Dynamical Systems: Chaos and Attractors

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Questions and Answers

What is indicated by a sensitive dependency on initial conditions in a dynamical system?

  • The system is chaotic. (correct)
  • The system is linear.
  • The system exhibits periodic behavior.
  • The system is unstable.

What does a continuous dynamical system utilize as a variable?

  • Discrete intervals
  • Randomized time steps
  • Fixed points
  • Time, t, as a continuous variable (correct)

What can be inferred if $ ext{n} ext{ ≳ } 45$ in the context of a chaotic system's predictions?

  • Behavior of the system stabilizes.
  • Predictions can be made reliably.
  • Orbits converge to a point.
  • Predictions become unfeasible. (correct)

In the context of dynamical systems, what is the purpose of varying parameters?

<p>To achieve qualitatively different behaviors (B)</p> Signup and view all the answers

How is the prediction window increased from 45 to 90 iterations?

<p>By reducing the initial error. (A)</p> Signup and view all the answers

What is one characteristic of discrete dynamical systems?

<p>They utilize maps represented as xn+1 = M(xn). (D)</p> Signup and view all the answers

What characterizes a chaotic attractor within a dynamical system?

<p>Orbits diverging exponentially. (C)</p> Signup and view all the answers

What is an example of a rounding error in a chaotic system known as the Hénon map?

<p>10^{-14} (B)</p> Signup and view all the answers

What is the term used to describe the path followed by a point in a dynamical system as time progresses?

<p>Orbit or trajectory (C)</p> Signup and view all the answers

A flow can be reduced to what form in a discrete dynamical system?

<p>A map of dimension N - 1 (A)</p> Signup and view all the answers

What type of dynamic behavior is represented by attractors that exhibit limit cycles?

<p>Orbits maintain a constant separation indefinitely. (B)</p> Signup and view all the answers

In the context of a dynamical system, what does the term 'strangeness' refer to?

<p>The geometry of the attractor. (D)</p> Signup and view all the answers

What does 'deterministic' mean in the context of dynamical systems?

<p>The systems follow specific equations and produce predictable outcomes. (C)</p> Signup and view all the answers

Which function represents partial measurement of a dynamical system state $X(t)$?

<p>g(t) = G(X(t)) (C)</p> Signup and view all the answers

What mathematical concept helps in understanding chaotic behavior in dynamical systems?

<p>Chaos theory (B)</p> Signup and view all the answers

What is indicated by a situation of long-term unpredictable behavior in a dynamical system?

<p>Chaos (C)</p> Signup and view all the answers

What condition indicates that a system exhibits chaos based on the Lyapunov exponent?

<p>λ &gt; 0 (A)</p> Signup and view all the answers

What does the Kolmogorov-Sinai entropy measure in a chaotic system?

<p>The complexity and randomness of the system (D)</p> Signup and view all the answers

How is the probability of visiting each cell calculated in the Kolmogorov-Sinai entropy method?

<p>By summing the relative frequencies of all trajectories (B)</p> Signup and view all the answers

If all trajectories in a chaotic system evolve closely and only one cell is occupied, what is the value of Sn?

<p>0 (C)</p> Signup and view all the answers

In the Lyapunov exponent formula, which component contributes to measuring the separation of orbits?

<p>ln|M' (xn)| (C)</p> Signup and view all the answers

What is the relationship between the number of occupied cells (Nn) and Kolmogorov-Sinai entropy?

<p>Sn grows logarithmically with Nn (C)</p> Signup and view all the answers

Which scenario describes a regular system within the context of K-S entropy?

<p>Provably finite number of occupied cells (C)</p> Signup and view all the answers

What is the main significance of the limit in the definition of the Lyapunov exponent?

<p>Measures the average divergence of trajectories (C)</p> Signup and view all the answers

What describes the stability of a fixed point when the derivative is less than 1?

<p>The fixed point is stable and attracts nearby points. (D)</p> Signup and view all the answers

In a delay coordinate phase space, what does the function Y represent?

<p>It is a function of an increased dimensionality of the system state. (D)</p> Signup and view all the answers

What is the behavior of the fixed point when the derivative is equal to 0?

<p>The fixed point super-stable and shows little sensitivity to perturbations. (A)</p> Signup and view all the answers

What is the effect when M is sufficiently large in the context of delay coordinate systems?

