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Stability and Bifurcation Routes in a Delayed Quadratic Map with Recursive Filtered Memory

This study analytically and numerically characterizes the stability and bifurcation routes of a delayed quadratic map with recursive filtered memory, demonstrating how delay and memory parameters govern fold, flip, and conditional complex-pair transitions within a three-dimensional polynomial framework.

Original authors: Hafidh Khoerul Fata, Yosephus Decupertino Sumanto, Robertus Heri Soelistyo Utomo

Published 2026-06-25

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CC BY 4.0

Original authors: Hafidh Khoerul Fata, Yosephus Decupertino Sumanto, Robertus Heri Soelistyo Utomo

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a system that tries to predict its own future based on two things: what happened just a moment ago, and a "memory" of what happened over the last few moments. This paper studies a mathematical model of such a system, specifically one that behaves like a bouncing ball on a trampoline that has a slight delay and a memory filter.

Here is a simple breakdown of what the researchers found, using everyday analogies.

The Setup: A Delayed Trampoline with a Memory Filter

The authors created a mathematical model (a "map") to describe how a value changes from one step to the next. Think of this value as the height of a ball bouncing.

The Delay ( $k$ ): Imagine the ball doesn't react to the trampoline immediately. It waits one split-second before bouncing back. This is the "delay."
The Memory Filter ( $\eta$ and $\mu$ ): Now, imagine the trampoline doesn't just remember the last bounce, but keeps a running average of the last few bounces.
- $\eta$ (Strength): How much weight the trampoline gives to this memory.
- $\mu$ (Update Rate): How quickly the memory updates. If $\mu$ is high, the trampoline only remembers the very last bounce. If $\mu$ is low, it remembers a long history of bounces, smoothing out the noise.

The researchers wanted to know: When does this system stay stable (bouncing in a predictable pattern), and when does it go crazy (chaos)?

The Two Main Ways the System Can "Break"

In math, when a stable system becomes unstable, it usually crosses a "tipping point." The paper discovered that there are two distinct ways this system can lose its stability, depending on how the memory is tuned.

Route 1: The "Flip" (The Period-Doubling Ladder)

This happens when the memory feedback is positive (like a helpful friend encouraging you).

The Analogy: Imagine you are walking on a tightrope. If you wobble too much, you don't just fall; you start wobbling back and forth in a wider pattern. Then, you start wobbling in a pattern that takes twice as long to repeat. Then four times as long.
What the paper says: When the memory is positive, the system loses stability by "flipping." It goes from a steady rhythm to a rhythm that repeats every 2 steps, then 4, then 8, and eventually becomes chaotic (random). This is the classic "period-doubling" route to chaos.

Route 2: The "Spin" (The Invariant Curve)

This happens when the memory feedback is negative (like a friend who pushes you in the opposite direction of your movement).

The Analogy: Instead of wobbling back and forth, imagine a spinning top. When it gets unstable, it doesn't just fall over; it starts tracing a perfect circle in the air.
What the paper says: When the memory feedback is negative enough, the system doesn't flip. Instead, the stable point turns into a smooth, rotating loop (an "invariant curve"). The system starts spinning around a center point rather than jumping between fixed spots.

The Key Discovery: Memory Decides the Route

The most important finding of this paper is that the memory filter doesn't just make the system "more" or "less" stable; it changes the type of instability.

The "Memory Update Rate" ( $\mu$ ): This is like the speed at which the memory refreshes. The paper found that changing this speed doesn't change where the system sits (the fixed point), but it completely changes how the system vibrates (the multipliers). It's like changing the tension on a guitar string: the note (pitch) stays the same, but the way the string vibrates changes.
The "Sign" of the Memory: The researchers proved that if the memory feedback is positive, you get the "Flip" route (chaos via doubling). If the feedback is negative, you get the "Spin" route (chaos via rotation).

The "Fold-Flip" Coincidence

The paper also found a very specific, rare mathematical condition where the "Flip" route and the "Fold" route (where the system suddenly appears or disappears) happen at the exact same time. They call this a "degeneracy." It's like a tightrope walker who, at a specific moment, is both about to fall off the rope and about to start spinning at the exact same time. This is a complex mathematical boundary that the authors mapped out precisely.

Summary

In short, the authors took a complex equation involving delays and memory, simplified it into a 3D model, and proved that:

Stability depends on a mix of delay and memory strength.
The "Update Rate" of the memory controls the vibration style of the system.
The "Sign" of the memory (positive or negative) acts as a switch: It decides whether the system will crash into chaos by flipping (doubling its rhythm) or by spinning (rotating in a loop).

They used computer simulations to confirm that their math predictions matched the actual behavior of the system, showing that these two different paths to chaos are real and distinct.

