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Fractal Math, Chaos Theory & Research => Mandelbrot & Julia Set => Topic started by: kjknohw on April 09, 2017, 09:01:35 PM




Title: z^fickle + c
Post by: kjknohw on April 09, 2017, 09:01:35 PM
We all know about c -> z^n + c for some n. Lets add a wrinkle: n is max(2,2+round(-k*log(abs(z)))) for some k>0. This will produce discontinuities in the fractal when n jumps but it also introduces a lot of interesting phenomena. Furthermore, we can multiply by a constant: z -> f(n)*z^n + c. It's natural to set f(n) = exp(0.5*x*x/k), so that abs(f(n)*z^n) is continuous. There is a transition around k = 0.32, lets start with k = 0.3, which superficially looks like the m-set.

The jumps in n quickly produces strange effects such as scrambling the "period n-folding" rule. Note the 3 fold symmetry and 12-fold symmetry deeper in for a power 4 minibrot:
(http://nocache-nocookies.digitalgott.com/gallery/20/4325_09_04_17_7_59_38.png)


This one can't seem to make up it's mind:

(https://lh3.googleusercontent.com/5kbEGpVfWfOJVPx8RUqDFrmw3YSTwnMrjliHfZV4TjYNNMO2xZVD1bg1g6ZY-2LT67EmPEr7DnrMY7iGz8U1hBz-pXLA_jFVmRLoAG2gVjCG2RtsBV_h6ABrBVFsjUo8PHZSCREpxonzwB_VKVwJIPmh6rSO9ZETCdrtI4r_lOcZKGmreKjM_5XXEnmG5FfhrPTWN9AYj1OUOCdv_JRu-wRjJ9eWZ_ZmPY8ThMcNYG0yov7UZg4eBI88aDwC0_2g46xB_SU7giUuZtBaUUrKnCr_Y7lcs1vXna9BM6el5bn82LC4xLPPESqEHR87sKcjSqKYUd-5HHlxyw5QJBlLhIar58cFmeOR4nVRilr8GJjMSpQCMXzQRtmGCCbGg__haqE4mAKaepoau1-GNSPRbaXkEW_mZ0V055YLORXvFm43WrJu54AqFJmrucPR7wy9hGgA5Q-zs_H0NN3Oe04ck2oyv3xSQ_3WtJcEOlIHpvcoUWv-B3CwkZ9u2afsB1cq6x4edn15Lb4797So-_6YWIw1LQ9PrATj_HycH9mfl0Zvua1l534eoVtMbeVOqHGsk2YHvEXytNBeHdWaV0sMUottRY0RfCm-pfGoKKtpr2IqqeevI5k=w640-h446-no)

Zooming in, the mandelbrot set becomes a collage of various features. Most of these cannot be unseen (these features will repeat themselves when we find deeper baby mandelbrot sets):

(https://lh3.googleusercontent.com/jA0VC5XgwDtYpcWoGIb55GI11Z48aLhpygAlFJYVQLPXVzGFEPfbP8Mah5zXSHSigHQrfYgw3wOrCfrxVC8ak0PHQyyOl4d25voAU6JSweiZFzGKdFOUCxPtMV9mf71CEqCVKvzycebLDwlO_RbRqiXxYtCQQl9bINEK9OMGF4wRBO4pOVNrVdwQdUmAi8p50OKMFnZRhBeZt5XxecjEKcctttx41KG1NjsOurtbEK2KgKuv3wCb8Yy2tWNu5ZPiG7T5yYoPFlIt4W2HlNL-XHSOPs2D-MDcDM6r_QcMv13_8IogRPYLXA0Job4xZTujZz7_NK4MYr6B1dc6u_hSq4N2oJv12sGVDAs_WwBRL5xRQhoswz_gZtlUAVudNwIm4lW9mFmRaefyMDyvC1hUnrzc0tGIrrGh0rMaZ0ObuEDbmLRLosPqx5EGIDEQKQACKNL-Ymba5CxWpxdhHahXqOqo1K-beFiHIatrXMXtjWeKLLSIg4H0HKi-qcNcYyTojJF7rYOI6J14QXXgHemzQlrLDM954BaDNBKxYfv6VSnRDNnHtdlq5Ktuugm3zJDlxjSSVzLp-IUU7SQSHQ1mz3r6klpfeLnjyTo62ZgUH6hgRh74_Fs=w640-h446-no)