<p>It reproduces a qualitatively similar attractor structure to X. (A)</p> Signup and view all the answers

In the logistic map, what happens to the population if r is set to 0?

<p>The population will eventually die out to zero. (B)</p> Signup and view all the answers

Which of these describes the maximum value of the function f in the logistic map?

<p>It is found at r = 4. (D)</p> Signup and view all the answers

What happens to the derivative at a fixed point when it is equal to 1?

<p>The fixed point is meta-stable, with possible instability from small disturbances. (A)</p> Signup and view all the answers

In the context of the Lyapunov exponent, what does it signify?

<p>It represents the long-term average divergence of orbits. (A)</p> Signup and view all the answers

What condition characterizes a conservative system in the context of attractors?

<p>J(X) = 1 everywhere (A)</p> Signup and view all the answers

In the context of the damped harmonic oscillator, what is the nature of its attractor?

<p>A fixed point (C)</p> Signup and view all the answers

What defines a strange attractor?

<p>It has a fractal dimension (B)</p> Signup and view all the answers

What happens to orbits in the phase space as the initial condition differences approach zero?

<p>They are limited and tend towards a similar state (B)</p> Signup and view all the answers

For the van der Pol oscillator, what type of attractor is observed?

<p>A limit cycle (A)</p> Signup and view all the answers

What is the possible implication of a Jacobian determinant signifying dissipative behavior?

<p>J(X) &lt; 1 in some region (C)</p> Signup and view all the answers

Which of the following best describes the function of the Hénon map?

<p>It illustrates local structure typical of fractals (D)</p> Signup and view all the answers

What describes the characteristic of the attractor of a damped harmonic oscillator in terms of dimensionality?

<p>It has a dimension of 0 (C)</p> Signup and view all the answers

What does the notation |∆(t)| ↗ exp imply about the behavior of orbits with small initial condition differences over time?

<p>Orbits exhibit exponential divergence (B)</p> Signup and view all the answers

What dimension does a limit cycle possess in terms of attractor classification?

<p>Dimension 1 (B)</p> Signup and view all the answers

What does Kolmogorov-Sinai entropy measure in a dynamical system?

<p>Changes in entropy over time (B)</p> Signup and view all the answers

What condition indicates a chaotic system as per Kolmogorov-Sinai entropy?

<p>K &gt; 0 (D)</p> Signup and view all the answers

In the limit of cells of size 0, what does the limit of K-S entropy approach?

<p>0 or a positive value (B)</p> Signup and view all the answers

If all occupied cells have identical probabilities in a chaotic regime, how is K-entropy expressed?

<p>As the Lyapunov exponent summed over dimensions (A)</p> Signup and view all the answers

What happens to K if all trajectories are fixed points in a dynamical system?

<p>K = 0 (A)</p> Signup and view all the answers

What is the relationship between the initial state and K when considering a random system?

<p>K = log M if initial state is considered (D)</p> Signup and view all the answers

What does the parameter K in Kolmogorov-Sinai entropy indicate?

<p>The growth rate of entropy (B)</p> Signup and view all the answers

How is the limit of K defined for a growing number of iterations P?

<p>As the limit of the difference of entropy normalized by Pτ (C)</p> Signup and view all the answers

Flashcards

Dynamical System

An area of mathematics that deals with deterministic equations representing the time evolution of a system.

Continuous Dynamical System (Flow)

A dynamical system where time, t, is a continuous variable. It's represented by a set of differential equations.

Orbit or Trajectory

The path taken by a point in a continuous dynamical system over time.

Discrete Dynamical System (Map)

A dynamical system where time, n, is discrete and represented by instants. It's defined by a map or a set of equations relating the state at one time to the next.

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Orbit (for maps)

The sequence of points generated by a discrete dynamical system.

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Poincare Section

A technique of reducing a continuous dynamical system in N dimensions to a discrete dynamical system in N-1 dimensions by taking a cross-section of the flow at regular intervals.

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Chaotic Dynamical System

A dynamical system with unpredictable long-term behavior, even when starting with very similar initial conditions.

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Qualitatively Different Behaviors

Different qualitative long-term behaviors exhibited by a system with varying parameters.

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Attractor

A region in a system's phase space where trajectories converge over long time periods, regardless of the starting point. They represent the long-term behavior of the system.

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Dissipative system

A system where energy is not conserved and dissipates over time. This can be due to friction, heat loss, or other factors.

