Swarm simulationusing anti-Newtonian

forces

Swarm simulationusing anti-Newtonian

forces

Vladimir ZhdankinChaos and Complex Systems

October 6, 2009

Vladimir ZhdankinChaos and Complex Systems

October 6, 2009

OutlineOutline

Nature Overview of observed swarming

Computer simulation Swarming model Selected cases

No predatorSingle predatorMultiple predatorsBlack sheep

Conclusions

Nature Overview of observed swarming

Computer simulation Swarming model Selected cases

No predatorSingle predatorMultiple predatorsBlack sheep

Conclusions

Swarming in natureSwarming in nature Birds flocks Fish schools Insect swarms Mammal herds Human crowds

Birds flocks Fish schools Insect swarms Mammal herds Human crowds

Why swarm?Why swarm? Defense from predators

Confuses predator Improves perception Lowers predator success rate

Foraging Mating Navigation

Defense from predators Confuses predator Improves perception Lowers predator success rate

Foraging Mating Navigation

Predator optionsPredator options Form hunting packs

Divert a prey away from swarm and catch

Spread and surround the swarm Lead swarm into a trap

Form hunting packs Divert a prey away from swarm and catch

Spread and surround the swarm Lead swarm into a trap

How does swarming happen?

How does swarming happen?

Emergence Organization arises from repetition of simple actions

Each individual makes some decisions Chooses optimal distance from neighbors

Aligns with neighbors Reacts to obstacles

Emergence Organization arises from repetition of simple actions

Each individual makes some decisions Chooses optimal distance from neighbors

Aligns with neighbors Reacts to obstacles

SimulationSimulation Model each member as a particle (“agent”)

Model landscape as Cartesian plane Implement force laws

Long range attraction Short range repulsion Friction Anti-Newtonian force between predator and prey

Model each member as a particle (“agent”)

Model landscape as Cartesian plane Implement force laws

Long range attraction Short range repulsion Friction Anti-Newtonian force between predator and prey

Anti-Newtonian forceAnti-Newtonian force

Term coined by Clint Sprott Disobeys Newton’s Third Law

Newtonian forces are equal in magnitude and opposite in

direction Anti-Newtonian forces are equal in magnitude and equal in direction

Term coined by Clint Sprott Disobeys Newton’s Third Law

Newtonian forces are equal in magnitude and opposite in

direction Anti-Newtonian forces are equal in magnitude and equal in direction

Circular orbitCircular orbit

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Elliptical orbitElliptical orbit

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Precessing orbitPrecessing orbit

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n =1,2,3,...N-body anti-Newtonian

problemN-body anti-Newtonian

problem With more bodies, simplest choice is to have no force between similar agents

For swarming, can add Newtonian forces Attractive force between rabbits is natural

Force between foxes is not as obviousAttraction to form hunting packs?Repulsion to spread and surround rabbits?No interaction at all?

With more bodies, simplest choice is to have no force between similar agents

For swarming, can add Newtonian forces Attractive force between rabbits is natural

Force between foxes is not as obviousAttraction to form hunting packs?Repulsion to spread and surround rabbits?No interaction at all?

Equations of motionEquations of motion

Can adjust to give predator repulsion instead of attraction

Equation parametersEquation parameters

Agent parameters: Mass m Coefficient of friction b Priority p (scales force toward agent)

Force parameters: Long-range force power γ Short-range repulsion power α Usually γ = -1 and α = -2 works best

Agent parameters: Mass m Coefficient of friction b Priority p (scales force toward agent)

Force parameters: Long-range force power γ Short-range repulsion power α Usually γ = -1 and α = -2 works best

Trivial case (no predator)

Trivial case (no predator)

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Trivial case (no predator)

Trivial case (no predator)

Trivial case equilibria

Trivial case equilibria

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+aˆ v

Notes on trivial caseNotes on trivial case

Uninteresting approximation of nature

For complexity, add other terms: External potential Self-propulsion Noise

Or, introduce a predator…

Uninteresting approximation of nature

For complexity, add other terms: External potential Self-propulsion Noise

Or, introduce a predator…€

−∇φ

Single predatorSingle predator

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Single predator - chaotic

Single predator - chaotic

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Can the predator win?Can the predator win?

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Can the predator win?Can the predator win?

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Why not γ=0?Why not γ=0?

Why not γ=0?Why not γ=0?

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Older equations of motion

Older equations of motion

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Multiple predatorsMultiple predators

Two predators repelling

Two predators repelling

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Two predators attracting

Two predators attracting

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Predator packsPredator packs

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More complicated caseMore complicated case

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An unrealistic solution

An unrealistic solution

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A black sheepA black sheep

One swarm agent may have handicaps Injured, sick, or weak in nature Higher mass, friction, or priority in simulation

In nature, predators target these prey

Will it happen in simulation?

One swarm agent may have handicaps Injured, sick, or weak in nature Higher mass, friction, or priority in simulation

In nature, predators target these prey

Will it happen in simulation?

Black sheep - greater friction

Black sheep - greater friction

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Black sheep - higher priority

Black sheep - higher priority

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Black sheep - increased massBlack sheep - increased mass

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Emergence in simulation

Emergence in simulation

Swarming maneuvers Unified motion (away from predators) Splitting to confuse predators

Predator actions Diverting one agent away from swarm Capturing the black sheep

All of these come about from using the anti-Newtonian force

Swarming maneuvers Unified motion (away from predators) Splitting to confuse predators

Predator actions Diverting one agent away from swarm Capturing the black sheep

All of these come about from using the anti-Newtonian force

ConclusionsConclusions

Swarming behavior can be approximated by modeling swarm members as particles that obey simple force laws

The anti-Newtonian force plays a critical role in the swarm dynamics

Emergence is responsible for part of Nature’s complexity

Swarming behavior can be approximated by modeling swarm members as particles that obey simple force laws

The anti-Newtonian force plays a critical role in the swarm dynamics

Emergence is responsible for part of Nature’s complexity

AcknowledgementsAcknowledgements

Clint SprottClint Sprott

ReferencesReferences

Images of swarms in nature are from National Geographic: http://photography.nationalgeographic.com/

Images of swarms in nature are from National Geographic: http://photography.nationalgeographic.com/