural phenomena. This distribution of production can be considered an attractor of these systems. Its not clear how to alter these attractors, and caution should be taken to avoid changing parameters to the point of inducing chaos.
Living standards were greatly improved due to technological advances in the industrial revolution. However, over time as these advancements have continued it has become unclear whether they are completely beneficial. Technology can be thought of as a parameter that has fundamentally changed the behaviour of many systems, the climate for example, leading to chaos and uncertainty.
Overall, we should be aware of attractors within our systems, and be cautious in implementing changes to them, or risk descending into chaos.