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Conservative system

A system where energy is conserved, meaning there's no energy loss over time. Its dynamics are governed by reversible processes.

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Jacobian (J)

A mathematical measure of how much a system stretches or contracts in phase space. It's calculated as the determinant of the Jacobian matrix of the system's equations of motion.

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Fixed point attractor

A type of attractor that is a single point in phase space. The system eventually settles into this point.

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Limit cycle attractor

A type of attractor that is a closed loop in phase space. The system oscillates around this loop indefinitely.

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Strange attractor

An attractor that has a complex, fractal structure. Its dimension is not an integer, indicating self-similarity at different scales.

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Iterative map

A mathematical map that describes the evolution of a system's state in discrete time steps. It takes an initial state and applies a set of rules to find the next state.

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Dependency on initial conditions

The initial conditions of a system significantly influence its long-term behavior in dissipative systems with strange attractors. Small changes in initial conditions lead to large differences in the system's evolution.

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Differential equation

A mathematical expression that describes the rate of change of a system's state. It helps analyze the dynamics of the system.

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Sensitive Dependency on Initial Conditions

In a chaotic system, even a tiny difference in initial conditions leads to drastically different long-term trajectories.

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Predictability in Chaotic Systems

The power of chaos lies in its ability to make long-term predictions impossible due to exponential error growth. Even a small rounding error in a chaotic system can accumulate rapidly and overwhelm accurate predictions.

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Chaos and Exponential Divergence

A chaotic system shows sensitive dependency on initial conditions where, even with a tiny difference in starting points, two orbits diverge exponentially, creating unpredictable long-term behavior. This behavior is typically seen in systems with attractors that exhibit exponential divergence of trajectories.

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Types of Attractors

An attractor in a dynamical system can be a fixed point (orbits converge to a point), a limit cycle (orbits stabilize around a closed loop), or chaotic (orbits diverge exponentially and chaotically).

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Unpredictability and Chaos

In a chaotic system, the long-term behavior of a system can be unpredictable due to its sensitivity to initial conditions, regardless of how small the error is.

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Delay Coordinate Vector

The delay coordinate vector transforms a scalar measurement of a system into a multidimensional representation, capturing information across different time points. This method allows for the analysis of the dynamics and geometry of attractors, even when only partial information is available, by reconstructing a higher-dimensional embedding.

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Reconstructing Attractors with Limited Data

The delay coordinate vector helps us understand the dynamics and geometric features of an attractor, even when the system is not fully observable. By reconstructing the attractor in a higher-dimensional space using past values of the measurement, we can gain insights into the system’s complex behavior.

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Delay Coordinate Embedding

A method for analyzing complex systems by representing their state over time using a vector of values at different time delays. Each element in the vector represents the system's state at a specific time point.

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Delay Time (τ)

The time interval used to create the delay coordinate vector. It should be roughly the same as the typical time scale of the system's dynamics.

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Embedding Dimension (M)

The number of delay coordinates used to represent the system's state. A higher embedding dimension captures more of the system's complexity.

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Fixed Point

A point in the phase space where the system remains stationary, meaning its state does not change over time.

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Stable Fixed Point

A fixed point is stable if nearby points in the phase space are attracted towards it over time. Think about a ball rolling down towards the bottom of a bowl.

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Unstable Fixed Point

A fixed point is unstable if nearby points are repelled away from it over time. Think about a ball placed on top of an upside-down bowl.

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Super-stable Fixed Point

A fixed point is super-stable if it attracts nearby points very quickly. Think of a very deep and narrow bowl where the ball falls directly to the bottom rapidly.

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Meta-stable Fixed Point

A fixed point is meta-stable if a small change in the system parameters can cause it to become unstable. Think about a ball placed at a very shallow spot in the bowl, where a small push could make it roll away.

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Lyapunov exponent

A measure of the rate at which nearby trajectories in a dynamical system diverge over time. It quantifies the sensitivity to initial conditions, which is a key characteristic of chaotic systems.

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λ > 0 ⇒ chaos

The Lyapunov exponent being positive indicates chaotic behavior. As time increases, the trajectories diverge exponentially.

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λ < 0 ⇒ trajectories converge

The Lyapunov exponent being negative indicates that trajectories converge. This means the system is not chaotic.

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Kolmogorov-Sinai entropy

A measure of the complexity or chaotic behavior of a dynamical system. It quantifies the rate at which information about the system's state is lost over time.