Technical Summary: Stability and Bifurcation Routes in a Delayed Quadratic Map with Recursive Filtered Memory

Problem Statement
This study addresses the gap in the systematic characterization of smooth discrete-time models that simultaneously incorporate quadratic nonlinearity, one-step time delay, and recursive filtered-memory feedback. While delayed quadratic maps and memory-modified logistic maps have been studied separately, the specific coupling of explicit delay ( $k$ ) and recursive filtered memory ( $\eta, \mu$ ) within a single polynomial map remains underdeveloped. The primary challenge lies in distinguishing how these distinct feedback mechanisms influence fixed-point stability, multiplier structures, and the specific routes by which stability is lost (e.g., fold, flip, or Neimark–Sacker bifurcations).

Methodology
The authors formulate a delayed quadratic recurrence with first-order filtered-memory feedback:
$x_{n+1} = c - x_n^2 + kx_{n-1} + \eta z_n$
$z_{n+1} = (1-\mu)z_n + \mu x_n$
where $k$ is the delay strength, $\eta$ is the filtered-memory strength, and $\mu$ is the memory update rate.

To analyze the system, the authors:

Embed the System: They embed the scalar recurrence into a smooth three-dimensional polynomial map $F(x, y, z)$ by setting $y_n = x_{n-1}$ , allowing for explicit derivation of the fixed-point branch, Jacobian determinant, and characteristic polynomial.
Normalization: They apply an invertible linear scaling to normalize the quadratic coefficient to unity, ensuring the qualitative stability structure is preserved.
Stability Analysis: They apply the Schur–Jury criterion to the characteristic polynomial of the Jacobian matrix at the fixed point. This reduces the stability conditions to explicit inequalities in terms of the fixed-point coordinate $x^*$ .
Bifurcation Verification: They derive algebraic conditions for genericity (non-degeneracy, transversality, and non-resonance) to classify the unit-circle crossings as fold, flip, or Neimark–Sacker bifurcations.
Numerical Validation: They employ numerical continuations, Lyapunov spectra, Kaplan–Yorke dimensions, and parameter-plane scans to validate the analytical thresholds and observe post-critical attractor transitions.

Key Contributions

Explicit Stability Thresholds: The study derives closed-form stability inequalities in the fixed-point coordinate $x^*$ . It demonstrates that the delay strength $k$ and memory strength $\eta$ enter the fixed-point branch solely through the combined term $(k + \eta)$ , while the memory update rate $\mu$ influences the multiplier structure via the Jacobian but does not affect the location of the fixed points.
Separation of Bifurcation Routes: The authors explicitly separate the stability boundaries into:
- Real-Multiplier Boundaries: Identified as fold ( $\lambda=1$ ) and flip ( $\lambda=-1$ ) thresholds.
- Complex-Pair Boundary: The third Schur–Jury boundary is shown to be a candidate for a Neimark–Sacker bifurcation only under specific sign-dependent conditions on the filtered-memory feedback.
Admissibility Conditions: A critical contribution is the derivation of admissibility conditions for the complex-pair boundary. The paper proves that a Neimark–Sacker bifurcation is only possible if the filtered-memory feedback $\eta$ satisfies specific inequalities relative to $k$ and $\mu$ . In the regime of positive delay and positive filtered feedback, the complex-pair route is mathematically excluded, leaving the flip bifurcation as the sole route to instability.
Fold–Flip Degeneracy: The study identifies the locus $\eta = (2-\mu)(1-k)$ where the fold and flip thresholds coincide, defining a fold–flip degeneracy.

Results

Positive Feedback Regime: For $k > 0$ and $\eta \geq 0$ , the system loses stability exclusively through a flip bifurcation (period-doubling). The upper fixed-point branch becomes unstable at the flip threshold, leading to a cascade of period doublings, chaotic bands, and periodic windows. The complex-pair route is inadmissible in this regime.
Competing Feedback Regime: When the signs of $k$ and $\eta$ allow for competing feedback (specifically sufficiently negative $\eta$ ), the complex-pair boundary becomes admissible. In this scenario, the loss of stability is governed by a complex-conjugate pair crossing the unit circle, leading to invariant-curve-like dynamics (rotational motion) rather than period-doubling.
Role of $\mu$ : The memory update rate $\mu$ does not shift the fixed-point branch but significantly alters the stability boundaries and the admissibility of the complex-pair route by modifying the Jacobian's eigenvalues.
Global Dynamics: Numerical simulations confirm that the analytical thresholds accurately predict the transition from stable fixed points to periodic orbits, chaos, or invariant curves. The phase-space projections illustrate how the delay coordinate and filtered memory variable shape the geometry of the attractors.

Significance and Claims
The paper claims that the recursive filtered-memory state is not merely an additional coordinate but a fundamental mechanism that alters the multiplier geometry of the embedded quadratic map. The study establishes that delay and filtered memory do not simply shift the stability window; they determine which unit-circle mechanism (real multiplier vs. complex pair) is dynamically accessible. By explicitly separating the real-multiplier and complex-pair routes and providing rigorous admissibility conditions, the work clarifies the local bifurcation structure of this class of maps. The authors modestly note that their results are local and focused on fixed points along one-parameter continuations, leaving global bifurcation unfoldings, resonance tongues, and the breakdown of invariant curves for future investigation.