Changing k upward causes the different powers to interact and the mini-mandelbrots don't even resemble z^n mandelbrots (image for k=0.4):

(https://lh3.googleusercontent.com/YUKt0Wk5-XLw-_hHssA0H9Wkce8Ve8ztJxUGwsiJLQuE-uD5_qtzNP1Oa4uVLkJHYwtxqUPbAspmtL5Y9wFqmiw3uB3kGkrGVSvM3Dl463YVukZMatYQ7t-MSdGajoWWgZsd8tTA7ekw8chzltpS-mCNS9DkyS-HAUQjQDn6ITZ0294KJu943qUCbiUt8jSSVyjjaYw3GE-VGJPgwWnsa3hpc6kRcqmLin0wLcO1qUOGjZHuWNbTNtNtKCmPHl5iYKlz_zdwiAWNvDOhto1h-ktHaTUqVURMeDyfZa7uokkixX9UozjJNlOktYWVH3Y00DIEzHMaFU-eswxFNqPxEye7TG8wRC52AbfnT4T1ldqkElkJA_cFDbPATdzdbQ2Y35B3E6wLf_VHVpP3iLv683jVh-VupLYWTLtCdcHfQgnz2BCVLoAlHnFwTMTZ_BLq59oD7sKSr1_Zb7zNAepCZLLEQ-ajxcr9nxwoJ6z6ouT9tMbaLJZYhyT6OBxBS-nCy-c-tK0o9mrFKjqewUDEkx1z19oJbWWH13CTsQBDDJwVgFfKbrf3LvwCH_NljyXrFtKp--Y4KXcs3laVQa0ErZbrT7DvxfEQ6Rmkq1f3c0gHFxI=w640-h446-no)

Perhaps the most interesting parameters are at k back to 0.3 and f(n) = exp(1.1*x*x/k) instead of f(n) = exp(0.5*x*x/k). This produces a more open fractal with islands of activity buried inside mini-msets (values much above 1.1 destroy the existence of deeper mini m-sets in the first place):

(https://lh3.googleusercontent.com/o6EClAfFd01E4dQ8qZ3QItbSpfH30h1u5Lics0vgioWHw3CowpmrLjsk8ve3W4TkuOjEqPCKs1mpujHUN2MaVwvklfWyw5_qjjMVV90Ff3c6meK8HbwPSL_MWAdFkL1ilCJB4qOfbu_64c4iCgxyZrhCtiNHu1-cHKfW7pSmpb1Q8ODM6ZzinVjFNso_ZB74BXzFuJVWF6tRXeJr6Pa4EZOBxNzgX2l-YKyEAGSfaWwxFMVnv1sG7Ac-KSahXFBxSST26EO9RbWSKDLMdkQ_26NeB2Yq7UusPjrdzUr_PhgY_361ZsiGsvfjZx4LFH8gpfpmpXP2gwZ6kOTK07gjJDavqxDvWnl-t3DCdGFEtxIHEMG4WujIkR221FV1KpXa7tgYfTg0BnuSJBxn3vd7lJem9jdJ9EkCRb9sJO29qN_iF1Hjubc_BGFRmgFjGZQPv5WhOtjqXpVu1LaifsS0xwqwXqywmg0SAd9ce54L0kbJ0jNgWkeJQ57czN2aEYszsQWJV-PjeWh_IR_IkDh5_jujUb5tlRGBedfzUGFn3avec7pE8qRVRiNPweVSqs4gX_haqpoN0cqjqrKxgEoTcCjtpDrQG34dNlT3IJYMQoZoD4qOBRY=w1101-h768-no)

Attached is the ultra fractal formula file.