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Sn = 0 (fixed point or limit cycle)

If a system evolves to a fixed point or limit cycle, all trajectories eventually occupy one or a limited number of cells, making the entropy zero or minimal.

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Sn = log Nn (equal probability)

If the number of occupied cells increases over time, but their probabilities are equal, the entropy grows with the logarithm of the number of cells. This implies a chaotic or regular system with no clear trend in entropy.

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Kolmogorov-Sinai Entropy (KSE)

A measure of the information gain in a dynamical system during a single time step, calculated as the difference in Shannon entropy between consecutive steps.

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How does KSE distinguish between chaotic and non-chaotic systems?

The K-entropy, or KSE, is positive for chaotic systems; it signifies the system's inherent unpredictability and exponential divergence of trajectories over time. In non-chaotic systems, it remains zero, indicating deterministic and predictable behavior.

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What is the relationship between K-entropy and Lyapunov exponents?

K-entropy is equivalent to the Lyapunov exponent in chaotic systems. It measures the rate of exponential divergence of nearby trajectories. For a D-dimensional system K-entropy is the sum of all positive Lyapunov exponents.

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How do we calculate K-entropy for chaotic systems with exponential growth?

This represents the total amount of information generated by the system based on its initial state and the rate of divergence of trajectories. It can be understood as the sum of all positive Lyapunov exponents for a D-dimensional system.

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How does K-entropy apply to random systems?

For a random system, the entropy remains constant after the initial step, making K-entropy zero if we consider the initial state. However, when considering the evolution from the initial state, K-entropy equals the logarithm of the number of possible states (M). This is because the system is equally likely to be in any of the M possible states.

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What happens to K-entropy in a fixed point system or a limit cycle?

If all trajectories in a system eventually converge to a fixed point or a limit cycle, then the K-entropy becomes zero due to the lack of divergence and information gain.

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In addition to K-entropy, what other measures can help us understand chaotic systems?

Measures like fractal dimension, correlation dimension, and invariance measure provide additional insights into the complexity and dynamics of chaotic systems. They offer a more nuanced understanding of the system's behavior beyond simple entropy calculations.

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What is a dynamical system?

Dynamical systems are mathematical models that describe how systems evolve over time. Dynamic system studies describe the behavior of systems with varying initial conditions and system parameters.

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Study Notes

Dynamical Systems

  • Dynamical systems are areas of mathematics
  • They use deterministic equations that represent how systems evolve over time
  • Behaviors are qualitatively different depending on varying parameters
  • Some systems exhibit long-term unpredictable behavior

References

Continuous Dynamical Systems

  • Time, t, is a continuous variable
  • The equation dx(t)/dt = F[x(t)] represents the system's evolution.
  • This is also expressed as a vector of equations.
    dx(1)/dt = F₁[x(1), x(2),...,x(N)]
    dx(2)/dt = F₂[x(1), x(2),...,x(N)]
         ...
    dx(N)/dt = FN[x(1), x(2),...,x(N)]
    

Discrete Dynamical Systems

  • Time, n, is discrete (instants)
    • n ∈ {0, 1, 2,...}
  • A map describes the system as
    Xn+1 = M (xn)
        with
        Xn = (x₁), x₂),... , x(n)
        M = M₁, M₂, ..., MN
    
  • Orbits are similar to flows.

Flows to Maps

  • A flow of dimension N can be reduced to a map of dimension N-1 via a Poincaré section surface.

Orbti Complexity

  • Flows (N ≥ 3) may exhibit chaos.
  • Maps (N ≥ 2) may exhibit chaos.
  • Non-invertible maps can show chaos with N = 1

Logistic Equation

  • M(x) = rx(1-x)
  • Chaos is observed for specific ranges of r.

Conservative Systems

  • All points within an initial closed surface (dimension N-1) evolve to points on another closed surface (same dimension) in time t.
  • Volumes (V(0), V(t)) delimited by such surfaces remain equal.
  • Newton equations of frictionless body movement are examples.
  • Non-conservative systems do not conserve volume.

Conservative Systems (Formally)

  • Formally, the divergence theorem explains conservation of divergence as
    dV(t)/dt =∫∫∫ S_i FdN_x
    ∇ ⋅ F = Σ₁ₙ Fᵢ(x₁, ..., xₙ) / ∂xᵢ
    
  • A system is conservative if dV(t)/dt = 0; dissipative otherwise
    • In maps: conservative if J(X)= |det[M(X) / ∂x]| = 1
    • dissipative if J(X) <1 in some region

Attractors

  • Fixed points: orbits converge to a point, exponentially
  • Limit cycles: orbits get close to a difference in the order of △(0) and maintain the proximity indefinitely.
  • Chaotic: orbits diverge exponentially

Partial Measurement

  • Not all system state components (X(t)) are always accessible
  • Use g(t)= G(X(t)) for one component/ a scalar function of X(t) to learn attractor (geometry & dynamics)
  • Delay coordinate vector Y is developed from these g(t) values with delay T (~ characteristic time of g(t))

Delay Coordinate Vector

  • Integration back in time provides X(t-mτ)=Lm(X(t)).
  • Thus, Y= H(X), which is a function of the original phase space (X)
  • A large value of M yields a phase space Y with attractor structure similar to X

Stability of Fixed Points

Case Condition Description
Stable df(x)/dx <
Unstable df(x)/dx >
Super-stable df(x)/dx =
Meta-stable df(x)/dx =
Description Remarks
Lyapunov exponent Long-term average divergence of orbits

Logistic Map

  • A simplified model of annual insect population variation. Xn₊₁=rXn(1-Xn)
  • Conserves volume for 0 < r < 4
  • x* = 0, x* = 1-1/r are the attractors.

Logistic Map Stability

  • Study attractor stability by calculating λ= |r(1-2x)|
  • For attractors:
    • x* = 0 ⇒ λ = |r|
    • x* = 1 - 1/r ⇒ λ= |2-r|

Bifurcation Diagram

  • Transition from stable to unstable behavior

Quasi-periodicity

  • Periodic functions (continuous/discrete) with multiple frequencies
  • The frequencies aren't commensurate.

Route to Chaos

  • Transition steps: stable state, periodic orbits, limit cycle, quasi-periodicity, chaos

Symbolic Dynamics

  • Representation of a dynamical system's phase space by regions with labels.
  • Converts numeric trajectories into symbolic sequences.
  • Formal language (language structure) analysis of these sequences.

Feigenbaum Constants

  • Universal constants describing bifurcation rates in dissipative systems entering a chaotic stage.
  • rₘ - rₘ-₁ / rₘ₊₁ - rₘ → 4.6692....
  • dₘ / dₘ₊₁ → -2.5028....

Lyapunov Exponent

  • Measures the exponential divergence of trajectories in a dynamical system.
  • Positive value indicates chaos.
  • Zero value indicates non-chaos

Uni-Dimensional Flows (No Chaos)

Type Description λ
Attractor Stable fixed point λ < 0
Repeller Unstable fixed point λ > 0
Saddle Attracts on one side, repels on another both

Rare Cases in 1D;

Characteristic (λ) Case Observations
λ = 0 Fixed point, same signal left/right; Structural instability. Attractor and/or repeller can change type or disappear.

2D Saddle Point

  • λ₁ < 0, λ₂ > 0, or combinations
  • fixed point classification based on the signs of λ₁ and λ₂

Landscape as Phase Space of a 2D Dynamical System

  • Visualization of phase space using a topographic plot.

Lyapunov Exponents in Flows

  • The system is chaotic if at least one Lyapunov exponent is positive. (+,-,-), (+,0,-), (0,-,0): fixed point, limit cycle, quasi-periodic torus, chaotic)

Kolmogorov-Sinai Entropy

  • Entropy measurement of dynamic system complexity/chaotic behavior
  • Probability calculation (relative frequency) for the system visiting each cell;
  • If the number of occupied cells gets bigger, the entropy grows. a higher value reflects more complex behavior

Other Measures of Dynamics (Chaos)

  • Fractal dimension
  • Correlation dimension
  • Invariance measure

Concepts

  • Complexity is correlated to information.
  • Entropy measures randomness; not complexity.

Coupled Fingers Oscillation

  • A real-world example of coupled dynamical systems.

L-systems

  • A class of rewriting systems that create strings/images of symbols.
  • Alphabet
  • Axiom
  • Rules

L-system with Turtle Graphics

  • Used to generate fractals
  • Drawing shapes iteratively

Other Measures of Dynamics (Chaos)

  • Fractal dimension
  • Correlation dimension
  • Invariance measure

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